摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1857-1864 Photoinduced Electron Transfer in a-Helical Poly(L4ysine) carrying RandomIy Dist r ibu ted Donor-Acce pto r Pa irs A Kinetic and Conformational Statistics Investigation Basilio Pispisa* and Mariano Venanzi Dipartimento di Scienze e Tecnologie Chimiche, Universita ' di Roma ' Tor Vergata ', 00133 Roma, Italy Antonio Palleschi Dipartimento di Chimica, Universita' di Roma 'La Sapienza ', 00185 Roma, Italy The photophysics of protoporphyrin IX (P) and 1-naphthylacetic acid (N), covalently bound to &-amino groups of poly(L-lysine) (PL) in a [PI : [N] molar ratio of 0.25, were investigated as a function of pH. Differential circular dichroism spectra and polarized fluorescence data on PNPL and blank samples (NPL and PPL) suggest that the amide bond in the chromophore linkages slows down the internal rotation of the aliphatic side-chains of the polypeptide. The conformational mobility of these linkages is further reduced by varying the pH from 7 to 11, the a-helical conformation of the backbone chain making the whole structure stiffer.Steady-state fluorescence, transient-absorption spectra and time-decay measurements indicate that quenching of excited naphthalene chiefly results from interconversion to the triplet state when the polymeric matrix is in a random coil conforma- tion (pH ca. 7) and from intramolecular electron transfer, P -+ 'N*, when it is in an a-helical conformation (pH ca. 11).The kinetic law of these processes, based on a two-state model for the polypeptide matrix, is presented.The specific rate constant of the photoinduced electron transfer is 3.1 x lo7 s-' (25"C),in very good agreement with that obtained from the lifetimes of naphthalene fluorescence in a-helical PNPL and NPL (pH ll),i.e. 2.7 x lo7 s-' . PNPL solutions exhibit very little exciplex fluorescence, whatever the pH, suggesting a relatively large average separation distance between the chromophores, in agreement with the results of a conformational statistics analysis on the fully ordered PNPL. The probability distribution of centre-to-centre distances between the chromophores, as obtained by adopting a rotational isomeric state model of the probe linkages, allowed us to reproduce the experimental fluorescence decay curves and estimate the parameters governing the electron- transfer process . Photoinduced intramolecular electron transfer in biomimetic systems is a subject of considerable interest because of its implications in several fields, such as photochemistry, polymer chemistry, biology and biotechnology.'v2 Among the different materials used, synthetic polypeptides carrying cova- lently bound fluorophores are well suited for obtaining basic information on this type of process because they are both structurally simple and well defined.3 Direct contact between donor (D) and acceptor (A) species is generally believed to be a requirement for charge separa- ti~n,~and several experiments support the idea that a through-bond interaction between D and A is more effective than a through-space intera~tion.~ It has been recently shown, however, that electron transfer in a-helical polypeptides carrying covalently bound PA pairs chiefly occurs by the latter mechani~m,~ the semirigid helical conformation being excluded to take part in the process, at least in the nanose- cond timescale.We have recently reported the photophysics of proto-porphyrin IX (P) and 1-naphthylacetic acid (N), covalently and randomly attached to &-amino groups of poly(L-lysine), showing that the conformational equilibria in solution of PNPL system affect the excited-state processes of the bound chromophore~.~ We now present the results of the calculations on the dis- tribution of the interchromophoric distances in the or-helical PNPL.For the sake of comparison with computed values, we also summarize here some of the experimental material pre- viously reported, together with new results on the solution properties of PNPL and on the quenching process of excited naphthalene. The probability distribution of centre-to-centre distances between the N and P molecules in the fully ordered PNPL was evaluated by a conformational statistics analysis, making use of a rotational isomeric state model of the chromophore linkages. We were thus able to reproduce the fluorescence decay curves and to get a reliable estimate of the parameters governing the electron transfer process, P --* IN*, that occurs only when PNPL attains an a-helical conforma- tion.Experimental Materials The following abbreviations are used throughout the text. Protoporphyrin IX : P ; naphthyl chromophore : N ; poly(L-lysine): PL; poly(L-lysine) carrying both N and P groups: PNPL; poly(L-lysine) carrying P or N group only: PPL or NPL, respectively. PNPL, PPL and NPL samples, which have the chromo- phores bound to &-amino group of the side-chains of poly(L- lysine), were prepared using analytical-grade reagents. A typical procedure was as follows. To a water/DMF (24:76, v:v) mixture at O"C, containing 1.13 x monomol of poly(L-1ysineeHBr) (Sigma), 1.13 x mol of 1-naphthylacetic acid (Aldrich), 3.39 x mol of N(Et), (C. Erba) and 1.13 x lop4mol of protoporphyrin IX (Aldrich), 3.39 x mol of l-ethyl-3-(3'-dimethylaminopropyl)car-bodiimine hydrochloride (Nova biochem) were added.After stirring for 2 h at 0°C and standing at room temperature for about 18 h, the mixture was lyophilized and then dissolved in 50 ml of water. The solid residue was filtered and the solution dialysed against 0.01 mol dm-3 HC1, using pre-treated dialy- sis tubes (A. Thomas Co., Philadelphia). After freeze-drying, PNPL was obtained, with a yield of 75%. Blanks were pre- pared following the same procedure, and the yields were 38% (PPL) and 62% (NPL). The amount of bound P and N molecules was determined by both 'H NMR and absorption measurements, the agree- ment being better than 10%. The naphthalene content is 8.7% in NPL and 5.2% (w) in PNPL, while the porphyrin content is 4.1% in PPL and 4.2% (w) in PNPL, the molar ratio between the chromophores in PNPL being thus [PI :[N] = 0.25.Elemental analyses were in good agreement with these results. Found: C% 47.25, 46.65 and 45.86; N% 15.04, 15.83 and 15.26; H% 7.48, 7.71 and 7.33 for NPL, PPL and PNPL, respectively. Calculate: C% 48.21, 45.13 and 47.83; N% 15.86, 16.78 and 16.10; H% 7.79, 7.89 and 7*26for (C,8N202H,0XC60N2H12 * HC1)10 9 (C40N604H44) (C60N2H12 HC1)77 and (C18N202H20)4(C40N604H44)(C60N2H12HC1)68, respectively. -Porphyrin molecules were linked to &-amino groups of poly(L-lysine) by only one pendant carboxylic group, the ratio of the NMR signals between the porphyrin B-protons (6 x 3.2)7 and the 6-protons of the substituted side-chains of the polypeptide (6 x 2.6) being around 1 : 2.The major portion of the solid residue of the syntheses (see above) is, instead, formed by poly(L-lysine) chains cross-linked by both carboxylic groups of porphyrin. Bis-Tris Propane? (pH = 6.5-9) and Caps$ (9.5-11) buffers (Sigma) were employed in a concentration of 0.01 mol dm-3. All measurements were carried out on freshly prepared solu- tions, using doubly distilled water. Methods All fluorescence experiments were carried out in quartz cells, using solutions that were bubbled for about 20 min with ultrapure nitrogen. Steady-state fluorescence spectra were performed on a Hitachi MPF-2D fluorimeter and fluores- cence anisotropy measurements were performed on a SPC Fluorolog-SPEX apparatus, equipped with Glan-Thomson polarising prisms.Time-decay measurements were carried out using an excimer pumped dye laser, frequency doubled on a KDP crystal to produce 5 ns pulses of a few tens of mJ, tuned at 285 nm. The emisssion was focussed on a double mono- chromator tuned at 340 nm, with a resolution of 2 nm. The output of a 56DUVP photomultiplier was fed into a Tektro- nix R7912 AD digitizer (bandpass 400 MHz). The decay curves were fitted by a non-linear least-squares analysis to exponential functions by an iterative deconvolution method, using a conventional PC. Time-resolved emission spectra were obtained by the same apparatus, using a boxcar inte- grator with a time window of 5 ns. Transient absorption spectra were carried out by a flash- photolysis set-up, the pulsed excitation (308 nm) being achieved by a Xe/HCl excimer laser (Lambda Physik EMG 50E).The pulse width was about 15 ns, the laser energy <10 mJ per pulse, and the delay time <1 ns.The light (150 W Xe lamp) was examined through a Baird-Tatlock monochro- mator, a Hamamatsu R928 photomultiplier and then cap- tured by a Tektronix DSA602 transient digitizer. 'H NMR spectra were recorded on a Bruker AM 400 instrument operating at 400.135 MHz. The spectral width used was 14 ppm and a residual water resonance of 4.77 ppm was assumed for chemical shift calibration. Absorption spectra were recorded on a Jasco 7850 appar-atus, and circular dichroism (CD) measurements were carried out on a Jasco 5-600 instrument with appropriate quartz cells. The other apparatus has already been rep~rted.~,~ t Bis-Tris Propane is 1,3-bis[tris(hydroxymethyl)methylamino] propane.$ Caps is 3-(cycIohexylamino)-l-propanesulfonicacid. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Results and Discussion Differential Circular Dichroism and Polarized Fluorescence Spectra Circular dichroism spectra of both blanks (NPL and PPL) and PNPL show that in all cases the coil-to-a-helical tran- sition in the poly(L-lysine) matrix is shifted towards lower PH.~@)This shift is greater than 0.5 pH units in the case of PNPL, indicating a remarkable efficiency of the bulky, apolar N and P groups in increasing the stability of the a-helical structure in aqueous solution.Although the chromophores are bound to the main chain by a rather long spacer, i.e. [-(CH,),-NH-CO-CH,-naphthalene] or [-(CH,),-NH-CO-(CH,),-porphyrin ring], they experience a rotational mobility which is more hindered than that of polymer-free molecules, probably due to the amide bond in the substituted side-chains and intramolecular hydrophobic interactions between the probes. Fig. 1 illus- trates the differential circular dichroism (DCD) spectra of NPL and PNPL against PL at pH = 11.1, i.e. under condi- tions where the samples are in a-helical conformation. Naph- thalene exhibits extrinsic CD bands in the 'Bb and 'Cb absorption region (220-190 nm),3(a)79 despite the fact that it is bound far away from the chiral C" atom.This is a strong indication of hindered conformational mobility of the side- chains carrying N, which is enhanced by the presence of P molecules in the chain, because the rotational strength is defi- nitely larger in PNPL than in NPL. The lack of exciton split- ting suggests, however, that naphthyl groups are not regularly arranged around the ordered polymeric matrix. Exciton state in aromatic poly(a-aminoacid)s is normally achieved only when the chromophores are bound to the chiral atom in the backbone chain by the shortest possible linkage^.^'") By contrast, at pH 7, DCD spectra show only a weak (negative) background rotation below 200 nm, as one would expect owing to the flexibility of the disordered chains. The results of polarized fluorescence measurements on N (Aex = 280, Aem = 337 nm) and P (Aex = 503, A,, = 625 nm) in NPL, PPL and PNPL samples are fully consistent with the aforementioned data.Table 1 lists the relative variation of the anisotropy coefficient r, according to eqn. (l), where subscript h and c denote a-helical and coil states. On inspection of Table 1, it appears that the conforma- tional mobility of the probe linkages decreases on going from pH 7 to 11, the a-helical conformation of the backbone chain r 180 200 220 240 260 A/n m Fig. 1 Differential circular dichroism (DCD) spectra of (a) NPL and (b) PNPL against PL at pH 11.1 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Relative variation of fluorescence anisotropy coefficients N P sample ,pb ,p.c NPL 1.3 - PPL - 0.8 PNPL 4.9 2.0 a K = (ih-rc)/i, and r = [(I,,-ZJ(Zll331 nm; ' A,, = 503, A,, = 625 nm.+ 2ZJ; * A,, = 280, A,, = stiffening the whole structure. Furthermore, the observation that the fluorescence anisotropy coeficient of PNPL is larger than that of NPL or PPL indicates that the internal rotation of the probes in the former sample is more hindered than in the latter ones, i.e. that intramolecular interactions between N and P molecules are more effective than those between the same type of chromophores in the blanks. Steady-state and Time-resolved Fluorescence Steady-state fluorescence spectra show that exciplex emission (ca.420 nm) is minor, as illustrated in Fig.2, where the emis- sion spectra of PNPL at the two extreme conditions of pH 7 and 11 are reported. The spectra consist mostly of monomer fluorescence, with both the peak position and intensity of the exciplex being almost insensitive to pH (Fig. 2, insert). The results of time-resolved emission spectra by laser excitation at 285 nm (not shown) are consistent with this finding in that a broad emission peak at around 420 nm, characterized by a fast rise time (7'x 6 ns) and a strictly monoexponential time decay of 60 ns, was observed within the whole range of pH explored. The rate constant of exciplex formation was evaluated by eqn. (2),where @NpL = 0.25, @)pNpL = 0.1g4 and zo = 35 ns, the former being the quantum yields of N in the blank (NPL) and in PNPL at pH 7, and the latter the excited-state lifetime of naphthalene in NPL at the same pH.[The value of @NpL is quite similar to that of the free molecule (0.23),4*1'(b) indicat-ing the absence of photophysical interactions between the chromophore and backbone chain.] From the results, k, = 9.0 x lo6 s-', i.e. the rate of exci- plex formation is more than one order of magnitude slower than the timescale of local side-chain fluctuations, as deter- mined by 13C NMR relaxation measurements on the CBof X 20 .-0 $0 1859 Table 2 Fluorescence quantum yields of N and P chromophores at different pH values" A,, = 340 nm A,, = 625 nm PH %I? @)rNPL WPPLb %NPL 7.0 1 1 1 1 9.0 0.96 0.77 0.96 0.74 10.0 0.88 0.45 0.96 0.50 11.0 0.83 0.39 0.89 0.35 a They are normalized to the quantum yield at pH 7; A,, = 280 nm.Blank samples. helical poly(y-benzyl-L-glutamate).lo According to the afore- mentioned DCD and polarized fluorescence data, this is very likely because the P and N groups in the polypeptide chain experience a rather rigid arrangement. Where the fluorescence quantum yields of P and N in PPL, NPL and PNPL solutions, measured at 625 and 340 nm, respectively, are normalized to those at pH 7, the values of a',reported in Table 2, are obtained. The main inference to be drawn from this Table is that the quantum yields of both P and N in the blanks (WppLand WNpL)decrease by less than 20% on going from pH 7 to 11, while those in PNPL (WpNP,j lowers by more than 60% within the same range of pH.It appears, therefore, that quenching of N* by P effectively occurs as the amount of ordered polypeptide matrix increases, a process that cannot be ascribed to a Forster-type N* -+P energy transfer,l' otherwise increasing quantum yields of P emission would have been observed.12 Consistent- ly, the excitation spectra of P (Aern = 625 nm) do not exhibit any variation under pH changes, within the whole wave-length region explored (240-340 nm). We next investigated the time-resolved fluorescence as a function of pH. Fig. 3 shows typical decay curves of excited naphthalene in PNPL solutions at pH 7.7 and 10.8. No sig-nificant change was observed on varying sample concentra- tions within two orders of magnitude (10-5-10-3 mol dm- 3), which rules out the occurrence of interchain effects. The curves are described by a three-component exponen- tial decay, i.e. (z) = xiaizi (i = 1 to The first 3).3(')94(b)*'3 component is very short (d2 ns), partially overlapping the laser profile.The resolution time of our apparatus is, in fact comparable to early events, thus resulting in an inaccurate detection of the initial fluorescence decay following laser pulse. While deconvolution techniques can be, in principle, employed to successfully recover lifetime components 2-3 '5 0.4 c. .-!I 300 340 380 420 460 500 I A/nm 0 50 100 150 200 Fig. 2 Steady-state fluorescence spectra (Aex = 285 nm) of PNPL at time/ns pH (a) 7.0 and (b) 11.0.The insert shows the exciplex emission at the Fig.3 Time decay of naphthalene fluorescence in PNPL (Ae,,, = 340 same pH values. nm) at (a)pH 7.7 and (b)10.8. Curve (c)is the laser profile. 1860 35 30 v)F 25 c, 20 15 PH Fig. 4 Variation of the middle component of naphthalene time decay, z2 0, and of a-helical fraction of poly(L-lysine) in PNPL, fh (---), as a function of pH. +,70 (NPL). times shorter than the excitation pulse length,14 in our case an unavoidable scattering contamination, probably due to the polymeric nature of the samples, gives rise to a severe distortion of the first few tenths of channels of the decay profile. As a result, the very first part of the time decay curve cannot be fully analysed.However, the (7) values are insensi- tive to this effect because the other lifetime components refer to kinetic processes that take place on a much slower time- scale. The long-time component (z3= 60 ns) is pH-independent and reminiscent of the decay behaviour of exciplex emission, as observed by time-resolved emission spectra (see above). The middle component, z,, is the only one that varies with pH, thus appearing to be sensitive to the interchromophoric distances and hence to the structural fea- tures of the polypeptide. This is shown in Fig. 4,where the 2-helical fraction in the polymeric matrix (fh) is also reported. It was obtained by the conventional expression bobs = CAEh fh + (1 -fh)AcC], where the differential molar absorption coefficients, A&, measured at 222 nm, are as follows: AE~= -9.3 and AE~0.8 dm3 mol- ' cm-'for the helical and coil = form of PNPL, respectively.The same sigmoidal trend of z2 us. pH was observed by lowering the temperature from 25 to 8°C. By contrast, the time decay of N* in NPL, zo, remains nearly constant when the pH is increased, as shown in Fig. 4. Comparison between the relative quenching eficiencies of steady-state (Q$pL/QNpL) and time-resolved ((z)/zo) fluores-cence is good, within experimental errors, at all values of pH investigated. This result, illustrated in Fig. 5, suggests that a fast quenching process or (strong) ground-state interactions are absent. [@:ApL is the overall quantum yield, as given by 0.6' ' ' ' ' ' ' ' ' ' ' 7 a 9 10 11 12 PH Fig.5 Relative quenching efficiencies, from (a) time-resolved (z)/zo and (b) steady-state @p&/DNPLmeasurements, as a function of pH J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 @FApL= QE + QpNpL, where QE = QNpL -a,,,, = 0.06 (PH 7) is the quantum yield of the exciplex. Note the good agree- ment between k,, as obtained by eqn. (2), and k, = @&' = 0.06/6 x low9= 1 x lo7 s-', z' being the characteristic time of exciplex formation]. The time decay of P in PNPL (z" = 20 ns; Lex = 285, Lem = 625 nm) is insensitive to pH changes, while the quantum yield at 625 nm definitely decreases as pH increases. Since the quantum yield of P in the blank (PPL) is only slightly affected by pH (Table 2), one may safely conclude that the protoporphyrin ground state is involved in the quen- ching process of excited naphthalene.This, in turn, suggests that or-helical PNPL experiences a P -,N* electron-transfer process, an hypothesis confirmed by the results of transient absorption spectra, as shown below. Transient Absorption Spectra Transient absorption spectra of the PNPL solution at pH 6.8 [Fig. 6(a)] exhibit the characteristic triplet-triplet (T-T) absorption of protoporphyrin at around 430 nm and the bleaching of the Soret band at 400 nm.' 5(u) Instead, at pH 11 the T-T absorption is absent while there is a new band at ca. 460 nm [Fig. 6(b)]. Since the same spectra of bound naphthyl or protoporphyryl chromophore alone show the T-T absorp- tion,15 whatever the pH (not reported), one may conclude that interconversion to the triplet state is an effcient process, except for PNPL at high pH values where the quenching of N* by P predominates.Transient kinetics following laser flash excitation of PNPL at pH 11 suggest that the absorption at 460 nm is due to a long-lived species, with a slow decay (T',~ = 5.5 p).This finding is reminiscent of that found in flash-photolysis studies on functionalized, water-soluble porphyrins, where the absorption at 450-460 nm was assigned to a charged porphy- rin moiety, i.e. P*+.'6*1 In addition, kinetic experiments by oxygen saturation following laser excitation of PNPL solu-tions show that at pH 7 the second-order rate constant for bleaching the 430 nm absorption is 3 x lo9 dm3 mol-' s-', a value which agrees quite well with those reported for triplet quenching by 0, .15(')-16 Instead, at pH 11 the rate constant I I qd -0.2 0.05 I "' I -0.05 1 300 400 500 600 700 l/nm Fig.6 Transient absorption spectra of PNPL solutions at pH (a) 6.8 and (b) 11.0 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 for bleaching the 460 nm band is one order of magnitude smaller, i.e. 3 x lo8dm3 mol-' s-'.'' To summarize, besides minor exciplex formation that occurs regardless of the polypeptide conformation, quenching of excited naphthalene in PNPL takes place by two competi- tive, pH-dependent mechanisms. One is interconversion to the triplet state, with rate constant k,, and the other intra- molecular electron transfer from the porphyrin ground state to the singlet excited state of N (AGO FZ -0.6 eV"), with rate constant k,.The former process predominates at pH ca. 7 while the latter at pH ca. 11. A similar competition between triplet and (singlet) charge-separated states has been recently reported for a tetraphenyl-porphyrin-quinone system.20 Kinetics Interconversion to the triplet state and exciplex formation account for quenching of N* when PNPL is randomly coiled. However, as pH increases PNPL experiences an intramolecu- lar electron transfer, that eventually becomes the predomi- nant process. The role of the ordered polymer thus appears to be rather subtle: the rigidity of the structure allows the N and P groups to be sufficiently accessible so that the quen- ching can occur while maintaining sufficient separation to prevent the immediate charge recombination reaction.The aforementioned quenching pathways of N* in PNPL, under the extreme conditions of coil (c) and a-helix (h) state, are summarized in Scheme 1 while the kinetic law of these processes, based on the two-state model for the polymeric matrix [eqn. (3)], is given by eqn. (4),on the assumption that light excitation does not perturb the equilibrium conditions of the coil-to-a-helix transition, i.e. k&, = k,*/k,*. [The a-helical fraction is then given byf, = k,/(kd + k,)]. L(N-P)~-kd (N-p)h (3) kw In eqn. (3), (N-P)c(h)denotes PNPL in the ground state and k, and k, the specific rates for the forward and reverse reac- tion, while in eqn.(4),[A,*(h,]= [N*-P],,,,, where [A,*ch,] is the population of A* in the coil or helical state, respectively, and the subscripts of the specific rate constants refer to the processes illustrated in Scheme 1. \b107 k*, /.ixl (N*-P)E 3( N*-P), Scheme 1 Since (A*) = (A,*) + (A:), one can also write: where kobs (=(z)-') is given by eqn. (6a)or (6b),depending on which step is rate-determining.4(b)*21When the fastest process is the equilibrium A,*+A,*, and hence (A*) = [(k,/kd) (A:) + (A:)] because the coil and helical substates are in equilibrium at each pH, one gets eqn. (64, otherwise, i.e. when electron transfer is faster than any other process and hence one may safely assume that (A:) x 0, eqn.(6b). By plotting (/cobs -k, -kE) as a function Offh, within the pH range 7-11, a straight line is obtained, in full agreement with eqn. (64. This is shown in Fig. 7. From the least-squares analysis of the data, the intercept is k, = (1.2 & 0.3) x lo7 s-', in fairly good agreement with the rate constant for radi- ationless population of triplet states of aromatic molecules," and the slope (k, -kJ = (1.9 f0.3) x lo7 s-l (25°C). The specific rate constant for charge separation is then k, = (3.1 & 0.5) x lo7 s-', which compares very well with the electron transfer rate constant estimated from the lifetimes of N in PNPL and NPL at pH 11, i.e.(z; -z; ') = 2.7 x lo7 s-at 25 and 1.8 x lo7 s-' at 8°C.The activation energy is thus 4.0 kcal rno1-l and the activation entropy around -13 cal mo1-' deg-'. According to the foregoing results, 'gating' effects in the reaction investigated can be ruled out,21 i.e. k, 4 k:. This implies that k, represents the actual specific rate of the intra- molecular electron transfer, in agreement with the kinetic theory of helix-coil transition in polypeptides,22 for which the initial rate constant for helix growth in a helix region at the ends of a chain is of the order of 10" s-1,22,23although it drops to a range of 107-1010 s-' 24-26 at midpoint tran- sition (fh = 0.5) because the elementary step of growth of a helix is a diffusion-controlled proce~s.~~.~~ Finally, the (N--P+) -,ground-state rate constant, evalu- ated by the decay kinetics of the radical species absorbing at 460 nm (see above), is k, z 1.4 x lo5 s-'.The larger rate constant of (singlet) charge separation over charge recombi- nation probably reflects the inverted character of the highly exergonic charge-recombination step, as opposed to the normal character of the charge-separation process,2o in agreement with Marcus A ,-A0& 0 , ,-A' 0 0 I' , 0 0.25 0.50 0.75 1.oo ftl Fig. 7 (kobs-k, -kE) as a function of the a-helical fraction in PNPL, according to eqn. (64. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Conformational Statistics The problem of the flexibility of the probes linkages and the random distribution of the chromophores in the polymer chain was then addressed. If a wide distribution of lifetimes results, comprising all decay values, it is reasonable to expect that each lifetime component changes as pH varies. In fact, this is not the case.Only the middle component, z2, is seen to decrease from about 30 to 18 ns on going from pH 7 to 11, while both the longer component, z3, and the shorter one are unperturbed. This finding suggests that each lifetime rep- resents a different decay process, distinctly separated from the others, an idea consistent with the observation that the varia- tion of z2 within pH 7-11 is too large compared with that found in NPL (ca. 4 ns) to be solely ascribed to the confor- mational transition.To check the foregoing hypothesis and evaluate to what extent the distribution of fluorescence lifetimes may affect the kinetic model based on a multiexponential discrete deconvol- ution, we have undertaken a conformational statistical analysis on the a-helical PNPL, using the method described in the Appendix for evaluating the interprobe distances dis- tribution function, P(R).The fluorescence decay intensity was then calculated by eqn. (7), where (k, + k, + kf) are the rate constants of eqn. (64 for the casef, = 1, as that under exami- nation. The hypothesis, here is that (i) electron transfer does not depend on the number of bonds between N and P groups, otherwise it would have been also measured in the random coil sample. Instead, it depends exponentially on the separation distance of the reactants, eqn.(8),27-31where /? is a parameter associated with the attenuation length of the wave functi~n,~~*~~*~~ R, = 3 A is the van der Waals contact and k, is the specific rate when R = R,. (ii) Besides exciplex formation which occurs with rate constant kE, the donor P group does not introduce any additional quenching mechanism of naphthalene fluorescence other than electron transfer, P -+ IN*. (iii) The natural fluorescence decay of the bound naphthyl molecules is monoexponential, occurring with rate constant k,. P(R)[exp -(k, + k, -k kE)t] dR (7) Regardless of the functional form of P(R), the integration of eqn. (7) can be approximated to a sum taken over all R values from zero to the maximum interchromophoric dis- tance, R,,,, when the whole integration range, 0 to R,,,, is divided into N equivalent subintervals of increment SR = R,,,/N, i.e.32 ~(t)= I, 1 P(R)[exp -(k, + kf + kE)t]6R (9) c In our case, N z 700, corresponding to SR = 0.1 A.There-fore, after excitation of naphthalene in the a-helical PNPL by a &pulse radiation, the emission intensity for the fraction P(R)SR of macromolecular chains, which have an interprobe distance between R and R + 6R, decreases according to eqn. (9). However, the laser profile needs also to be considered because the excitation pulse is not extremely short-lived. Therefore, the fluorescence decay I(t) must be deconvoluted to the excitation profile L(t’-t) at time t’, according to eqn. (lo),where K is a scaling factor.Itheor(t’) K 1P(R)GR Yt’ -t)[exp -(k, + kf + kE)t] dt= r (10) 120 r 100 t A A -a-.-c c 3 r 40 0 50 100 150 200 ti me/ns 20 I 1 , I 0 50 100 150 200 time,’ns Fig. 8 Comparison between calculated [eqn. (lo)] and experimental fluorescence decay curves of ordered PNPL (pH 11). The weighted residuals are shown in the insert. From the result, the plot of Fig. 8 is obtained, where the experimental decay curve of PNPL at pH 11 is compared to the theoretical one, eqn. (lo), that makes use of the following parameters: k, = (6.9 ~t0.7) x lo6 s-I, kE = (9.0 2 0.9) x lo6 s-’,and R, = 3 A, while P(R)is given by eqn. (A2) in the Appendix. The agreement between the two curves is satis- factory.Moreover, if comparison is made between three experimental decay curves, within pH 10.6-11.0, and those calculated by eqn. (10) using the aforementioned parameters, one obtains /? = 1.0 f0.1 A-’ and k, = (9.2 f0.2) x 10l2 s-’. These values compare very well with those reported for electron transfer in proteins where a non-adiabatic long-range process is thought to occur.3c~27-31 (kf) = 1 P(RPRkf(R) (1 1)5 Finally, the average electron transfer rate constant, evalu- ated by eqn. (ll), is close to that experimentally determined, i.e. (4.0 f0.4)x lo7 s-’. This finding clearly indicates that the kinetic data are well described by a narrow distribution of interprobe distances, as that implicitly assumed in the multiexponential decay analysis of the model used.According to Fig. 9, where the functions kf(R), eqn. (8), and P(R), eqn. (A2), are reported, the average interchromophoric distance for which the electron transfer has the highest probability of occurring is ca. 12 A, which corresponds approximately to three helical turns. This is also approximately the centre-to- 1.2r (b) -6x10-’ n wQ 0.8i -I \(a)I/ \ I ... I 0.4 tt h s ot -0 CL 0 10 20 30 40 50 60 RIA Fig. 9 (a) Probability distribution of interchromophoric distances, P(R), for the case in which the contacts between non-bonded atoms shorter than 2.5 A were discarded for steric reasons; (b) distance dependence of the electron transfer rate constant, reported as k,/k,; (c) resulting function, as given by eqn.(11). The figure 2 x lo7 is a scaling factor, and the area under this curve gives the ratio (k,)/k, . J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 centre distance of other donor-acceptor pairs covalently bound to a-helical polypeptides, exhibiting both an electron transfer rate constant (ca. 5 x lo7 s-'; 20°C) and an elec- tronic driving force (-0.4 eV) of the same order of magnitude as that of P-N in PNPL.3(') This result can be taken as a further indication of a through-space mechanism, otherwise the rate constant would have been much larger in both ca~es.~(').~ Conclusion Three major conclusions are drawn from the present study. First, the relaxation time for the coilea-helix transition in PNPL is <20 ns.Secondly, electron transfer occurs only within a rigid frame, i.e. when the a-helical conformation of the backbone chain stiffens the whole structure. This also implies that the process takes place primarily by a through- space (or through-solvent) mechanism. Thirdly, the proposed treatment of fluorescence data is effective in reproducing the experimental decay curves of randomly distributed pairs in the a-helical PNPL. It makes use of the rate constants of the exciplex model for the kinetics, and of a rotational isomeric state model of the chromophore linkages for the distribution function of the donor-acceptor distances. We thank Prof. M. DAlagni for preparing the samples, and Drs. F. Elisei, P.Morales and L. Nencini for collaboration in transient-absorption and fluorescence-decay measurements. This work was supported in part by the National Research Council (CNR) and in part by MURST (Rome). Appendix To evaluate the distribution of interchromophoric distances, we began by considering PNPL as formed by macro-molecular units of 272 residues, carrying randomly distrib- H-N h-H@+, 10 x ;?a- C ,'C L XC T o=c Xll II 0 aT-H C=O Fig. A1 Molecular representation of a poly(1ysine) chain carrying protoporphyrin IX (P) and naphthyl (N) chromophores covalently bound to the &-amino groups. The features of the ensemble examined are such that each chain contains 272 L-IYS residues in a-helical con- formation ($ = -57', Y = -47', o = 180°), with 4 P and 16 N molecules (see text).The intramolecular centre-to-centre distance for a given conformation m of the side-chains carrying P and N, and a given configuration i, is denoted as R(m, i) = IR(m, i). The angles of internal rotation are indicated as za(6 = 1-17). Hydrogen atoms are omitted for clarity. 1863 Table A1 Relative energies and statistical weights of the rotational isomeric states of the donor and acceptor chromophore linkages" bond rotational rotat. isomeric energy angleb state' (En,jsId (stat. wt.),,, jse +60 0 1XlO XI -60 0 1 180 0 1 +60 0.5 exp(-OS/RT)XI 1 x2 -60 0.5 exp( -0.5JRT) 180 0 1 +60 0.5 exp( -0.5JRT) x12 x3 -60 0.5 exp(-OS/RT) 180 0 1 +60 0.5 exp(-OS/RT)XI 3 x4 -60 0.5 exp( -0.5/RT) 180 0 1 x14 x5 +60 0 1 -60 0 1 180 0 1 180 0 1x15 x6 x16 XI +60 0 1 -60 0 1 180 0 1 -Xa +60 0.5 exp(-0.5JR T) -60 0.5 exp( -OS/RT) 180 0 1 x17 x9 +90 0 1 -90 0 1 ~ ~~ The value of both relative energies, Ers,js (kcal mol-I), and sta- tistical weights of the rotational isomeric states are based on confor- mational analysis data of compounds containing similar structural elements.35 Dihedral angles, as illustrated in Fig.Al. 'The defini- tion of the angles conforms to IUB-IUPAC recommendations. Relative energy in kcal mol-'. The statistical weight is given by exp(- E,,JRT) or exp(- EjJRT). uted protoporphyrin IX and naphthalene probes, i.e. 4 P and 16 N, in agreement with both the experimentally determined stoichiometry and the average degree of polymerization of the sample used.We then define a configuration i as a polymer unit in which the relative position of the side-chains carrying N and P chromophores is fixed, each configuration being thus characterized by a given value of Ap-N. The fre- quency of the occurrence of configuration i,S,, is given by the ratio of li over 1 = cili,where li is the number of times that the event occurs, and the probability of the relative distances between the C" atom carrying the naphthyl moiety and that carrying the porphyrin moiety (AP-J by pi = lim (li/r) for 1 + a.This probability was calculated by the expression I(A)k+ -(A)k I < where the subscripts refer to the kth and (k + 1)th iterative process, each process taking into account a number of units lo5 larger than the preceding one.We next evaluated the probability distribution of the centre-to-centre distances between the chromophores bound to a-helical poly(L-lysine), making use of the geometrical parameters shown in Fig. Al, where the intramolecular dis- tance for a given conformation m, when the configuration is i, is written as R(m, i) = IR(m, i) I, while each bond in P chromophore linkage, denoted in the following as j, is num- bered from 1 to 9, and each bond in the N chromophore linkage, denoted as r, from 10 to 17. We adopted a rotational isomeric state for the probe linkages, in which rotation around each single bond is restricted to a few, highly populated low-energy isomeric states.The actual distribution of conformations is thus assumed to be well represented with this restriction. Table A1 lists the isomeric states, relative energies and statistical weights for the donor and acceptor linkages in the polypep- tide.35 By discarding both the conformations g+g- and g-g+ for the aliphatic portion of lysine side-chain, owing to their very low probability, and the conformations in which the centre of the chromophore is <7 8, from the helical axis, owing to steric hindrances, the total number of conformers of the probe linkages is ztot= 802,256, i.e. 1,508 for P and 532 for N linkage. Owing to this large number of side-chain confor- mations, each configuration i is coupled with a distribution of interchromophoric distances, whose probabilities are given by eqn.(Al), z1 (stat. wt.), P(R),= Zto11 (stat. wt.), (A1) m= 1 where z is the number of conformers having a given value of (R),, and (stat. wt.), = n,(stat. wt.),, nj(stat. wt.)j,.. The pro- ducts are taken over all bonds r and j of N and P chromo-phore linkages, respectively, while (stat. wt.),, and (stat. wt.)js, are the statistical weight assigned to bond r when populating the s state, and to bond j when populating the s’ state, respec- tively (Table A 1). The interprobe distance distribution func- tion, P(R), can be then written as eqn. (A2), and is illustrated in Fig. 9(a) for the case in which the contacts between non- bonded atoms shorter than 2.5 8, were discarded for steric reasons.References F. C. De Schryver, N. Boens and J. Put, Adv. Photochem., 1977, 10,359; G.L. Closs and J. R. Miller, Science, 1988,240,440. Biochemical Fluorescence: Concepts, ed. R. F. Chen and H. Edel- hoch, Marcel Dekker, New York, 1976, vol. 1 and 2; Per-spectives in Photosynthesis, ed. J. Jortner and B. Pullman, Kluwer Academic Publishers, Dordrecht, 1990. (a) M. Sisido, S. Egusa and Y. Imanishi, J. Am. Chem. SOC., 1983, 105, 1041; (b) M. Sisido, R. Tanaka, Y. Inai and Y. Imanishi, J. Am. Chem. SOC., 1989, 111, 6790; Y. Inai, M. Sisido and Y. Ima- nishi, J. Phys. Chem., 1990, 94, 8365; (c) Y. Inai, M. Sisido and Y. Imanishi, J. Phys. Chem., 1991,95, 3847. W. Schuddeboom, T.Scherer, J. M. Warman and J. W. Verhoe- ven, J. Phys. Chem., 1993,97, 13092. J. Liu, J. A. Schmidt and J. R. Bolton, J. Phys. Chem., 1991, 95, 6924. (a) B. Pispisa, A. Palleschi and M. Venanzi, Trends Phys. Chem., 1991, 2, 153; (b) B. Pispisa, M. Venanzi and M. D’Alagni, Bio-polymers, 1994,34,435. T. R. Janson and J. J. Katz, in The Porphyrins, ed. D. Dolphin, Academic Press, New York, 1979, vol. IV, ch. 1. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 8 B. Pispisa, A. Palleschi and G. Paradossi, J. Phys. Chem., 1987, 91, 1546. 9 H. H. Jaffe and M. Orchin, Theory and Applications of Ultra- violet Spectroscopy, Wiley, New York, 1962, ch. 13. 10 (a)A. Allerhand and E. Oldfield, Biochemistry, 1973, 12, 3428. (b) D. F. Eaton, Pure Appl.Chem., 1988,60, 1107. 11 T. Forster, Ann. Phys. (Leipzig), 1948, 2, 55. 12 B. Pispisa, M. Venanzi, A. Palleschi and G. Zanotti, J. Mol. Liquids, in the press. 13 K. S. Schanze and K. Sauer, J. Am. Chem. SOC., 1988,110,1180. 14 D. O’Connor and D. Phillips, Time Correlated Single Photon Counting, Academic Press, London, 1984, ch. 6. 15 (a) R. S. Sinclair, D. Tait and T. G. Truscott, J. Chem. SOC., Faraday Trans. I, 1980, 76, 417; (b) P. K. Chatterjee, K. Kamioka, J. D. Batteas and S. E. Webber, J. Phys. Chem., 1991, 95,960. 16 G. S. Nahor, J. Rabani and F. Grieser, J. Phys. Chem., 1981,85, 697. 17 B. Kr. Manna, D. Sen, S. Ch. Bera and K. K. Rohatgi-Mukherjee, Chem. Phys. Lett., 1991,176,191. 18 V. Bruckner, K-H. Feller and U-W. Grummt, Application of Time-Resolved Optical Spectroscopy, Elsevier, Amsterdam, 1990, ch. 3. 19 N. Mataga and T. Kubota, Molecular Interactions and Elec- tronic Spectra, Marcel Dekker, New York, 1970, ch. 9. 20 J. A. Schmidt, A. R. McIntosh, A. C. Weedon, J. R. Bolton, J. S. Connolly, J. K. Hurley and M. R. Wasiliewski, J. Am. Chem. SOC., 1988, 110, 1733. 21 B. M. Hoffman and M. A. Ratner, J. Am. Chem. Soc., 1987, 109, 6237; S. A. Wallin, E. D. A. Stemp, A. M. Everest, J. M. Nocek, T. L. Netzel and B. M. Hoffman, J. Am. Chem. SOC., 1991, 113, 1842. 22 G. Schwarz, J. Mol. Bid, 1965, 11,64. 23 H. Morawetz, Adv. Protein Chem., 1972,26, 243. 24 S. Inoue, T. Sano, Y. Yakabe, H. Ushio and T. Yasunaga, Bio-polymers, 1979, 18, 681. 25 B. Bosterling and J. Engel, Biophys. Chem., 1979,9, 201. 26 J. A. McCammon, S. H. Northrup, M. Karplus and R.M. Levy, Biopolymers, 1980, 19,2033. 27 R. A. Marcus and N. Sutin, Biochim. Biophys. Acta, 1985, 811, 265. 28 M. D. Newton and N. Sutin, Annu. Rev. Phys. Chem., 1984, 35, 437. 29 G. McLendon, Acc. Chem. Rex, 1988,21,160. 30 Faraday Discuss., 1982,74. 31 J. R. Winkel and H. B. Gray, Chem. Rev., 1992,92,369. 32 G. Lew, Macromolecules, 1993,26, 1144. 33 J. R. Miller, T. T. Calcaterra and G. L. Closs, J. Am. Chem. SOC., 1984,106,3047. 34 P. J. Flory, Statistical Mechanics of Chain Molecules, Wiley Interscience, New York, 1969. 35 C. A. McWherter, E. Haas, A. R. Leed and H. A. Scheraga, Bio-chemistry, 1986,25, 1951. Paper 3/06267H; Received 20th October, 1993

ISSN:0956-5000    

DOI:10.1039/FT9949001857

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1865-1874 Irreversible Thermodynamic Coupling between Heat and Matter Fluxes across a Gas/Liquid Interface -Scott C. Doney Advanced Study Program, National Center for Atmospheric Researcht P.O. Box 3000, Boulder, CO 80307, USA The governing equation for irreversible thermodynamic coupling phenomena, the dissipation function, is derived for a generalized gas/liquid interface following two routes and is shown to differ significantly from previous results (L. F. Phillips, Geophys. Res. Lett, 1991, 18, 1221; J. Chern. SOC.,faraday Trans., 1991, 87, 2187). The magnitude and direction of the coupled heat-mass effect is then computed for an idealized interface based on results from the kinetic theory of evaporation and condensation.The characteristic parameter for the coupling behaviour is the heat of transfer (93, the ratio of the conductive heat to mass flux arising from the cross-terms in the irreversible thermodynamic formulation. 9: for interfacial transport is shown to have a value of -0.43RT for both liquid-vapour and liquid-gas mixture systems. The coupled or cross-effect mass flux forced by the temperature field occurs from the cold fluid to the warm, counter to the conductive heat flux. This work suggests that the coupled mass fluxes across a gas/liquid interface are generally weaker than the direct, pressure-driven mass fluxes and are much smaller, by approximately a factor of 20, and in the opposite direc- tion from earlier model predictions.1. Introduction The transport of mass between gas and liquid phases is a complex problem involving both the physical chemistry of the gaspiquid interface as well as the fluid dynamic properties of the two fluids, in particular the presence and nature of the boundary layers adjacent to the interface. In addition to being of interest as a strictly theoretical question, the mass flux between gases and liquids, termed either evaporation/ condensation or gas exchange depending on whether the species is a bulk or trace constituent in the liquid, has numer- ous applications in fields as diverse as chemical engineering,' aerosol science,, and 0ceanography.~9~The fundamental thermodynamic and kinetic processes influencing gas exchange rates have been discussed for some time,'s5 but many important qpestions still remain.6 Recently, Phillips',' has re-examined the topic of gas exchange in terms of Onsager's irreversible thermodynamic This work7,', is focused primarily on under- standing the flux of CO, between the atmosphere and ocean, but is relevant to other fields as well.The res~lts~,*,~~ suggest that the coupling of heat and mass fluxes at the gas/liquid interface arising due to irreversible thermodynamic pheno- mena is an important factor in determining the magnitude and, in some cases, even the direction of the gas flux at a gas/liquid interface. In this paper, a development is presented of irreversible thermodynamics as it applies to the flow of trace gases and solvent across a gaspiquid phase boundary.The entropy pro- duction rate and dissipation function that govern the irre- versible heat-mass coupling phenomena are derived in Section 2 following two different routes. The two approaches are shown to yield an equivalent result, which differs signifi- cantly from that of previous work.' The proper equations describing the heat and mass coupling are presented in Section 3. The direction and approximate magnitude of the irreversible coupling effect are then computed for a simple gas/liquid interface based on results from the kinetic theory of evaporation and condensation (Section 4). For most systems, the irreversible effects are shown to be smaller and in t The National Center for Atmospheric Research is sponsored by the National Science Foundation.the opposite direction from those predicted by previous mod- elling studies.',' Fluid boundary layers near the interface are not considered in these calculations, which focus strictly on the phase interface itself; the bulk fluids outside of the inter- face region are assumed to mix at an infinite rate. The role of irreversible phenomena such as thermal diffusion in the inter- facial boundary layers is explored in a companion paperI3 with particular emphasis on the atmosphere-ocean system. Irreversible thermodynamic processes are distinguished from idealized, reversible processes by the production of internal entr~py,~,' that is '*14 dSi = 0; reversible dS, > 0; irreversible (1) Unlike in a reversible system, irreversible processes occur at a finite rate and involve finite gradients in temperature T, chemical potential ,u, and electric potential.The field of irre- versible or non-equilibrium thermodynamics developed as an extension of classical thermodynamics, which deals strictly with closed systems at equilibrium, and is particularly useful for studying open systems involving the transport of heat, matter and electricity.' Two key properties for characterizing irreversible processes are the production rate of internal entropy o = dSJdt and the related dissipation function To. The dissipation function quantifies the amount of dissipated energy that could have been utilized for useful work in a reversible process.In general, the dissipation function is equal to the sum of a set of fluxes Ji times forces Xi Ta= 1JiXi I The fluxes Jiinclude such things as heat, mass and charge while the forces result from differences or gradients in tem- perature, chemical and electrical potentials. A basic tenet of irreversible thermodynamics is that for slowly evolving systems close to equilibrium the fluxes Ji can be written as a linear combination of the forces" Xi Ji= c LiiXj (3) i where Lij are phenomenological coefficients linking .Ii and Xi. [Eqn. (3) may also be written in the reciprocal form, Xi = xiBijLj, where Bij are the corresponding jump 1866 c~efficients.'~] The choice of appropriate flux-force pairs (Ji and Xi) to use in the linear flux eqn.(3) is guided by the dissipation function; not any choice of flux-force pairs will produce a consistent set of flux equations.'.'' The phenom- enological coefficients are independent of the forces Xi, but may depend on temperature, pressure or system composi- tion. ''Two constraints apply to the phenomenological coef- ficients: the first, known as Onsager's relationships, states that the matrix of phenomenological coefficients is sym- metric,' i.e. Lij = Lji;the second results'' from the condition that a > 0 leading to Lii2 0 and LiiL, 2 Le .The criteria for whether a system is close enough to equilibrium to apply eqn. (3) can be difficult to address when chemical reactions are involved, but the assumption of linearity generally holds for most transport problems (e.g.diffusion, heat conduction).'* '' The exchange of heat and mass across a gas/liquid phase boundary can be written in terms of generalized irreversible thermodynamic flux-force pairs, eqn.(3). The conductive heat flux J,, sometimes referred to as the sensible or measurable heat flux, and mass flux Ji are often parametized using linear, phenomenological relationships such as J, = -C,pc,AT (4) (5) where AT and APi are the temperature and partial pressure differences between the two phases, p and c, are the density and specific heat for one of the phases, and Ki is the Henry's law coefficient relating the equilibrium liquid concentration of species i to the gas partial pressure. C, and Ci are empiri- cally derived transfer coefficients or transfer velocities and are dependent on the specific configuration of the system.For many systems, C, and Cican be specified as a function of the turbulence level in either the gas or liquid phases. The empirical flux relationships for J, and Ji [eqn. (4) and (5)] can be partitioned into a thermodynamic forcing term and a kinetic term, much like the generalized flux-force form, eqn. (3), for a single flux-force pair. For the conductive heat and gas flux equations [eqn. (4) and (5)], the thermodynamic Xj and kinetic Lij terms are the interfacial temperature and partial pressure differences and the transfer coefficients, respectively. In contrast to many irreversible thermodynamic problems, the kinetic or phenomenological coefficients L, linking the fluxes and forces for gas-liquid exchange may be controlled by both molecular and turbulent transport pro- cesses.Eqn. (4) and (5) are developed for systems involving a single forcing term. Irreversible thermodynamic theory, however, suggests that when there are multiple force terms, for example, coexisting temperature and partial pressure dif- ferences, additional cross phenomena may arise due to coup- ling between the different flux-force pair^.^-'' That is, one would expect mass fluxes driven solely by the temperature difference across the interface and a conductive heat flux due to partial pressure differences. The remainder of the paper examines the importance of these cross-effect terms right at the gas/liquid interface; a more general discussion including thermal and concentration boundary layers away from the interface region is presented elsewhere.' 2.The Dissipation Function for the Gasbiquid Interface 2.1 Traditional Derivation We are interested in the fluxes of heat and matter (solvent and/or trace gas) between a liquid and gas phase under J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 steady-state conditions. For environmental and chemical engineering applications, the gas phase is often a mixture of a bulk carrier gas, trace gas species and vapour from the solvent. Below, the generalized dissipation function is derived for heat and mass phase transfer, applicable for both bulk solvents and trace species. For simplicity, the gas phase is assumed to behave as an ideal gas but this is not necessary for the derivation.First, consider an idealized gas-liquid system where the bulk properties are uniform in both the liquid (subsystem 1) and the gas phases (subsystem 2), the rate of change in the bulk fluid properties is slow relative to the timescale of inter- est, and all of the gradient in temperature and chemical com- position between the two fluids is isolated to an arbitrarily small interface region (Fig. 1).The difference in the properties across the interface is expressed using the A notation (A,, = a, -ul), and fluxes are defined as positive when heat and mass are transferred from the liquid to the gas (see Glossary for notation and units). This simplified model of the gas/liquid interface, although clearly unrealistic in detail, is useful for developing in a general sense the irreversible thermodynamic equations for exchange between the two phases.The system shown in Fig. 1 is, according to the nomencla- ture of Hasse," a discontinuous system, and as shown in the Appendix, the general form of the dissipation function Ta per unit area A for a discontinuous system can be written as T is the mean temperature across the interface, hi,2 is the partial molar enthalpy of species i in the gas phase, and pi is the chemical potential of species i. Ji is the molar flux of i per unit area from the liquid to the gas, and is the conduc- tive, or measurable, heat flux absorbed by the gas phase. The total heat transferred from the liquid to the gas, Jq,2+xihi, Ji, includes terms for both the conductive heat flux and the enthalpy flux associated with the mass exchange.gas (subsystem 2) ;" liquid (subsystem 1). Fig. 1 Schematic for heat and mass exchange in a generalized gas/ liquid system. The bulk properties of the liquid (subsystem 1) and gas (subsystem 2) phases are TI,Pi,l and T, ,Pi,z, respectively. The two subsystems exchange mass fluxes .Ii and a total heat flux J,, which includes components from both conduction and enthalpy transport, and the conductive heat fluxes into the liquid and gas are given by Jq,l and .Iq,2, respectively. Heat and mass fluxes are defined as posi-tive when the transport results in a decrease in the quantity in the liquid and/or an increase in the gas phase.J. CHEM. SOC. FARADAY TRANS.,1994, VOL. 90 Under most circumstances, the conductive heat loss from the liquid Jq,' does not balance the heat absorbed by the gas Jq,2, and the difference between the two is equal to Jq, 2 -Jq.1 = -1Avhi Ji (7)i A"hi, the heat of vaporization or latent heat of i (for solutes the heat of solution), is generally positive. When water, for example, evaporates from a surface (Ji> 0) the cooling of the bulk liquid Jq,l is balanced, at first approximation, by the latent heat flux carried by the water vapour flux away from the interface into the gas phase, and the conductive heat flux in the gas .Iq, is near zero.The general form of the dissipation function can be trans- formed to one more applicable to gas-liquid exchange by specifying the relationship for the chemical potential for each species between the two phases. For a dilute solute the chemical potential in the liquid and gas phases are given by15 pi,' = ,u;(sol, T,)+ RT, 1n(mi,,/mol kg-') (8) Pi,2 = PP(g, T2)+ RT2 WPi,2/atm) (9) where p'(s01) and po(g) are the standard states for an ideally dilute solute and ideal gas, respectively, mi.' is the molality (mole solute per kg solvent) of species i in solution, and Pi, is the partial pressure of species i in the air. To simplify the discussion any non-ideal behaviour in either the gas or solu- tion phases is neglected; a more complete treatment could be derived using activity coefficients and gas fugacities but that is beyond the scope of this paper. The partial pressure and molality of gas i can be related at equilibrium via Henry's law Pi = Kimi (10) where Ki is the Henry's law coefficient on a molality scale.Combining eqn. (8)-(lo), it can be shown that for constant T &(sol) -pp(g) = RT In K, (11) Similar relationships to eqn. (8) and (9) can be developed for the solvent using the chemical potential for the solvent po in an ideally dilute solution po(sol, T') = &(sol, Tl)+ RT, In xo (12) and Raoult's Law in place of Henry's law Po = P;xo (13) The chemical potential for the solvent is defined using the solvent mole fraction, xo,and the equilibrium solvent vapour pressure at Tl over pure solvent, P;.Note that the standard states for solute (Henry's law) and solvent (Raoult's law) are defined in a different manner and are applicable over different ranges. For an ideally dilute solute, the standard state chemi- cal potential is defined for mi = 1 where each solute molecule interacts as if in contact only with solvent molecules ;I5,l6 Henry's Law is applicable for small mi (mi-,0). In contrast, the chemical potential for a solvent is defined as the pure solvent at the specified P and T,and Raoult's law is valid for xo close to l.16,17 The standard state of an ideal gas is defined at a partial pressure of 1 atm and T = 273.15 K. The dissipation function for liquid-gas exchange can be found by substituting the expressions for the chemical poten- tials for trace gases and solvents into eqn.(6) and then sim- plifying using eqn. (1l), the relationship between &'(sol) and p;(g), and eqn. (10) (Henry's law) and eqn. (13)(Raoult's law) Using the van't Hoff relationship' h(s)p= relating the temperature derivative of chemical potential at constant pressure to the partial molar enthalpy, a relation- ship for A(&'/T)can be derived ClP(g, T2) PP(g, Tl) 'hdT (16)T2 Tl ---=-lr T2 The partial molar enthalpy in the gas phase, hi,B, is a slight function of temperature (%) = cp,i (17)P where cp, is the molar heat capacity of i at constant pressure, which for a small temperature range can be integrated about hi,2 Eqn.(16)can then be integrated to give: where eqn. (6) and the fact that: ln(T + AT)/T = AT/T; AT << T (20) have been used to cancel the two terms involving cP.l7Sub-stituting back into eqn. (14) leads to the final form for the dissipation function for exchange across the interface The contribution to the total heat transfer d@ from the enth- alpy of the transported material cancels out of the final equa- tion. Using the liquid, rather than the gas phase, as the reference system leads to the same result. Note that from the definition of Jq, and Ji as positive when flow is into the gas phase, both terms in eqn. (21) are always positive, as they must be since the internal entropy can only increase for an irreversible process.' For most systems, it can be assumed that the partial pres- sure difference directly across the gas/liquid interface is small relative to the mean partial pressure and eqn.(21) can be further simplified to To AT APi -= --J,,2 -RT C -Ji (22)A T i Pi 2.2 A1terna tive Reversible- path Approach The entropy change dS,,, for a system can be separated into two components dS,,, = dSi + dS, (23) where dS, is the entropy change due to the exchange of matter and heat between the system and the surroundings and dSi is the entropy change arising from internal pro- cesses.' ' For a reversible process, ds, is exactly zero, eqn. (l), while dS,, and thus dSsys, can be positive, negative or zero depending on the process. The entropy change of a closed system undergoing a reversible process can be computed from an integral over the process:16 J dS,,, = J %; reversible where dq,,, is the heat transferred reversibly from the sur- roundings to the system; ds,,, associated with a reversible process, therefore, is zero only for an adiabatic system.dS,,, should not be confused with the entropy change of the uni- verse dSuniv,which includes the entropy change for both the system and its surroundings and is zero for a reversible process. Owing to the fact that entropy is a state function and thus path independent, the dS, for an irreversible process required for the dissipation function, eqn. (2) can be determined from the initial and final states of the system alone; the computed value of dS, should be the same irrespective of the path used in the derivation. In the Appendix, the overall system is chosen to be closed and adiabatic, dS, = 0, and dSi is found as the sum of the subsystem entropy changes dSl and dS, from the irreversible heat and mass transfer.For a reversible path such as given in eqn. (25), dSi for each step is zero and the entropy change for the whole path dSsys, which is equal the dSi for the corresponding irreversible pr&&sses, is just the sum of the individual values of dS, for each step. Consider the simple system consisting of a pure liquid and J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 state variable and can therefore be expressed as the sum of the enthalpy changes for the individual steps of eqn.(25). Using eqn. (17) along with the fact that (8hi/aP),= 0 for an ideal gas AVhi = cp AT + 0 + Avhi,(TI,P~) (29) Inserting eqn. (7) and eqn. (29) into the entropy production eqn. (28) cancels the terms starting with c,, A,hi and leaving behind the same entropy production equation as pre- sented earlier from the more traditional approach. Again, the terms arising from the enthalpy transport cancel according to the final dissipation function. 2.3 Comparison with Previous Results Phillips' uses an alternative approach from that in the Appendix for calculating the entropy production for the irre- versible process of transferring a solute, in this case CO, gas, across the gas/liquid interface. The method involves construc- ting a reversible path by which a mole of solute is transferred from phase 1 to phase 2 via a gas phase.The temperature and partial pressure of the solute are adjusted from T,, P, (phase 1) to T, ,P, (phase 2) reversibly in the gas phase CO,(phase 1, P,, Tl)+CO,(gas, p,, TI) its vapour at different pressures and temperatures. A reversible path can be written for the condensation of an +'1,CO,(gas, 7i) infinitesimal amount of vapour dn dn(vapour, P, , T,) +dn(vapour, P, , Tl) -+ dn(vapour, PI, TI) --+ dn(liquid, P,, Tl) (25) The entropy change for the path is given as the sum of the dS, associated with each reversible step16 where the vapour is assumed to be an ideal gas and the tem- perature variation of c, is taken to be small.During reversible, isothermal condensation (gas transfer) a quantity of heat equal to A, hi times the mass flux is released, and this heat must be transferred to the surroundings to maintain a constant temperature. The third term in eqn. (26) reflects the heat of vaporization of the liquid to the vapour evaluated for constant TI and P,. The entropy production due to the heat flux between the liquid and vapour also needs to be included in eqn. (26). For isothermal, reversible heat transfer, the entropy change is equal to dS, = dq/T. If an amount of heat dq, is removed from the vapour and dq, added to the liquid, the entropy production is, therefore Combining eqn. (26) and (27) and rewriting in terms of fluxes J,, 2, J,,l and Ji and for small AT and AP [eqn. (20)] Recall from eqn.(7) that the heat flux from the vapour is related to the heat flux into the liquid via the heat of vapor- ization, J,,l = J,, ,+ A, hiJi. Like entropy, enthalpy is also a + CO,(gas, P, 9 T2) -+ CO,(phase 2, P,, T,) (30) The entropy changes computed for each individual step along the path are summed to arrive at the net entropy change for the irreversible process leading to the same end state. If done correctly the reversible path approach (Section 2.2) leads to the same dissipation function [eqn. (22)] as the traditional method. The dissipation function derived above for heat and mass fluxes across the gaspiquid interface [eqn. (22)] differs from that of Phillipst7-' Ta AT ATA T -c,, , -)Ji (31)T in two important respects: the inclusion of the extra term c,, AT/T and the definition of the heat flux J, .Both of these differences arise in part due to the treatment79' of the con- ductive heat fluxes Jq,l and Jq, and their relation to the heat of vaporization A, hi. In the derivation by Phillips,8 the entropy change attrib- uted to the final step of the path in eqn. (30), where vapour (or solute) is condensed onto the liquid with the same T and Pi,is set to zero; it is stated that 'the entropy lost by the [vapour] in the last step is similarly balanced by the entropy gained by transfer of the heat of solution to [the liquid], so these processes do not contribute to the rate of entropy production'. As discussed above, however, the entropy change associated with the condensation step is not zero, having a value of Avhi,(T1,P!)(T[eqn. (26)].The earlier derivation' also does not explicitly address the contribution to the total heat flux J, from the enthalpy of the transported mass, hiJi, and thus does not distinguish between the gas (J4,,) and liquid (J,,l) conductive heat fluxes. As a result, can- cellation of the heat capacity term in eqn. (28) does not occur leaving an extraneous term c, TJ, AT/T in the dissipation t Note that the dissipation equation from ref. 8 has been written in a slightly different form where the limit for small values of AT and APi has been taken. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 function [eqn. (9) in ref.81. The extra term propagates throughout the calculation, affecting the magnitude and even the sign of the thermodynamic forcing in the coupled case. The problems associated with the earlier derivation' are compounded further because the heat flux in the dissipation function'* is assigned to be Jq,l rather than J,, 2. The differ- ence between Jq,' and J,, eqn. (7)is quite large under most circumstances, and, as discussed in more detail below, this error leads to a considerable overestimation of the magnitude of the irreversible thermodynamic coupling between heat and mass fluxes for the gas/liquid interface. 3. Coupling of the Heat and Mass Fluxes Following eqn. (3), the dissipation function for exchange across the liquid/gas interface eqn.(22) can be expressed using the force-flux notation, where the driving forces are simply the fractional temperature and partial pressure differ- ences across the interface and where the fluxes are the conductive heat Jq, and mass Ji fluxes into the gas phase." For slowly evolving systems close to equilibrium the fluxes Jq,2 and Ji can be expressed as a linear sum of the force^^.'^ Xi (33) I Ji = Li,X, + 1LijXj (34) i The phenomenological coefficients L,, and Lijare the transfer coefficients for heat and mass across the interface in the uncoupled case. The cross-terms L,, and L,, are the transfer coefficients resulting from the coupling between the heat flux and the chemical potential difference and the mass flux and temperature difference across the interface, respectively.Addi- tional mass-mass coupling terms LiAi #j)can also arise due to coupling between the fluxes and forcing terms for different species. For a single-component system, eqn. (34) states that the evaporation and/or condensation rate Ji is controlled by two factors: (35) The corresponding conductive heat flux equation is The cross-effect or coupling coefficients Li, = Lqi can be related to the primary mass transfer coefficient Lii in the fol- lowing manner. Fix the temperature (and partial pressures of all other species in a multi-component system) as constant across the interface, i.e. X, = 0 and Xj+i= 0. Setting X,= 0 in eqn. (35) and eqn. (36) leads to L.=qfLii (37)qi where the proportionality term q:, the heat of transfer, is defined as the ratio of the conductive heat and mass fluxes in the gas phase for the case where AT = 0 qT = (38)(9)AT=O, APj#i=O The flux equation for Ji [eqn. (35)] can then be rewritten solely in terms of the uncoupled coefficient Liiand the heat of transfer qf (39)Pi Comparing eqn.(39) with the traditional, empirical gas exchange parametrization eqn. (5), one can see a strong simi- larity between the two formulations in that each can be decomposed into a thermodynamic and a kinetic component. Neglecting the cross-effect term in eqn. (39) for a moment, it can be shown that the two formulations are identical if one sets the phenomenological coefficient Lii equal to Pi Cip/RTK,. The effect of the irreversible thermodynamic coupling between the mass transfer and AT enters eqn.(39) as an additional term in the thermodynamic driving force for mass transfer. The magnitude of this perturbation depends on the size of AT/T and AP/P as well as the value of qf relative to RT. The heat of transfer qf (J mol-') reflects the conductive heat flux J,, set up in the gas phase by a mass transfer of a mole of species i under isothermal conditions. To a first approximation, q: is zero for gas-liquid exchange because the bulk of the heat released or absorbed during the phase transfer comes from the liquid rather than the gas phase; that is Jq,l x A, hi J, and J,, x 0 [eqn. (7)]. As shown in the next section, because of second-order effects a small fraction of the heat required for the phase transfer does come from the gas phase, and the qf for a generalized liquid/gas interface is about the same magnitude or slightly smaller than RT.Since for many systems of interest the fractional temperature differ- ence is smaller than that for partial pressure, we expect the irreversible cross-effects to appear as a minor correction to the traditional gas-exchange formulation. 4. Kinetic Theory of Evaporation and Condensation Irreversible thermodynamics by itself does not constrain the magnitude of the coupling effect across the gas/liquid inter- face; for that one must turn either to experimental and/or other theoretical results. Results from work on the kinetic theory of evaporation and condensation are used in this section to estimate the size of qf for a simple gas/liquid system.The kinetic theory calculations should be viewed as correct in general terms but not in detail since they are devel- oped from an idealized molecular model of the interface and gas phase, one much simpler than that found in real applica- tions. They do, however, indicate the direction and relative magnitude of the heat-mass coupling phenomena for the gas/ liquid interface. A liquid/vapour interface undergoing non-equilibrium phase change is characterized by a transition region, some- times called a kinetic or Knudsen layer, in the gas phase where the temperature and vapour pressure deviate, due to the presence of the liquid surface, from the values extrapo- lated to the interface from the gas phase" (Fig.2). The thick- ness of the kinetic layer is of the order of a few mean path lengths, and the details of the molecular behaviour in the layer can often be reduced to two simple parameters, the macroscopic jumps in temperature and partial pressure. These jumps are equal to the difference between the values extrapolated to the interface from the gas continuum, T(0) P,(O),and the liquid surface values, T,Pi,s, where Pi, is the saturation pressure at T,.'9-21 The macroscopic jumps are generally reported in fractional form molecular I layer I Ti I I I layer Iiquid/gas i nterfaceT(O)yPi01 Ts,Pi,s Fig. 2 Schematic of the kinetic or Knudsen layer near the gaspiquid interface.The partial pressure Piand temperature T in the kinetic layer deviate from their values, PJO)and T(O),extrapolated to surface from the gas continuum. The magnitude of this deviation is chara-terized by the macroscopicjumps E, and E~,i, the fractional property difference E, = [do) -aJa, . The macroscopic jumps for the kinetic layer can be com-puted, using kinetic gas theory, as a function of the heat and mass fluxes across the interfaces and are zero only when the vapour and liquid are in thermal and chemical equilibrium.20 The kinetic theory results for condensation and evapo-ration, derived from a microscopic, statistical mechanical view, can be shown to be equivalent to the macroscopic, irre-versible thermodynamic formulation for the gas/liquid inter-face.20The dissipation function Ta for a phase interface is, in full detail, quite complex,22 but it can be simplified greatly using several reasonable assumptions : a planar surface (no curvature terms), no viscous pressure effects, surface tension independent of temperature and composition and no sur-factants.The final two assumptions lead to no excess storage of energy or mass along the interface, and thus require that the fluxes of mass Ji and total heat J, be continuous across the interface. The dissipation function then reduces to a similar form as presented earlier :I8 Ta----E, Jq,2 -C RT E~,i Ji A I 4.1 Liquid-Vapour System: One Component The constitutive relations for the macroscopic jumps in the simplest system, the half-space problem involving fluxes between a pure liquid and its vapour, are given by Cipolla” as and 1 RT Ep,i = -Pi, piui Jq.2 -Bii -Pivi Ji (43) where vi is the most probable speed where Mi is the molecular weight.Bqq,Bqi,pi,, and Bii are the dimensionlessjump coefficients that relate the heat and mass fluxes to the jump The macroscopic jump equa-tions can be derived from the interface dissipation function [eqn. (41)] and are the reciprocal of the traditional flux equa-tions presented in Section 3 [eqn. (35) and (36)]. The Onsager relations hold for the jump coefficients (i.e. Bqi = &), and the matrix of jump coeffcents /3 can be related to a matrix of phenomenological coefficients 1 by 1 = 8-’, where both matrices are in non-dimensional form.” J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Evaporation or condensation from the liquid surface per-turbs the distribution of gas molecules in the kinetic layer near the interface, and Cipolla2* calculate the following numerical values for the jump coefficients /3 using a varia-tional principle on the linearized Boltzmann equation /344 = 1.030; /3ii = 2.125; /3,i = pi, = 0.447 (45) These jump coefficients are applicable for planar surfaces, slow evaporation/condensation rates, ’ and Maxwellian mol-ecules; correction factors may be required for non-monoatomic gases or different forms of molecular interactions leading to a possible reduction in the magnitude of E, and cp, .23 These /3 values do not necessarily hold for transport to and from small aerosol particles since the pertur-bation to the molecular distribution around the particle is much smaller in this case.24The /3 values are very similar to those reported by Labuntsov and Kryukov” and are within about 10% of the results from the more simple approach given by Barrett and Clement.24 From eqn.(42) and (43), it is apparent that for constant ,heat or mass fluxes the macroscopic jumps E, and E~ are ~ inversely proportional to the vapour pressure and the most probable speed, vi. The preferred systems for experimental study, therefore, are not surprisingly liquid metals such as mercury that have both a high molecular weight and low vapour pre~sure.~~,~’For comparison, the predicted macro-scopic temperature jumps for a conductive heat flux Jq, = lo00 W m-2 are 6 x and 4 x respectively, for water vapour at 289 K (Pi = 0.032 atm) and mercury at 400 K (Pi= 0.00132 atm).The jump equations can be inverted to give the relation-ship between the heat and mass fluxes and the macroscopic jumps (46) RT-J.= -1 iq E t -l..~.PiVi ti p. t (47) where for the liquid-vapour system of Cipolla et ~1.~’ I,, = 1.069; Iii = 0.518; l,i = li, = -0.225 (48) Note that the coefficients for the cross-effects, Eiq and lqi, are negative resulting in heat or mass fluxes opposite in direction, for the same sign jump, from that expected for the primary effects. For example, when the vapour pressure in the gas is higher than that of the liquid, .sp> 0, the mass flux is nega-tive, condensation onto the liquid surface, while the conduc-tive heat flux resulting from the cross-effect is positive; a pressure jump with no accompanying temperature jump leads to a net heating of the gas phase.The ratio of the heat and mass fluxes in this case is deter-mined by the heat of transport, which for the liquid vapour interface can be found from 4: = RT Iqi/lii 4: = -0.434RT (49) The heat flux due to coupling q:Ji can be compared with the latent heat flux A, hiJi associated with Ji. For most liquids, the heat of vaporization can be approximated to A, hi z 10RT (Trouton’s or cu. 20 times larger than @Ji. Note also that Ahi is generally positive while 4: is negative.Combining eqn. (49) and Trouton’s rule with the constraint that the latent heat flux is balanced by the vapour and liquid conductive heat fluxes, eqn. (7)leads to Jq,l = Ahi Ji -0.434RTJi x lORT Ji -0.43RTJi (50) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Based on the kinetic theory results, about 4% of the heat required for the phase change comes from the vapour phase with the remaining 96% from the liquid. Note that this result does not hold for aerosols where, because of the small heat capacity and large thermal conductivity of the liquid droplet, the latent heat is supplied almost entirely from the gas phase.4 Another useful quantity that can be defined from the kinetic theory results is the ratio liq/lqqRT, which is equal to the ratio of the mass to heat flux driven by a temperature difference across the interface From eqn.(48), the mass-heat coupling parameter c? is -0.210/RT. When the vapour is warmer than the liquid but has the same partial pressure, the vapour is both cooled due to conduction and made more supersaturated by an evapo- rative flux generated from the non-equilibrium cross-effects. Interestingly, taking A" hi = lORT the liquid is also cooled by conduction, but since the heat capacity of liquids is generally much larger than that of gases the vapour will cool more rapidly. If the temperature contrast between the vapour and liquid phases is maintained by some outside heat source, the system will evolve to a steady-state where the cross-effect evaporative flux is balanced by condensation from the super- saturation and where the magnitude of the supersaturation E~,can be computed from the jump equations by setting Ji to zero.4.2 Liquid-Gas Mixture System: Multiple Components The analysis of Cipolla et a!.'' can be extended to include gas mixtures where some of the gases can be non-condensable or and where the pressure and molecular speed terms are now the total pressure P = XiPi and the mean speed u = Xixi, ui , respectively. The macroscopic jump coefficients B for a gas mixture are, with the exception of the mass coefficients Bii, altered only slightly if at all from their values in the pure liquid-vapour system, eqn. (45). The mass jump coefficient for species i, pii, varies inversely with both xi and xi, and the predicted jump E~,for a fixed mass flux is constant with Pi.That is, larger pressure jumps are required for the same mass flux when the partial pressure of i is smaller or when the ability of i to enter the liquid is reduced. The irreversible thermodynamic coupling effects for the multi-component system can be defined in the same fashion as for a single-component system with the parameters 4: and c? [eqn. (38) and (51)]. Additionally, coupling between the mass flux of one species and the pressure difference of another may also arise in the multi-component case. The mass-mass coupling is characterized by the parameter rn; (55) It is helpful to visualize the parameter dependences for the coupling parameters.Fig. 3 and 4 present the results, for a two-component mixture, of varying: (1) the mole fraction of vapour x1 in a vapour-gas mixture; and (2) the condensation coefficient x1 in an equimolar mixture, x1 = x2 = 0.5, x2 = 1. In both cases the total pressure P is kept constant and the molecular weights of the two gases are set equal, MI = M,. The modified The curves shown in Fig. 3 are valid for either a condensable only slightly soluble in the liquid pha~e.'~~~~~~~ or non-condensable second gas species. jump coefficients are: 16 Xi xi, and xk, are the molar fractions of species i and k in the gas phase and 6, is the Kronecker delta, 13~~(~+~)= 0. Ai/2 is the ratio of the thermal conductivity for species i to the total thermal conductivity, which Bedeaux et d.'*set equal to the mole fraction. xi is the condensation coefficient for i, the boundary condition for gas molecules at the interface reflec- ting the fraction of impinging molecules that condense onto the liquid surface.A value of x equal to 1.0means that all of the gas molecules condense on impact while x = 0 signifies a non-condensable gas that cannot cross into the liquid. The constitutive equations including multiple gas species become The heat of transfer q: does not vary with either the mole fraction or condensation coefficient of species 1(Fig. 3 and 4). The presence of additional gas species, therefore, does not invalidate the conclusions given above regarding the magni- tude or direction of the mass flux driven by a temperature difference across the interface.The mass-heat coupling parameter cT decreases with both x1 and x1 (Fig. 3 and 4) for similar reasons to the variation seen in pii. The interfacial 1o-~ I 1111111~ I I11111 I n- I 10-4 1o-2 lo-' 1oo X1 Fig. 4 Plot of dimensionless, irreversible coupling parameters, (a) -q:/RT, (b)m;l, (c) -c:/RT and (6)rnX us. condensation coefi- cient x1 for a two component mixture. The total pressure P is keptconstant and the molecular weights of the two gases are set equal, M, = M,. mass-mass coupling parameter mT2 has (Fig. 3) a maximum value of 0.043 at x1 = 1 and decreases linearly with xl. The mass-mass cross-effect flux of species 1 is at most 4.3%of J, and only when the mole fraction of species 1 is large.my, also decreases with x1 as would be expected (Fig. 4). The second mass-mass coupling parameter in Fig. 4, mf,, reflects the flux of species 2 forced by a cross-effect from species 1 and is inde- pendent of x1. 5. Discussion and Summary Recent theoretical ~ork~?~ has suggested that the transfer of mass between gas and liquid phases can be strongly modified by irreversible thermodynamic coupling of the mass flux with the temperature field. The irreversible coupling phenomena can be deduced from the form of the dissipation function for heat and mass transfer across the liquid/gas interface. The dissipation function has been rederived in Section 2 following two different methods, the traditional one presented, for example, in de Groot and Mazurg and Haase" and the reversible path approach outlined by Phillip~.~.~ The two techniques, when correctly implemented, lead to the result given by eqn.(22). An earlier derivation by Phillip~~.~ is shown to lead to an incorrect dissipation function in part because it does not properly treat the contribution to the total heat flux J, from the enthalpy of the transported mass, hiJi, and thus does not explicitly separate the gas (Jq, and liquid (Jq,1) conductive heat fluxes. The governing equation for gas-liquid mass flux can be divided into a kinetic and a thermodynamic component, and it is shown (Section 3) that the effect of irreversible thermody- namic processes is to modify the thermodynamic term to include an additional term q:AT/T in the mass transfer equation, eqn.(39). The heat of transfer q: is the ratio of the conductive heat to mass flux arising from the cross-terms in the irreversible thermodynamic formulation and is the char- acteristic parameter for understanding the coupling pheno- mena. The strength and direction of the temperature-driven mass flux is estimated for an idealized interface in Section 4 based on kinetic theory arguments for the transition region near a liquid surface. In this idealized system, q: is shown to have a value of -0.43R T for both pure vapours and gas mix- tures. The sign of 4: is such that the coupled or cross-effect mass flux forced by the temperature field occurs from the cold fluid to the warm, counter to the conductive heat flux.The relative weights of the pressure (RT) and cross-effect temperature (-0.43RT) terms in the mass-transfer equation, J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 eqn. (39), are comparable, but since for many systems, the fractional temperature difference AT/T will be smaller than that for pressure, the irreversible thermodynamic coupling effect should generally appear as a correction rather than as the dominant term in eqn. (39). The results presented here differ significantly from an earlier analysis of coupling effect^,^,' where the coupling effect is predicted to be of the same magnitude or larger than the uncoupled, pressure-driven mass flux.As discussed above, however, the incorrect replacement of Jq,,by Jq,lin the phenomenological flux equations, eqn. (35) and (36), leads to the erroneous conclusion2 that the heat of transfer 4: is equal to the enthalpy of vaporization AVhi.This error is very sig- nificant and invalidates many of the conclusions of the earlier ~ork.~+','~For example, the enthalpy of solution of many common gases in water is quite largei5 (10-20 kJ mol-'), a factor of 4 to 8 times RT at room temperature (RT29, = 2.5 kJ mol-'). Using AVhi for q:, therefore, leads to an over-estimation, by ca. 20 for evaporation/condensation and ca. 8-16 for gas exchange, of the importance of irreversible ther- modynamic phenomena for liquid-gas mass exchange. In addition, because AVhiis generally positive while 4: is nega-tive, the fluxes driven by cross-effects are in the opposite direction from those presented here. Real applications of mass-transfer between gas and liquid phases commonly involve mixtures in the gas phase, and the .~~kinetic theory results of Cipolla et ~1 offer a method for computing, at least for a very idealized molecular system, the size of irreversible effects for mixtures.An examination of eqn. (52) and Fig. 4, however, shows that the condensation coefficient xi plays an important role for slightly condensable gases and there has been little mention, to this point, of what is a reasonable range for this parameter. The flux of gas impinging on a liquid surface can be computed from kinetic theory, and xi can be written as the fraction of this kinetic flux that actually enters the liquid' Barrett and Clement24 argue based on experimental work on liquid metals25 that xi for vapour above a planar, liquid surface should be near 1.0.The discussion for trace gases, however, becomes more complicated. Noyes et aL6 estimate xi from laboratory measurements of the maximal transfer velocity eqn. (56). They find that for N, dissolving into various solvents, xi values range from 4 x lo-'' to 1 x and are approximately proportional to the equilibrium con- centration of N, in the solvent. The flii and c: values deter- mined for the gas mixture case are quite sensitive to the choice of xi, and this factor needs to be explored further..~~The kinetic theory results of Cipolla et ~1 are symmetric with respect to the direction of the mass flux Ji; that is the same values of fl are predicted for evaporation as for conden- sation. As discussed by Barrett and Clement,24 however, the mass flux during condensation to a planar surface can be greatly reduced due to the presence of non-condensable gas. The net mass flow towards the liquid surface leads to a build- up of non-condensable gas near the interface, and the resulting pressure gradient causes an additional impedence to the mass transfer, one that can overwhelm the impedance caused by the phase change alone.27 This phenomenon, which arises not in the kinetic layer but rather due to macroscopic effects in the gas-continuum region, can be appreciable in the two-surface problem where a non-condensable gas is trapped between an evaporating and a condensing ~urface.~~.~~ To examine this effect further requires a method for coupling the kinetic layer with the gas-continuum.13~'9 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 For many practical applications, the major resistance or impedance for mass transfer between gas and liquid phases is not interfacial but rather is due to the formation of boundary layers in one or both of the fluids where the concentrations and temperatures deviate from their bulk values.' The inter- face-controlled system addressed in this paper is a special case of the more general mass-transfer problem in which the two bulk phases are assumed to be infinitely well mixed.8 The focus of this paper has been on the role of irreversible process at the interface itself, and the more general discussion of irre- versible thermodynamics in the boundary layers is presented e1~ewhere.l~Briefly, however, when the mass flux is domi- nated by the resistance of one or both of the boundary layers, the flux through the kinetic region is set external to the inter- face, and the jump equations, eqn.(53), then provide a steady- state estimate of the macroscopic interface jumps. Values of zp and E, are computed for typical air-sea ~0nditions.l~ Experimentally, irreversible coupling phenomena often appear in this fashion where the magnitude of the effect is calculated from steady-state property distributions forced by fixed mass and heat fluxes.25 Appendix Irreversible Thermodynamics and the Dissipation Function The derivation for the dissipation function To is included both for completeness and because it differs from that pre- sented elsewhere.8 A number of excellent references are avail- able for irreversible thermodynamics, and the reader is referred to these texts for a more detailed treat-ment.9,10,11,14,29The entropy production rate u can be for- mulated for either a single, continuous phase or for multiple, discontinuous (heterogeneous) phases.Consider an adia-batic, closed system made up of two subsystems, 1 and 2, separated by a membrane but which can exchange heat d@ and matter dn, (Fig.1). The subsystems are each assumed to be uniform or isotropic in internal properties (e.g. tem-perature, concentration). Let do be the total heat transported from subsystem 1 to subsystem 2, consisting of both the con- ductive or measurable heat dq and the enthalpy of the trans- ported matter hidni. Assuming only PV work (i.e. no electrical or magnetic work), the changes of the internal ener- gies dU of the two subsystems can be described by dU1 = TI dS1 -PI dV1 + 1pi.1 dni,, (All 1 dU2 = T2 dS, -P2 dV2 + 1pi, 2 dni, 2 (A2) 1 where T is absolute temperature, P is pressure, I/ is volume and pi is the chemical potential of species i. Eqn. (Al) and (A2) can be rearranged to solve for the entropy changes The internal energy changes dUl and dU2 can also be expressed using external variables dU, = d@l -Pl dV1; dU2 = d@2 -P2 dV2 (A5) For a closed, adiabatic system, the only exchanges of heat and matter are between the two subsystems d@= -d@1 = dcD2 (A6) and and the entropy change for each subsystem can then be written as The total entropy change dS = dSl + dS2 is or AT dS = --T2 d@ -i A(;) dni using T being a mean temperature and AT = T2 -T, The total heat term do can be expanded into the measur- able heat and enthalpy released from subsystem 1 and absorbed by subsystem 2 dcD = dql + 1hi,l dni = dq, + 1hi, dni (All) 1 i and the difference in the conductive heat transferred between the two subsystems is equal to dq2 -dq1 = 1(hi,l -hi, 2) dni (A 12) 1 The conductive heat terms dql and dq, are equal only when either the transfers of material dni are zero or when the molar enthalpies in the two subsystems are equal (for example, if both subsystems are gases, h, x h,).Note that enthalpies, hi, and thus the total heat @ do not have absolute values and must be referenced to a standard state. The general form for the dissipation function T dS/dt can be found by combining eqn. (A9) and (All) and differentiat- ing with respect to time (A13) where A is the area between the two subsystems, Ji is the molar flux of i per unit area from subsystem 1 to 2, and J,, is the measurable heat flux absorbed by subsystem 2. Eqn. (A13) is identical, barring different sign conventions, with the formulation of Haasel' for the time rate of entropy pro- duction for a discontinuous system.For subsystems where the system would be homogeneous if the membrane were to be removed (i.e. gas-gas or miscible liquid-liquid subsystems but not for the gas-liquid system), the term A(pJT) can be eliminated using where (pi),is the chemical potential of species i at constant temperature. Setting subsystem 2 as the reference state, eqn. (A 13) reduces to : and the dissipation function can be expressed simply as a function of the measurable heat and matter fluxes in one of the phases. In developing eqn. (A13)it is assumed that both phases are isotropic, with concentration or temperature gradients between the phases being isolated to the membrane separat- ing the two subsystems.In a real gas-liquid system, however, both phases often contain finite thickness boundary layers near the interface where properties diverge from their bulk values. Following the discussion above, a volumetric dissi- pation function Ta, (J s-' m-3) can be derived for a contin- uous region (single phase) of variable composition.' ',14 In one-dimension Ta = ---J,+ChiJi)-xTJi-(dL;) (A16) TId'(dz dz T where J, and Ji are defined as positive in the z direction. Following the discussion above, To, can be rewritten as: J dT dTa = -3 --1Ji -(p.)T dz dz IT This work was motivated by discussions of the NCAR/ NOAA CMDL carbon cycle group and by the encour-agement of D. Schimel and W. Large. I would also thank D.Glover, S. Green, R. Wanninkhof, and R. Keeling for their comments on earlier drafts of the manuscript. This work was funded by an NCAR Advanced Study Program Postdoctoral Fellowship. Glossary value of a in the liquid phase value of a in the gas phase value of a for solvent standard state of a a2 -a1 flux of a, positive from liquid to gas extrapolated interface value of a from gas contin- uum value of a at liquid surface macroscopic jump of a across gas/liquid interface area, m2 jump coefficient, various, dimensionless empirical transfer velocity, m s-' molar flux of i, mol m-2 s-l conductive heat flux, J m-s -' total heat flux, J m-2 s-' Henry's law coefficient, atm kg mol- ' phenomenological coefficient, various, dimension- less molecular weight, kg mol- pressure, atm partial pressure of i, atm gas constant, J mol-' K-' entropy, J K-internal entropy, J K-entropy flow from surroundings, J K-' absolute temperature, K internal energy, J volume, m3 thermodynamic driving force, various mass-heat coupling parameter, mol J-concentration of i, mol mP3 molar heat capacity, J mol-K-' specific heat capacity, J kg-'K-' molar enthalpy of i, J mol-' molality of i, mol kg- ' mass-mass coupling parameter mol of i, mol J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 conductive heat, J heat of transfer, J mol-most probable molecular speed, m s-' mole fraction of i heat of vaporization (solution), J mol-' total heat, J entropy production rate, J K-' s-volumetric entropy production rate, J K-' s-' m-3 m-' K-' thermal conductivity of i, J s-mass density, kg mP3 chemical potential of i, J mol-' chemical potential, constant T,J mol-' condensation coefficient of i References 1 P.V. Dankwerts, Gas-Liquid Reactions, McGraw-Hill, New York, 1970. 2 E. J. Davis, Aerosol Sci. Technol., 1983, 2, 121. 3 P. S. Liss and P. G. Slater, Nature (London), 1974, 247, 181. 4 P. S. Liss and L. Merlivat, in The Role of Air-Sea Exchange in Geochemical Cycling, ed. P. Buat-Menard. D. Reidel, Hingham, MA, USA, 1986, 113. 5 T. G. Theofanous, in Gas Transfer at Water Surfaces, ed. W. Brutsaert and G. H. Jirka, Reidel, Boston, USA, 1984. 6 R. M. Noyes, M. B. Rubin and P. G.Bowers, J. Phys. Chem., 1992,%, 10oO. 7 L. F. Phillips, Geophys. Res. Lett., 1991, 18, 1221. 8 L. F. Phillips, J. Chem. SOC.,Faraday Trans., 1991,87,2187. 9 S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962 (reprinted by Dover, New York, 1984). 10 A. Katchalsky and P. F. Curran, Nonequilibrium Thermodyna- mics in Biophysics, Harvard University Press, Cambridge, USA, 1965. 11 R. Haase, Thermodynamics of Irreversible Processes, Addison-Wesley, Reading, USA, 1969 (reprinted by Dover, New York, 1990). 12 L. F. Phillips, Geophys. Res. Lett., 1992,19, 1667. 13 S. C. Doney, J. Geophys. Rev., submitted. 14 K. S. Forland, T. Forland and S. K. Ratkje, Irreversible Ther- modynamics, John Wiley, New York, 1988. 15 I. N. Levine, Physical Chemistry, McGraw-Hill, New York, 1983. 16 D. D. Wagman, W. H. Evans, V. B. Parker, R. H. Schumm, I. Halow, S. M. Bailey, K. L. Churneg and R. L. Nuttall, J. Phys. Chem. Ref: Data, 1982,11, Supp. 2. 17 W. Stumm and J. J. Morgan, Aquatic Chemistry, John Wiley, New York, 1981. 18 D. Bedeaux, J. A. M. Smit, L. J. F. Hermans and T. Ytrehus, Physica A, 1992,182,388. 19 D. A. Labuntsov and A. P. Kryukov, Int. J. Heat Mass Transfer, 1979, 22, 989. 20 J. W. Cipolla, Jr., H. Lang and S. K. Loyalka, J. Chem. Phys., 1974,61, 69. 21 D. A. Edwards, H. Brenner and D. T. Wasan, Interfacial Trans- port Processes and Rheology, Butterworth, Boston, 1991. 22 D. Bedeaux, in Aduances in Chemical Physics, ed. I. Prigogine and S. A. Rice, 1986,64,47. 23 P. N. Shankar and M. D. Deshpande, Phys. Fluids A, 1990, 2, 1030. 24 J. Barrett and C. Clement, J. Colloid Interface Sci., 1992, 150, 352. 25 J. Niknejad and J. W. Rose, Proc. R. Soc. London, A, 1981, 378, 305. 26 J. W. Cipolla, Jr., H. Lang and S. K. Loyalka, in Rarefied Gas Dynamics, ed. M. Becker and M. Fiebig, DFVLR-Press, Porz- Whan, Germany, 1974, vol. 11, F. 4-1. 27 T. Soga, Phys. Fluids, 1982,25, 1978. 28 J. C. Haas and G. S. Springer, J. Heat Transf., 1973,95,263. 29 H. J. V. Tyrrell and K. R. Harris, Diffusion in Liquids, Butter-worths, London, 1984. Paper 3/05897B ;Received 30th September, 1993

ISSN:0956-5000    

DOI:10.1039/FT9949001865

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1875-1894 Thermodynamic Properties of 0-6 rnol kg -Aqueous Sulfuric Acid from 273.15 to 328.15 K Simon L. Clegg School of Environmental Sciences, University of East Anglia , Norwich, UK NR4 7TJ Joseph A. Rard Earth Sciences Division, Physical Sciences Department, Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA Kenneth S. Pitzer Department of Chemistry, University of California, Berkeley, CA 94720, USA Generalised equations are presented for an extended form of the Pitzer molality-based thermodynamic model, involving an ionic strength-dependent third virial coefficient. Compatibility with the established formulation is retained. Osmotic coefficients, emf measurements, degrees of dissociation of the HSO, ion, differential enth- alpies of dilution and heat capacities for aqueous H2S0, from 273.15to 328.15 K, 0-6.1 rnol kg-' and at 1 atm pressure have been critically evaluated.Treating this solution as the mixture H+-HSO,-SOz--H,O, and using hydrogen sulfate dissociation constants from the literature, the model parameters were fitted to the data yielding a self-consistent representation of activities, speciation and thermal properties together with the standard poten- tials of four electrochemical cells and standard-state heat capacities of the SO;-ion as functions of tem-perature. The model equations represent the experimental data accurately (without the use of mixture parameters OHSO4, so4 and t,hHSo4,so4,,,), and should yield values of the osmotic coefficient that are suitable for use as an isopiestic standard over this temperature and molality range.The new model will also enable improved prediction of the properties of mixed acidic sulfate systems. 1. Introduction terms of the molalities of the dissolved species, which may be Aqueous sulfuric acid is a major industrial chemical, and its ionic or neutral solute^.^ The model treats strong electrolytes thermodynamic properties have been studied extensively over as fully dissociated in solution. In addition to earlier work on many years.' Recent evaluations include those by Staples' the thermodynamics of aqueous H,S04," the model has and Rard et d3(osmotic and activity coefficients at 298.15 been used successfully in numerous geochemical applica- K), Bolsaitis and Elliott4 (partial pressures) and Zeleznik' tions.(properties of aqueous and solid phases, excluding vapour Raman spectral studies have shown that the first disso- +pressures). Sulfuric acid is also an important component of ciation of sulfuric acid (H,SO, g H + HSO,) is essentially and of complete at <40 mol kg-' (14 mol dm-3) and 298.15 K (seeatmospheric aerosols,6 notably in the ~tratosphere,~ brines. A knowledge of the sulfate-hydrogensulfate equi-also Section 3.5).14 However, this is not the case for the librium is required to calculate solubilities and partial pres- second dissociation reaction involving the hydrogensulfate sures of volatile acids such as HC1 and HNO, in acidified ion, whose dissociation must be considered explicitly : sulfate mixtures, and for relating hydrogen-ion activities in HSOi(aq,+ SO:,,) + Hi,) (1)seawater and estuarine waters to pH measurements.8 For practical applications, a treatment of aqueous H2S04 thermodynamics that readily generalises to solution mixtures is required.The Pitzer ion-interaction model9 has previously been applied to represent evaluated osmotic coefficients, emf and enthalpy data for that However, to achieve where KHso&nol kg- is the thermodynamic dissociation the accuracy that is often required, and to incorporate more constant of HSO, in solution, rn and a denote molality and recent experimental work on the osmotic coefficient, heat activity, respectively, and yi is the activity coefficient of capacity and hydrogensulfate dissociation constant, a more species i.detailed and comprehensive treatment is worthwhile. The basic model equations for osmotic (4) and ionic activ- Here we utilise an extended form of the Pitzer ion-ity coefficients contain the cation-anion (ca) interaction interaction model to represent osmotic coefficients, vapour parameters E:),fi:), and C,, ,which are functions of tem- pressure, emf, enthalpy of dilution, heat capacity and degree perature and pressure. A series of additional parameters may of dissociation data from 0 to 6.1 mol kg-', T = 273.15 to be included to describe interactions between ions and any 328.15 K. The equations and parameters presented provide neutral species present, but these are not required here.an accurate and self-consistent description of these thermody- Three terms relate to the H', SO:-pairwise interaction; namic properties, and will form the basis of an improved in the first order in molality this effect is given by the sum model for solution mixtures at moderate temperature and 1/KHSo4+ pg)so4+./3g,!w4.In second or higher order, the molality. effect of each term is different; hence there is no objection to the inclusion of all three." The limiting effective KHS04at infinite dilution is the reciprocal of this sum. Since Sg),,, and2. The Model j3g.)so4are small in comparison to 1/KH,04, their effect is to The Pitzer ion-interaction model is based upon an expression modify the limiting KHSO4 only slightly from 0.0105 to for the excess Gibbs energy of the solution in terms of an 0.010424 mol kg- '.The flf2)terms used for 2 :2 and higher extended Debye-Huckel function and a virial expansion in charged electrolytes are inappropriate for this system and would only add to the redundancies.They are therefore omitted. In practical applications, several workers have employed empirical extensions of the model to represent better experi- mental results at high ionic strength. For example, Archer,' for the system NaCl-H,O, assumed an ionic strength depen- dence of the third virial coefficient, leading to an additional parameter, Cz',).Test calculations showed that this type of extension leads to a valuable improvement in quality of fit when applied to aqueous H2S04, and it has been adopted here.(The original C,, remains from the earlier formulation, but is now designated CL:).) It is important that the extended equations for aqueous H2S04 are also applicable to solution mixtures containing other components; indeed the extensions may also prove useful for other solutes. In Appendix I are given generalised equations of the extended model for excess Gibbs energy and osmotic and activity coefficients, for a solu- tion containing an indefinite number of ionic solutes. Equa- tions for the system H,SO,-H,O are given below: ln(yH) = + m(HS04X2BH, HS04 + "L, HSO4) + m(S04X2BH, so4 + "L, SO3 + m(H)m(HS04)CL, HSO4 + m(H)m(S04)CL, so4 { + 2m(Pb)'H, Pb) + m(HS04)m(S04)~HS04, sod,H (3) ln(yHs04) = + m(HX2BH, HSO4 + "i, HSO4) + m(H)m(HS04)CL, HS04 + m(H)m(S04)CL, so4 m(S04X2@Hs04, so4 + m(H)$Hso4, so*, H) (4) ln(7s04) = 49 f m(HX2BH, so4 + "i,so*) 4-and F= { + m(H)m(Pb)@h, Pb) + m(HS04)m(S04%S0,, so4 (7) Superscript T denotes 'total'. For clarity, ion charges are omitted from the species molalities in the above equations.In some electrochemical cells (Section 3.2), PbSO, or Hg,S04 are present at low molalities. To the first order, their influ- ence is accounted for uia an increase in the ionic strength of the solution (both salts); and in the case of PbSO, only, by the unsymmetrical mixing functions QH, pb , a$,pb and WH, pb given in the terms in braces in eqn. (3), (6) and (7). In the above equations I is the ionic strength (in mol kg- '), and A, is the Debye-Hiickel constant, as recently calculated by J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Archer and Wang.16 Here we represent their A, using a Chebychev polynomial, see Appendix 11. ArcherI7 has discussed the effect of differences between his Debye-Huckel constants and those from earlier evaluations, which are greatest at extremes of temperature and pressure, on representations of solution properties using the Pitzer model. Of relevance to the present study, we note maximum differences of 1%in Debye-Huckel slopes for apparent molal enthalpy (A,/RT) and 2.5% for apparent molal heat capacity (AJR) from values tabulated by Pitzer,' which are based on the equation of Bradley and Pitzer'* for the relative permit- tivity of water.The functions B,,, Bf,, C:,, aCa,@fa and W,, contain model parameters and (in some cases) unsymmetrical mixing terms, and are defined in Appendix I, as are the molality- dependent functions Z, g'(x) and h'(x). The coefficient rca (which appears in B,, and Bf,) is normally set constant for broad ranges of electrolytes, typically to 1.4 mol-'I2 kg'/, for 2 : 2 metal sulfates" and 2.0 mol-'12 kg112 for most other valence types. However, values for individual cation-anion combinations can also be assigned, which is the approach we take here. An analogous coefficient o,,also appears in func- tions h(x)and h'(x). Model equations for apparent molal enthalpy (L+/J mol-') and heat capacity (C$/J mol-' K-') may be obtained by partial differentiation of the excess Gibbs energy expression with respect to temperature, with pressure and molality held constant: L4 = -T2 {a[G""/(n, T)l/aT)/m(H,SO4) (8) 0c;= ~$ + aL+/aT (94 = ct0-T{2a[Ge"/(n, T)]/aT + Ta2[Ge"/(n, T)]/aT2}/m(H,S04) (9b) where G'" is the excess Gibbs energy of the mixture, n, the number of kg of solvent (water) and m(H2S04) the stoichio- metric molality of sulfuric acid.The excess Gibbs energy per kg of solvent can be expressed in terms of the activity and osmotic coefficients, yielding for pure aqueous H2S04 : G'"/(n,RT) = 3m(H,SO4)[ln(y*) -k 1 -4stl (10) for ideality defined on the molality basis, where R is the gas constant (8.3144 J mol-' K-'). The stoichiometric mean activity coefficient y* and osmotic coefficient $st of H2S04 are related to the quantities in eqn.(3), (5) and (6) by: At = $Cm(H+) + m(HS0,) + m(SO:-)I/C3m(H,SO4)1 (11) Yi = 7; ~so4C~~~+~12~~~~:-~/~~C~~~2~0,)13)(12) For pure aqueous solutions of strong electrolytes the equa- tions for Lo and C$ are straightforward, and have been pre- sented many times before for the model without the additional parameter C',',).9*20*21Archer gives equations which include C',',) for thermal properties of pure aqueous solutions of 1 : 1 electr01ytes.l~ Similar equations for aqueous H2S04 would be extremely complicated, as the dissociation of HSO, varies with temperature, and so introduces extra differentials. We therefore differentiate eqn. (10) numerically to obtain the required quantities in the expressions for L@ and C$,using centred finite difference formulae incorporating either four (for a/aT) or five (for C2/8T2)terms.The step size was set to 5.0 x lOP3T,where T is the temperature of the measurement. 3. TheData Sulfuric acid and its aqueous solutions have been the subject of thermodynamic investigation for at least a century, resulting in a very large body of experimental measurements. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1877 Much early work is summarised in compilations of have generally used molar masses of H,O and H2S04 given Bichowsky and Rossini,22 and Timmermans.', Sources of by the authors of the various studies, otherwise we have used data have been compiled by Staples and Wobbeking' and the following values from the 64th edition Staples et ~1.'~Previous evaluations of the thermodynamic of the CRC Handbook,29 based upon the 1969 IUPAC properties of aqueous H,SO, include those of Giauque et recommendation^:^^ M(H,O) = 18.0152 g mol-' and u1." (0-100% acid and for solid hydrates; T < 300 K), Pitzer M(H2S04)= 98.07, g rnol-'. et a/." (0-6 mol kg-'; temperatures at, or close to, 298.15 Conversions of temperature scales to the current ITS-90 K), Staples' (0-28 mol kg- ; 298.15 K), Rard et aL3 (0.1-27.7 are tabulated by Goldberg and Weir.31 In general, the 6T mol kg-; 298.15 K) with later impr~vements,'~,'' and applicable to the experimental temperatures, for example Zeleznik' (0-100% acid and for solid hydrates; T < 350 K).-0.014 K at 25 "Cfor IPTS-48 and -0.006 K for IPTS-68,31 Vapour4iquid equilibrium in the H2S04-H,O system, with are of the same order as the accuracy of temperature control an emphasis on high temperatures, has been evaluated by which is typically 0.01-0.02 K. Also, the changes in most Gmitro and Vermeulen28 and more recently by Bolsaitis and thermodynamic properties (4,emf etc.) with temperature are Elli~tt.~ low enough that the change A, due to any temperature cor- The aim of the present study is to provide an accurate and rection, is small relative to the precision of measurement and self-consistent description of aqueous solution activities and the fit of the model. Therefore, where experimental tem-thermal properties of aqueous H2S04 from 273 to 328 K and peratures have been quoted in "C they have been converted molalities up to 6 mol kg-', within a framework (the Pitzer to absolute values simply by adding 273.15 K.Note that model) that allows ready extension to more complex mix- absolute temperatures given by Covington et uL3' and Beck tures. We have attempted to be comprehensive in our con- et u1.33,34are based upon an ice point of 273.16 K = O"C3' sideration of the available data, though we cannot claim In this instance we have subtracted 0.01 K from the tem- complete coverage. Measurements included in the present peratures given by those authors. study cover the period 1899 to the present, during which The data are discussed below.Tables 1-5 list, for each data there have been several revisions both to atomic masses and type, the concentration and temperature ranges of measure-temperature scales. Changes in atomic masses only affect ment, the numbers of experimental points, which measure- molality in the fifth significant figure, and these molalities are ments were rejected as being in error and the relative weight often quoted to only three or four figures. For consistency we assigned to each dataset. Table 1 Availability of isopiestic (iso) and direct vapour pressure (vp) data for aqueous H,SO, no. of molality/mol kg- ' T/K observations* method standard wr rejected" N ref. 1.673-21.65 298 33 (13) is0 NaOH 1.o 11 1 37 2.083-4.354 298 12 is0 NaCl 1.o 0 2 38 0.019-4.349 298 18 is0 NaCl 0.25 3 3 39 0.091-2.830 298 23 is0 KCl 0.2510.75 1 4 40 0.091-4.3 74 298 28 is0 NaCl 0.25/0.75 2 5 40 0.195-3.136 298 53 is0 KCI 0.25 0 6 41 1.918-22.63 298 20 (10) "P -1.o 1 7 48 0.073-2.871 298 13 "P -1.o 9 8 50 13.88-27.74 298 3 (0) "P ---9 115 7.326-12.58 298 9 (0) "P ---10 51 0.346-4.361 298 44 is0 NaCl 1.o 0 11 26 4.349-19.33 298-409 146 (24)' VP -0.1 2 12 47 1.133-40.78 29gd 12 (3) VP --3 13 49 2.09 1-4.3 55 298 16 is0 NaCl 1.o 0 14 43 0.141-0.1 70 298 4 is0 KCl 1.o 0 15 42 -P0.442-0.487 298 3 is0 KCl 1.o 0 16 0.189-4.175 323 44 is0 NaCl 1.o 5 17 116 1.450-4.096 273 8 is0 NaCl 0.25 2 18 55 1.033 273 1 is0 urea 0.25 0 19 55 1.14-9.56 273-373 99 (61) VP --61 20 117 95.217-23.79/ 203-250 "P ---21 52 -h4.026-4.420 298 4 is0 NaCl 1.o 0 22 N is the dataset number, referred to in the figures.Note also the work of Giauque et aLZ5who have evaluated solvent and solute activities (and other properties) of aqueous H,SO, over the entire mole fraction range. Glueckauf and Kitt,"* using a bithermal isopiestic technique, have obtained osmotic coefficients to 76 mol kg-'. Their values were not included in our calculations because they reported that it was necessary to normalise their measurements to lower molality results from other studies. Osmotic coefficients relative to the NaCl isopiestic standard were calculated using eqn. (7) and (36) of Archer.' Note the following errors in Archer's eqn. (36):lines three and four should read: Also, the coefficient b3,12 should have the value 0.06622025084.The second number in parentheses gives the number of data points within the fitted molality range 0-6.10 mol kg-'; the figure in the 'rejected' column refers only to those points within the fitted range. Molalities of the rejected data for each reference: 1.673-4.376, 5.002, 5.144;37 0.0187, 0.0456, 3.815;39 0.0909, 0.0908, 4.374;40 2.239;,* 0.073-1.282, 2.468;50 3.399, 5.490;47 0.3037, 0.2576, 0.1894, 2.1157, 3.1911;'16 1.450, 2.186 mol kg-'.55 Molalities below 1.0 rnol kg-' are given the lower weight of 0.25. Molality and temperature ranges are those of the experimental determinations. Only five values (two of which were rejected) were used here, taken from the interpolated 298.15 K isotherm in Collins' Table 3.47 Data for some other temperatures given graphically.J. A. Rard, work in progress. Vapour pressures also measured along the freezing curve of sulfuric acid tetrahydrate and hexahydrate (argued by Zhang et ~21.~~to have composition H2S04.6.5H,0). Representative data presented only in graphical form, and also as fitting equations. J. A. Rard and D. G. Archer, unpublished data. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 2 Availability of emf data for cells I-IV no. of cell molality/mol kg -' T/K observations' int.b wr rejected' N ref. I 0.1000-8.272 278-328 77 (69) 1.o 0 1 33 I 0.0073-0.096 298 5 1.o 0 2 32 I 0.01945-0.997' 298 19 19 3 59 I1 0.0050-0.050 298 2 -2 4 119 I1 0.1003-7.972 278-328 54 (47) 1.o 7 5 34 I1 0.00734.096 298 13 1.od 0 6 32 I1 0.0506-2.386 298 7 1.o 0 7 66 I1 0.0506-8.207 298 5 (4) 1.o 0 8 120 I1 0.0010-0.0 10 298 5 -5 9 121 I1 0.0050- 1.04 1 298 6 1.od 2 10 67 I1 0.1000-4.Ooo 298 7 1.o 2 11 122 I11 0.0536-3.499 298 10 1.o 2 12 123 IV 0.00104.020 273-323 25 1.o 12 13 63 'See first sentence of footnote" in Table 1.Molalities of the rejected data for each reference: 1.872, 5.767 (278 K); 0.1003, 1.872 (318 K); 0.1003, 1.872, 5.767 (328 K);34 0.02506, 0.2529;67 2.0, 4.O;lz2 0.5154, 1.036;'23 0.001 and 0.002 (all temperatures); 0.005, 0.02 (273 K).63 Emfs reported in international volts. 'Measured values (Table 1 of ref. 59) using Hamer's preferred methods (4, 5 and 6) of electrode preparation. For molalities <0.04 mol kg-', relative weights are reduced to 0.5 as the model-calculated emf is sensitive to the amount of Hg,SO, assumed to be present (see text).Table 3 Availability of enthalpy data (enthalpies of dilution) for aqueous H,SO, no. of molality"/mol kg- TIK observationsb wr rejected N ref. (> 100 Wt.%o)-O.506' 298 72 (10) 1.o 4 1 70 6.423-0.001 298 25 (24) 2.23 0 2 71 3.679-0.003 29 8 45 0.86 7 3 72 0.050-0.003d 29 8 11 0.045 6 4 73 30.860-6.07' 253 10 (1) --5 70 0.005-6.0f 303-598 ---6 75 Range given for initial molality, m,. See first sentence of footnote a of Table 1. Molalities (m,)of the rejected data for each reference: 1.508, 1.040, 0.726, 0.506;70 0.0251 (two points), 0.0125, 0.00627, 0.003 13 (two points);73 0.003 05, 0.00504, 0.00508, 0.0174 (three points), 0.0846.72 See also corrections given by Giauque et aLZ5Differential enthalpies of dilution were calculated as Adi,H= L*(m,) -L*(m,) = AAqH,O) (Table 3 of Kunzler and Giauque7').Reported mol dm-3 concentrations were converted to molality using densities compiled by Sohnel and N~votn$."~'The single data point within the fitted range was not used. Experiments carried out at 7-40 MPa, and therefore not relevant to the present study. Table 4 Availability of heat capacities for aqueous H,SO, no. of molality/mol kg- T/K observations' wr rejected' N ref. (>1OO ~t.O/o)-1.149~ 298 75 (13) 1.0 70 0.563-0.052 298 9 1.o 81 2.230-0.044' 298 13 1.0 76 1.013-0.103 298 8 0.50 80 1.013-0.103 328 8 0.50 80 1.013-0.103 313 8 0.50 80 (100 wt.%)-0.035 293 37 (20) 0.25 77 1.O 1 3-0.103 283 8 0.50 80 30.869-4.508 253 11 (2)d 70 9.2468 214-300 11 125 8.5385 230-3 19 14 125 6.9377 213-296 11 125 55.509 284-305 4 126 27.754 239-306 12 126 18.503 182-298' 12 127 18.503 244-296' 8 128 13.881 251-305 9 128 Socolik7* tabulates the specific heats (to three figures) of Savarizky (unreferenced) at 295.65, 313.15, 333.15 and 353.15 K from 6.06 to 100 wt.% H,SO,.'See first sentence of footnote 'of Table 1. Molalities of the rejected data for each reference: 1.586, 1.231;70 0.0444, 0.0713, 0.1748, 0.5515;76 0.1035, 0.1781 (T 2 298.15 K);" 0.0347.77 See also corrections given by Giauque et aLZ5'These heat capacities are on a (g H,O)-' basis.76 This is not stated in the paper, and caused Zeleznik' to reject their results as erroneous.These two data points not used. Includes supercooled solutions. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1879 Table 5 Availability of degree of dissociation data (HSO, eSO:-+ H+) for aqueous H,SO, ~ ~ ~ ~~~ molality/mol kg-' method" T/K no. of observationsb wr rejected N ref. 0.01-0.304 1 300 -10 10 1 129 0.024.00' 2 298 10 10- 2 130 0.000254.050 3 298 7 1.o 0 3 131 0.00090-2.634 4 298 13 1.o 0 4 86 0.264-42.62 2 298' 18 (9) 1.o 3 5 84 0.050-40.15" 2 273-323 38 (16) 1.010.3 2 6 14 0.052-29.26 5 298' 12 (6) 1.o 2 7 85 dissociation step at very high molalities (H,SO,= Hf + HSO,). 3.1 Vapour Pressure and Isopiestic Measurements Available osmotic coefficients and vapour pressures relevant to the present study are summarised in Table 1.Early (and less reliable) data not included here, some of it from the 19th century, have been considered by Abe136 and are also listed in bibliographies such as that of Staples and Wobbeking.' Many of the osmotic coefficient were used in the published evaluation of Rard et with newer data26*42*43 (also included in Table 1) leading to minor revisions.26 Rard and Platford, have critically assessed the osmotic coeffi- cients of H2S04 at 298.15 K to 27 mol kg-', comparing the evaluation of Robinson and Stokes44 with the later work of Rard et and of Staples., A number of serious objec- ~1.~7,~ tions were raised to the latter study, concerning the use of freezing-point depression data (see Section 3.6), the overall goodness of fit, and circularity regarding the use of CaC1, as isopiestic standard.Because of this (osmotic coefficients of aqueous CaCl, being largely determined from isopiestic equi- librium with H2S04 solutions) such data are not included here. Aqueous NaCl was the standard for most of the isopiestic .~data listed in Table 1. In the work of Rard et ~1 and Rard26 the osmotic coefficients of NaCl were calculated using the equation of Hamer and WU.~~ Since their work, Clarke and Gle~~~and Archer' ' have published substantial critical reviews of the thermodynamics of aqueous NaCl that refine its osmotic coefficient. At 298.15 K their studies agree with each other to within 0.0006 in 4, but comparisons with the equation of Hamer and Wu show systematic differences of up to 0.003. All isopiestic data for which aqueous NaCl was the standard have therefore been adjusted to Archer's values of (see also footnotes to Table l),which we consider to be the most reliable.For measurements relative to aqueous KCl the best-fit equation of Hamer and Wu45 for &,, used by Rard et ~l.,~was also adopted here. Note, however, that we have not attempted to correct for the non-ideal behaviour of water vapour, which was not considered by Hamer and WU.~~We estimate this correction to be negligible at low molalities, and no more than 0.1% at high molality. The osmotic coefficients of H2S04 for which aqueous NaOH was the standard are based upon the evaluation of Rard et uL3 of and were taken directly from their Table 1.Water vapour pressures determined by Collins,47 and Shankman and G~rdon,~'Jones49 and Grollman and Frazer" have been used to calculate osmotic coefficients. Careful attention was given to the use of values of the vapour pressures of pure water compatible with the actual tem-perature scales used in the original measurements. For the References above are restricted to studies of the variation of a as a function of concentration, and do not include those whose sole aim is to determine the infinite-dilution value of KHS04.Kerker13, recalculated r from literature data (including those of Sherrill and Noyes' 31) but many values are grossly discordant with other work, and have not been included here. Methods: 1, spectrophotometry; 2, Raman spectroscopy; 3, conductance; 4, molar volume; 5, NMR.See first sentence of footnote of Table 1. Molalities of the rejected data for each reference: 0.52, 1.03;85 2.915, 1.377, 0.264;84 0.502, 0.504.14 Concentrations in mol drn-'. 'Temperature not specified. Presumably at a 'room temperature' close to 298 K. Measurements at 273.15 and 323.15 K were given relative weights of 0.30. Data were also reported in this study for the first last three studies, corrections were made for the non-ideality of the vapour phase by use of the second virial coefficient of water vapour as tabulated by Rard and Platf~rd.,~The resulting osmotic coefficients differ by <0.0002 (Shankman and Gordon4*) and <0.0005 (Grollman and Frazer") from those given previously by Rard et aL3 Such corrections were not made for the results of Collins47 because they are impre- cise around 298.15 K.We note that the dew-point determi- nations of Hepb~rn,~' for higher molalities than used here, are of quite low precision and deviate by up to 0.04 in d, from the evaluation of Giauque et al." and are probably unreli- able. The recent vapour pressure measurements made by Zhang et dS2at temperatures below 250 K are outside the scope of the present study. However, brief comparisons indi- cate that some of the data are inconsistent with H,O chemi- cal potentials tabulated by Zeleznik' and also with water activities calculated by us.53 For further discussion of both vapour pressure and isopiestic data, see Rard et aL3 and Rard.26 Relative weights (w,) for the different data sets are given in Table 1, and are reduced for the determinations of Robinson4' (for KCl as isopiestic standard) which are more scattered than his later measurements relative to NaC1,38 and for the results of Scatchard et d4'because they did not use replicate samples.For these authors the weights were further decreased for m(H,S04) < 1.0 mol kg-', where data are more scattered, and where agreement with the more recent work of Rard26 is poor, possibly owing to insufficient time being allowed to achieve isopiestic equilibrium in the earlier study. The data of Scheffer et deviate systematically from the evaluated osmotic coefficients of Rard et d3and all other isopiestic data at low molalities and were weighted 0.25.Olynyk and Gordon54 have noted that later experi- ments by the same group agreed more closely with the work of Scatchard et ~1.:' but the results of that redetermination do not appear to have been published. The data of Platf~rd~~ at 273.15 K appear to be reliable but are quite scattered, and have also been given a reduced weight. All direct vapour pressure and isopiestic data given non- zero weights are shown in Fig. 1-3. 3.2 EMF Measurements Sources of emf data are listed in Table 2 for the following electrochemical cells : (Pt)H, 1 H,S04(m) I PbSO,, PbO,(Pt) (cell I) E = E" + (RT/2F)ln[rn(Hf)2y~m(SO~-)y,0Ju~] (13) J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 I I Ill It' I symbol N x3 04 Q5 06 11 0 15 0 16 030 0.46 0.62 0.78 Jm/mol'12 kg-'I2 Fig. 1 Stoichiometric osmotic coefficient (4s,)of aqueous H,SO, at 298.15 K and low molality, plotted against [m(H,S0,)]"2. See also Fig. 2 for measurements at higher molalities. (-) Fitted model (Section 5). Key: symbols are related to the dataset numbers (N) in Table 1. 0.86 0.90 0.94 0.98 IIIII1 0.720 F 0' 3.77 3.76 symbol N x3 O 4C 4 5a" ~6 0.7411 v 12 I I I I 0.72 1.o 1.10 1.18 Jm/mol'12 kg-'I2 symbol N F:2 x3 04 C 45 a" 06 v7 as -11 # 14 1.8 1.9 2.0 Jmlmol 'I2 kg-'I2 (Pt)H2 I H2S04(m) I Hg,S04, Hg(Pt) (cell Ir) E = E" -(RT/2F)ln[rn(H+)2yirn(SO~-)yso4] (14) Hg, Hg,SO, I H2S04(rn)I PbSO,, PbO,(Pt) (cell 111) E = E" + (RT/F)ln[rn(H+)2y~rn(SO~-)yso4/a,] (15) PbHg(amalgam), PbSO, I H2S04(m)I H,(Pt) (cell IV) E = E" + (RT/2F)ln[rn(H+)2y~rn(SO~-)ys,4] (16) where E and E" are, respectively, the measured and standard emfs (in V) of the cell and F (96484.6 C mol-I) is Faraday's constant.Note that we do not consider here emf measure- ments involving electrolyte mixtures HC1-H,S0,-H2056,57 or NaHS04-Na2S0,,58 although the model could be used to account for the ion interactions that occur.l0 Similarly, emf measurements for cells involving the Ag,SO,/Ag elec-trode are not analysed because Ag2S04 is too soluble in aqueous H2S04 to yield meaningful results for pure aqueous H,S04 solutions.Early work of Hamer59 and Harned and Hamer6' has been reported by several authors to be inconsistent with modern data.32-34 Further comparisons by one of us (J. A. R.) suggested that a partial cause of the discrepancies, for at Jmjmol 'I2 kg-'l2 1.20 1.28 1.36 I I 1 11 (b) 0.83 x3 04-0.81a" -t15 g06 v7 ~a 0.79 . 11 v 12 # 14 0.77 1.50 symbol N +l1.45 02 C x3 1.40 Q5 v7 11 1.35 v 12 fl 14 A 22 1.30 ' 2.2 2.3 2.4 2.46 Jmlmol 'I2 kg -'I2 Fig. 2 Stoichiometric osmotic coefficient (4st)of aqueous H,SO, at 298.15 K to 6 mol kg-I, plotted against [m(H2S04)]112. For each of the four graphs, the plot (and $st axis) on the left are associated with the upper molality axis.See also Fig. 1 for low molalities. (-) Fitted model (Section 5). Key: symbols are related to the dataset numbers (N)in Table 1. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.25 -I 1 1 r I 0.7 1.1 1.5 1.9 dm/moll l2 kg-i2 Fig. 3 Stoichiometric osmotic coefficient (4$,)of aqueous H,SO, at 273.15 and 323.15 K, plotted against [m(H2S04)] 'Iz.(---) Fitted model at each temperature (Section 5). N: (0)17, (0)18, (4)19. least some of the measurements, might be systematic cyclic deviations introduced by their use of graphical smoothing and fitting equations to represent the experimental emfs, first as functions of molality and then as functions of temperature. Hamer59 reported the actual experimental data for only four molalities at 298.15 K for a comparison of a variety of types of electrode preparations.Those emfs for cell I exhibited both a large average deviation (> 1 mV) from the data of other workers, and a spread of ca. 1 mV (but 8-9 mV discrepancies for their preparation 7). The results of Hamer59 and Harned and Hamer6' are therefore not included in our analysis. The published emf data span the period 1914-1965. The change from international to absolute volts Cl.0 V (int.) = 1.OOO33 V (abs.)] occurred in 1948 and we have assumed that the results of all studies published after this year are in absolute volts, with the exception of the work of Beck et as pointed out by Covington.61 For measurements at low molalities of H2S04, dissolved PbSO, and Hg,SO, can contribute significantly to the total molality of the solution.The presence of the soluble sulfates in the cells is accounted for within the model by an increase in total ionic strength Z (assuming both salts to be fully dissociated) and sulfate molality m(SOz-). The solubility of PbSO, in aqueous H,SO, above 0.005-0.009 mol kg-' acid has been measured by Craig and Vinal at 273.15 and 298.15 K,62 and estimated for 0.001-0.02 mol kg- H2S04 (273.15-323.15 K) by Shrawder and C~wperthwaite~~ by the inter- polation and extrapolation of the measurements of earlier workers. There is reasonable agreement between the two sets of values. Equilibrium concentrations of PbS0, ,calculated from the tabulation of Shrawder and Cowperthwaite, are assumed to be present in the solutions in cells I, 111 and IV.For m(H,SO,) > 0.02 mol kg- ' the dissolved PbSO, concentra-tion is low enough (< 2.2 x 10-rnol dm -') to be neglected. Pitzer et al." note that the PbSO,/Pb electrode may be reli- able only for m(H,SO,) 2 0.005 mol kg-',therefore measure- ments for cell IV below this concentration have been rejected. Mercury@) sulfate is more soluble than PbSO, . Solubilities of 7.5 x lo-, to 1.1 x rnol kg-' at 298.15 K in 0.002 to 2.0 mol kg-' H,SO, are listed by Brown and Land,64 who combined their own measurements with interpolated values from the study of Craig et (0.001 to 3.6-4.2 mol dm-3 H,SO,, 273.15 and 301.15 K), which agree with the results of Brown and Land within experimental error.64 Test calcu- lations by us showed that the inclusion of dissolved Hg,SO, at its equilibrium molality in cells I1 and I11 yielded an improved model fit, and Hg2S04 molalities estimated by interpolation from the data of Craig et al.were therefore adopted at all temperatures. As the effect of correction for solubility of Hg,SO, on the emf was found to be significant only for m(H,SO,) < 0.04 mol kg-', the practical influence of its presence is restricted to 298.15 K. These corrections also proved quite sensitive to the concentration of Hg,SO, speci-fied. Therefore, in view of the very simple treatment of the effect of the dissolved salt, relative data weights for m(H2S04)< 0.04 rnol kg-' (for cell 11) were reduced by a factor of two.We also note that, for m(H,SO,) > 0.06 mol kg-', Hg' may be present as the complex ion Hg2(S04)(HS04)-.64 For cells of types T and 11, at 298.15 K, there are two and six data sets, respectively, that were given non-zero weights. It is possible for systematic deviations in E" (bias potential) to occur for different electrode preparations of the same type, because of subtle differences in the physical and chemical state of solid electrode material. The data were therefore examined for this by calculating E" [from eqn. (13) and (14)] for individual measurements from the observed emf and using the model to obtain the activity term. For cell I an average AEL of -0.78 mV was found between the results of Coving- ton et aL3, and Beck et For cell I1 there was agreement to within one standard deviation (in the mean value of our derived E") of the results of Beck et aL3, except in the cases of MacDougall and Blumer66 (ca.0.4 mV) and Trimble and Ebert67 (ca. 0.2 mV). To allow for these bias potentials, additional terms AEo were fitted, such that E"(experimenta1)= E"(true) + AE", for the emf data of Covington et (cell I), and MacDougall and Blumer66 and Trimble and Ebert67 (cell 11). The least-squares values of AEr were -0.78, -0.32 and -0.22 mV, respectively. Consis- tency in the variation of E" with temperature for cells I, I1 and IV was also tested, by fitting E" at each temperature indi- vidually. For cell I1 at 318.15 K 34 and cell IV at 285.65 K 63 small deviations for E" were found from the general trends (ca.0.1 mV), which were accommodated using AE" terms similar to that given above. The question of whether dissolved Hg,SO, gives rise to liquid-junction potentials within cell I1 has been examined by Dobson, who studied cells saturated with Hg,S04 using both glass and hydrogen-gas electrodes.68 There was no significant difference between the two sets of results, and it was con- cluded that there was no significant liquid-junction potential contribution to the emf in the molality range studied. Thus we made no corrections for this, although we note that the opposite view has been argued by Hamer.69 With the exception of the low-molality data for cell 11, all retained emf measurements have been given unit relative weight, and are shown in Fig.4. 3.3 Apparent Molal Enthalpies Sources of enthalpy data (differential enthalpies of dilution) are listed in Table 3. Experimental dilutions (m,-+m,) (in mol kg-') are about 30% for the work of Kunzler and Giauque," 15% or less for the determinations of Wu and Young7' and most of Groenier's but >96% for the results of Lange et and 70-80% for the other three experiments of Groenier. The most precise data are those of Wu and Young,71 who have also derived L" from their own work and that of the other authors given above. They did this by first estimating L* at low molalities using the mea- sured degrees of dissociation of Young and Blatq7, the enth- alpy of dissociation of the hydrogensulfate ion and estimates of apparent molal enthalpies, L*(H, H, SO,), (from Lo of Li,SO,) and L*(H, HSO,), and combining these generated 1882 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 I 1 I I I II2.0 (a) cell I 278 K (+ 0.20)1.8 ’ 293 K 298K symbol N-96& 1.6 v 10 0318 K A 13 02 .,O 328 K +71.4 ­.’4 ‘s 11 (-0.20),*do’-/.’ n5 0 08 1.2 I (-0.30) x 12 I I I I I I 0 1.o 2.0 3.0 0 1.o 2.0 3.0 Jm/mol”2 kg-’” -0.68 V 0.81 \ 0.64 t ’\* 0.600.79 i0.5610.77 t x‘ 1> ‘e 1.0 2.0 3.( 0.751, . I . . I . I ,j& %, Jm/mol’/2 kg-’I2 -0.75 0.2 0.6 1.0 1.4 1.8 Jm/mol’/2 kg-’I2 1 I I I 0.73 0.3 I 273 K (e)rv I (+0.125) 285 K 0.71 (+0.0625) 0.2 *-----298 K o.69 -cell I1 (298 K) >G-*MAI 1 I I I1 0.05 0.25 0.45 0.65 A/ 310 K (-0.0625)Jm/mol’ /’ kg-’l2 0.1 */ 323K 0 J I I I I 0.07 0.09 0.11 0.13 0.15 Jm/mol’/2 kg-’IZ Fig.4 Measured emf (E/V) of cells I-IV, plotted against [m(H2S04)]1i2. (a)Cell I; (b)cell 11; (c) cell 11, 298 K; (6)cell 111, 298 K; (e) cell IV. For clarity, in parts (a),(b)and (e) the emfs are offset by fixed amounts (V), indicated by the numbers in parentheses. Fitted model (Section 5). Key: symbols are related to the dataset numbers (N)in Table 2. values with integrated hL*/h,/m obtained from the experi- other data, and when the size of the dilution is taken into mental Adi,H using a chord-area plot. account, generally agree well with Wu and Young’s tabula- All dilution enthalpies were first assessed in a preliminary tion.However, test fits with our model showed systematic way by comparison with the L* of Wu and Young71 (their deviations of their values which were larger than the experi- Table 5). The enthalpies of dilution obtained by Kunzler and mental errors. These deviations were related to differences Giauque7’ at the lowest H,SO, molalities were found to between Wu and Young’s method of analysis to obtain L* deviate systematically from the results of other workers, and values generated by our model below about 0.01 mol probably because their calorimeter was optimised for concen- kg-’. In view of the empirical nature of Wu and Young’s trated solutions.25 Those points were rejected as being in estimates of L4 in this region, the four measurements for error, as were a few others with deviations of >40 J mol-’.which deviations from the model fit were >100 J mol-were The experiments of Lange et uE.,’~ and three of the measure- rejected. ments of Groenier” involve very large dilutions and the We note that Milioto and Sim~nson~~Oak Ridgeat lowest values of m2. Groenier’s three enthalpies of dilution National Laboratory have made many enthalpy of dilution show much larger deviations from values calculated from measurements for aqueous H,SO, from 0.005 to 6 mol kg-’, Table 5 of Wu and Young71 than the comparable data of from 303 to 598 K and at pressures of 7 to 40 MPa. Because Lange et al., and were therefore rejected. In percentage terms, our evaluation is restricted to a pressure of 1 atm, their the results of Lange et al.73 appear no less accurate than results were not included in our fits.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Relative weights were assigned to each dataset as l/a2 (comparisons made with differential enthalpies of dilution calculated from Wu and Young's tabulated Lo), normalised to unit values for the measurements of Kunzler and Giauque, see Table 3. All data given non-zero weights are plotted in Fig. 5 as aL+/aJm, corrected for the difference between this quantity and the measured -AdilH/(Jml -Jm,).This cor- rection, significant to the scale of this plot only for the results of Lange et aZ.,73is also shown, as is the range of dilution (Jm,to Jm,).Data were fitted as differential enthalpies of dilution, Adi,H= Lo(rn2)-L*(m,), calculated from eqn. (8) for initial (m,)and final (m2)molalities. 3.4 Apparent Molal Heat Capacities Heat capacity measurements for H2S04-H20 are sum-marised in Table 4. For completeness we have included all the data of Giauque and c~-workers~~*~~ even though many of their determinations are outside the molality range of the present study. Abe136 considers that, of measurements made prior to about 1945, only the data of Randall and Tayl~r,'~ andBir~n~~to a lesser extent Savarizky (tabulated by Soc01ik~~)are reliable. Biron's work has been used by Craig and Vir~al'~ in their study of aqueous H2S04 related to the lead storage battery, and that of Savarizky by Giauque et 0 0.1 0.2 aL2' to estimate temperature coefficients of partial molal properties.Here we have not used the work of Savarizky, since results were reported to only three significant figures, although Savarizky's values appear to be fairly consistent with other data at lower temperatures for molalities >1.0 mol kg- '. Apparent molal heat capacities, C:/J mol-' K-', are cal- culated from measured specific heats, c,/J K-' (g)-', by the equation : C: = [1000/m(H2SO4)](~,-c;) + M(H2S04)c, (17) where c;/J g-' K-' is the specific heat of pure water at the experimental temperature and M(H2S04) = 98.07, g mol-' is the molar mass of H2S04. From eqn. (17) it is clear that at low molalities (below ca. 1 mol kg-'), where c, and ci are almost the same, the calculated Cd are very sensitive to experimental error and to the choice of ci.Heat capacities for m(H,S04) < 1.0 mol kg-' have been measured by Hovey and Hepler," Larson et Randall and Taylor,76 and Bir~n.~~ The results of the first two studies used ci taken from Ke11.82 The data of Biron" at 293.15 K are treated by Craig and Vina179 as assuming a defined c; of 1.0 cal g-' K-'. It is unclear if Biron's experimental cp are relative to this value, which would then require a transform- ation of c,/cal 8-l K-' +cJcJ293.15 K)/l.O]. However, test 5t 1 0.2b1 , I , I . , , ,j 0.36 0.56 0.76 0.96 1.16 1.2 1.6 2.0 2.4 Jm/mo1'/2 kg-lI2 ,/m/mol''2 kg-'I2 Fig. 5 Differential of the apparent molal enthalpy with respect to [m(H2S0,)]112, calculated from enthalpies of dilution at 298.15 K, and plotted against the mean of the square roots of the initial and final molalities.Horizontal lines indicate the extent of the dilution Jm,--+ Jm2. Vertical lines indicate the extent of the correction -Adi, H/[(Jm, + ,/m2)/2] to the true differential as plotted, which was estimated from the tabulated Lo of Wu and Y~ung.~' Fitted model (Section 5). The dataset numbers (N) of Table 3 are: (*) 1,(a)2, (0)(-) 3, (A)4. 1884 calculations showed that this would have a negligible effect on the apparent molal heat capacity, so C: has simply been calculated from eqn. (17) assuming ci equal to 1.0 cal g-' K-'. Randall and Taylor76 used ci = 0.9979 cal g-' K-', equivalent to a molal heat capacity of water of 17.976 cal mol-' K-' (using the atomic masses current at the time).We .~~have followed Giauque et ~1 in adjusting Randall and Taylor's data (their Table 2) by the factor (@a1 mol-' K -')/17.976 before calculating C? . The measurements of Kunzler and Giauque7O at 298.15 K, for which m(H2S04) > 1.2 rnol kg-', were considered prior to the data at lower molalities, and apparent molal heat capacities were calculated using ci = 4.1796 J g-' K-', as listed in the CRC Handbook29 and due to Osborne et rather than the value recommended by Ke11.82 At 298.15 K, these c; differ by only 0.0003 J g-' K-', and the effect of this difference upon the calculated C$ is negligible. Unit relative weights were assigned to the data of Kunzler and Giauq~e,~' Larson et aLal and Randall and Taylor.76 The apparent molal heat capacities of Hovey and Hepler" at their lowest molalities and 298.1 5 K deviate systematically from those of other workers.These data were therefore given relative weights of 0.50 at all temperatures, and the values for the two lowest molalities for T 2 298.15 K were weighted zero. At the lower temperature of 283.15 K there was very good agreement with our model, even for the most dilute solutions, so all points were retained. The very early results of Bir~n~~appear to be consistent with other data, though there are no other measurements at the same temperature for com- parison. We have cautiously assigned them a relative weight of 0.25.All data given non-zero weights are plotted in Fig. 6. The infinite-dilution value of the apparent molal heat capacity, which varies as a function of temperature, is con- sidered further in Section 3.7. 3.5 Degree of Dissociationof the Hydrogensulfate Ion Studies of the degree of dissociation, a, are listed in Table 5. Considering the variety of methods used, agreement between these different datasets is quite good. As noted by Chen and Irish,84 the derivation of ionic concentrations from Raman peak intensities is somewhat ambiguous because some of the -_ 80 -100 I-' I r 1 I I I { 100 .- 60 - symbol 0 -8o - N 1 2 TIK 298 298 Y.-I- 40 - 460 + ' L 0 r 3 4 298 298 E" -40 x . sQA -20 c, 0 5 6 7 328 313 293 V 8 283 0.5 1.0 1.5 2.0 2.5 Jrn/mo1112kg -'I2 Fig. 6 Apparent molal heat capacity of H,SO, at different tem- peratures (see key), plotted against [m(H,S0,)]1'2.Note the use of three scales, with arrows indicating the datasets to which they apply. (-) Fitted model (Section 5). Key: symbols are related to the dataset numbers (N) in Table 4 and the temperature (T)of measure-ment. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 peaks overlap. At 298.15 K, a decreases steeply from 100% at infinite dilution to about 20% at 0.4-0.5 mol kg- ',thereafter rising to a broad peak of ca. 30% from 2 to 6 mol kg-' and then decreasing slowly. Speciations determined from NMR and Raman spectral data are least reliable for dilute solutions because of a lack of sensitivity, and two of the values of Hood and Reilly" for m(H,S04) < 2 mol kg-' have been dis- carded.However, the results of Chen and Irisha4 agree fairly well with values derived from molar volumes by Lindstrom and Wirtha6 to below 1.0 mol kg-I. Chen and Irisha4 discuss several other studies of a and note that values calculated from NMR shifts and partial molal volumes are dependent to some degree on the results of the earlier Raman spectral mea- surements of Young et because of the assumptions made in interpreting the data (e.g. an assumed relation between the chemical shifts of aqueous Hf and HSO, in NMR studies). At temperatures other than 298.15 K there are only the data of Young et ~1.'~One point at 273.15 K (0.504 mol kg-') was discarded, and the remaining measurements at 273.15 and 323.15 K assigned relative weights of 0.3.All data retained at 298.15 K were given relative weights of 1.0. Speciations of sulfuric acid over a range of molality and temperature, derived from fits of the Pitzer model to available thermodynamic data, have been plotted by Holmes and Me~mer.~~In the case of the 298.15 K parametrisation of Harvie et al." the model-generated a is consistent with the data listed in Table 5 to about 3 mol kg-',13 but at higher molalities predicts a degree of dissociation that is lower than the direct experimental values. Wirth" has represented data for H,SO, up to 2.89 mol kg-' at 298.15 K with a much simpler thermodynamic model than used here. However, in that study the values of a were taken from the determinations of Young et ~1.'~and Lindstrom and Wirth,86 and were not predicted by Wirth's model.Note that even where the model is fitted simultaneously to activity, osmotic coefficient and thermal data the speciation is not fully constrained, particularly its variation with tem-perature; thus it is worthwhile including the measurements referred to in Table 5, though a low weight has been assigned to the data as a whole (Section 3.8), reflecting the uncer- tainties in the measurements and their interpretation. All values of a given non-zero weights are plotted in Fig. 7. 3.6 Freezing-temperature Depression Measurements of the freezing temperatures of solutions yield the osmotic coefficient of the solution at the freezing tem- a0*410.2 0 0.5 1.0 1.5 2.0 2.5 Jrn/mol'iz kg-'I2 Fig.7 Degree of dissociation (a) of the hydrogensulfate ion, at dif- ferent temperatures, plotted against [m(H2S0,)] 'I2. (-) Fitted model (Section 5). The dataset numbers (N)of Table 5 are: (0) 4, (+) 5, (e) 3, (0) 6, (17)7, (all at 298.15 K); and (0) 6 (273.15 K), (*) 6 (323.15 K). J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 perature, via standard equations89 involving the thermal properties of pure ice and water. The osmotic coefficients can be adjusted to some other reference temperature, usually 298.15 K, using partial molal enthalpies and heat capacities. Staples' (in his Tables 18-27) lists 155 freezing-temperature measurements from 10 ~tudies,~'-~~but rejected 80 of these data.Staples suggested that the deviation of the 75 freezing- temperature points retained in his fit from osmotic coeffi- cients derived from emf measurements was due to experimental error in the freezing temperatures. RardZ6 and Rard and Platf~rd~~ came to a similar conclusion, suggesting that the precipitating solid phase could well be dilute H,SO, rather than pure ice. A comparison of 298.15 K osmotic coefficients, calculated from the freezing temperatures and molalities tabulated by Staples,' with the present model (fitted to isopiestic and emf data) showed positive systematic deviations of the order of 0.02 in 4 below 0.1 mol kg-', though there was reasonable agreement above this molality despite considerable scatter.A recalculation by using enthalpies of Wu and Young71 and heat capacities from sources listed in Table 4 yielded essen- tially the same results as obtained by Staples, and it was apparent that no conceivable error in the thermal data could account for the differences between the freezing points and other measurements. We therefore conclude that, below about 0.1 mol kg-', all the available freezing-point depres- sion data are systematically in error. No freezing-temperature data have been included in the present evaluation. 3.7 Dissociation Constant (KHso4)of the Hydrogensulfate Ion The value of KHSO4[eqn. (2)] is such that it is difficult to determine using methods employed for weaker acids, for example emf measurements."' The double charge on the sulfate ion complicates the extrapolation of activity coeffi- cients that is required, and renders this sensitive to model assumptions, especially from uncertainty in the value of the ion size parameter in the Debye-Huckel expression.58~'0' Many determinations of KHS04 have been made, and are sum- marised in a number of revie~s.''~-~~~ By the early 1960s it was clear that, at 298.15 K, KHSO4 lay between about 0.0102 and 0.0106 mol kg-'.'02 A previous application of the Pitzer model to aqueous H2S04 involved a value of 0.0105 mol kg-'.lo Evans and Monk"' have calculated KHS04 at 298.15 K Readnour and Cobble'07 based upon the enthalpy of solu- tion of Na,SO,(,, in aqueous HCl.Recently, Dickson et ~1.''~have conducted new measure- ments of the dissociation constant of HSO, in aqueous NaCl from 323 to 523 K. In their analysis, Dickson et al. included the KHSO4estimated by Pitzer et al.," enthalpies of reaction81*108and the heat capacities of HSO, and SO:-at infinite dilution as extrapolated by Hovey and Hepler8' and Hovey et al.,lo9 respectively, and other thermal data refer- enced by them. Dickson et obtained KHSO4= 0.010865 f-0.0005 mol kg-' at 298.15 K, in satisfactory agreement with other estimates considering that their analysis is prob- ably biased toward high temperatures. They also determined A,H" = -22.8 & 0.8 kJ mol-' and Arc; = -275 +_ 17 J mol-' K-l for the dissociation reaction, both at 298.15 K.The variation of the heat capacity change (Arc;) with tem- perature can be derived from eqn. (6) of Dickson et al. :lo4 Ar C; = -1962.617 71 1 + 9.486 301 486 92T -0.012 831 099 03T2 (18) [The constants in eqn. (18) are specified so as to retain close numerical agreement with the equation of Dickson et al., and do not reflect the actual accuracy to which Ar C; can be deter- mined.] The quantity Ar Ci is related to the infinite dilution values of the apparent molal heat capacities of the ions: Ar Ci = Ci(SO:-) + C;(H+) -C;(HSO,) (19) By definition Ci(H+) is equal to zero, hence: A, Ci = Ci(SO;-) -C;(HSO,) (204 EC:" -C;(HSO,) (20b) where C$ is the infinite dilution value of the apparent molal heat capacity of the acid, on the basis of complete disso- ciation, given in eqn.(9a). We have chosen to include qo,and its variation with tem- perature, as unknowns in our model. However, values of Cp can also be obtained from data for other electrolytes, since the apparent molal properties of ions are additive at infinite dilution. Gardner et a!."' have done this, using integral enthalpies of solution, and tabulate C$" of H2S04 from 273 to 373 K (their 298.15 K value is -295.4 J mol-' K-'). The analysis by Hovey and Hepler" of their own heat capacity data, and using C:' (-282.3 J mol-' K-' at 298.15 K) obtained by the same method in an earlier paper by Hovey etfrom their own cell data and those of Nair and Nanc~llas,~~ al.,'09 yielded -17.8 J mol-' K-' for Ci(HS0;) at 298.15 Hamer'05 and Covington et a/.58 While the activity coeffi- cient equations used by Evans and Monk"' are simpler than the Pitzer model expressions, these differences should have least effect on the calculated (stoichiometric) value of KHSO4 at low molalities. The results of Evans and Monk also suggest that KHSO4x 0.0104-0.0105 for a comparable ion size term [in the Pitzer model it is 1.2, see eqn.(6) and (7)] and empiri- cal constant Q of 0.3-0.6 mol- dm3. A further re-analysis of emf data by Mussini et ~1.''~ yielded values of 0.01043 & 0.00020 and 0.01039 & O.OO0 18 mol kg-l. Within the quoted uncertainties, these agree with earlier estimates obtained by other methods.lo2 In the present study we have therefore retained KHSO4= 0.01050 mol kg-' at 298.15 K.Different estimates of the standard enthalpy change (Ar H"/kJ mol- ') for the dissociation reaction at 298.15 K are listed by Young and Irish1O2 and by Dickson et ul.'O4 and range from -20.5 to -23.8 kJ mol-'. The previous applica- tion of the Pitzer model" yielded -23.47 kJ mol-' which was assumed to remain constant with temperature as heat capacities were not included in that fit. A significantly smaller value of A, H = -17.33 & 0.29 kJ mol- was reported by K. Combining these two values gives Arc; for the disso- ciation reaction equal to -282.3 -(-17.8) = -264.5 J mol-' K-', which agrees well with the value of Dickson et ~1.''~(-275 f17 J mol-' K-') referred to above. In developing the model fits described below, we have adopted the temperature variation of KHSO4 as determined by Dickson et al.,lo4combined with our choice of the 298.15 K value of the dissociation constant of 0.01050 mol kg-', as stated above.These yield the following expression for KHSO4 as a function of temperature: log(KHS0,) = 562.694 86 -102.5154 h(T) -1.117033 x 10-,T2 + 0.247 753 8T -13 273.75/T (21) where the first constant was adjusted slightly to give agree- ment with our choice of KHSO4. Note that the above equa- tion, adapted from eqn. (6) of Dickson et al.lo4 yields 298.15 K values of ArH" and Arc; of -22.7554 kJ mol-' and -274.8772 J mol-K-l, respectively. 1886 3.8 Weights Relative weights (w,) assigned in Sections 3.1-3.5 above are internal to each dataset and do not take into account the differing magnitudes of experimental error typical of each kind of measurement, for example about 1.5 J mol-' K-' in the case of C: compared with only 1.3 x V for the emf measurements.The absolute weight given to an individual data point in the model fit is therefore set equal to the rela- tive weight multiplied by a characteristic weight (w,) for each type of data. Initially these characteristic weights were set so as to give contributions to the total sum of squared deviations approximately in the ratio 3 :2 :1 : 1 : 0.25 (4 :emf: Adi,H : C$:a). As the fit of the model was refined, the characteristic weights for each type of property were recalculated as : where y and f are observed and fitted quantities, respectively, and N is the number of points (for which wr # 0) in each dataset.Some minor adjustments to w, were later made to reflect the consistency of the individual thermodynamic properties with the overall data set, reducing the character- istic weight given to degree of dissociation data (by 50%)and increasing that for AdilH (by 25%). The characteristic weights, w, ,finally assigned to each dataset are as follows: 4, 4.3 x lo4; emf, 4.4 x lo6; AdilH,5.8 x C$,3.7 x Q, 60. The standard errors equivalent to these w, values are =~(4) 0.0015, o(E)= 1.5 x lop4 v, o(AdilH) = 13.13 J mol-', o(C$)= 1.64J K-' mol-' and o(a)= 0.041. 4. Method The sets of parameters B',o,),pi:), C::),C::),a,,and Q,, must be determined as functions of temperature for the two ionic interactions H+-SOi-and H+-HSO,, so as to minimise the total weighted sum of squared deviations for the five mea- sured properties 4, emf, Adi,H, C! and a.This was done using a generalised non-linear least-squares fitting routine (E04FDF' "), first obtaining estimates of the parameters at 298.15 K from 4 and emf data, then extending the fit to 298.15 K thermal data and finally to emf and C! measure-ments at other temperatures. The use of different temperature functionalities for the parameters was explored. Sulfate-hydrogensulfate speciation is, of course, not known a priori and is calculated for every data point for each suc- cessive set of parameter estimates.This was done by deter- mining the zero of the following function/, which describes the distribution of the total molality of hydrogen ion [=2m(H2S04)] between free Hf [m(H+)] and HSO, [m(HSO,)] : +' = m(H+)m(SO~-)T/[K;l;so4+ m(H+)] -m(HS0,) (23) where ~Z(SO,)~is the total sulfate molality [=m(H,SO,)] and K&,, is the stoichiometric dissociation product of the hydro- gensulfate ion (and not the thermodynamic constant). Because the incremental change in parameter estimates may be very small from one cycle of the calculation to the next, and the fact that enthalpies of dilution, and especially heat capacities, are calculated as numerical differentials [eqn. (8) and (9)] it was necessary to determine speciation to full machine precision (typically one part in in the fitting program.5. Results A general equation for the temperature variation of model parameter P (where P = Br:), /3::), C(pa),C',',))that was found to J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 be satisfactory is: P = q1 + (T -~,){io-~q,+ (T -T,) x [10-3q3/2 + (T -T,)10-3q4/6]) (24) where T is in K and the reference temperature, T,, is 328.15 K. The equation for the temperature variation of E"/V, the standard emf of cells I, IT and IV, is: E" = rl + 10r2(1/T-l/To)+ 10-3r3 x [T In(T)-ToIn(To)] (25) where To = 298.15 K. The equation used to represent the infinite dilution value of the apparent molal heat capacity, C$"/J mol-I K-', is: CF = ~1 + (T -TO)S~+ O.I(T -T0)'~3 (26) All parameters determined in the model fit are listed in Table 6, together with their temperature ranges of validity and the standard errors for the parameters for eqn.(25) and (26) above. Assigned values of the coefficients acaand mcaare also ~,~~given in Table 6: note that ~t varies with temperature. ~ For convenience, 298.15 K values of all parameters are listed in Table 7. Fig. 8-12 show the residuals for each dataset, and indicate the deviations of the fit from the weighted data. Note that it was not necessary to include the mixing parameters 6HSo4, so4 and ~,b~~~~,so4, to represent accurately the experi- mental data. While some 'cycling' of the residuals is evident in the fit to the emf (Fig. 9) and AdilHdata below 1.4 mol kg-' (Fig.lo), most observations are fitted substantially within experimental error. Note that the patterns of residuals for the emf data are essentially random for cell I at 328.15 K and for cells I1 and I11 at 298.15 K, and systematic trends for other temperatures and cells are G2.5 x V. As a test, to determine whether activity data and thermal properties not taken at 298.15 K were biasing the results at 298.15 K, a separate fit to osmotic and emf data at 298.15 K only was carried out. There was no decrease in the sum of squared deviations, confirming the quality of the fit. For measurements of a at 273.15 and 323.15 KI4 we note a trend towards positive deviations at low mola- lities. However, the a data are too few in number to deter- mine whether this reflects real errors in the fitted model, and 298.15 K data from the same source show a similar pattern of residuals. The residuals of the osmotic coefficients are random for the direct vapour pressure measurements [Fig.8(c)], as they are for isopiestic data at 273.15 and 323.15 K [Fig. 8(d)]. There are slightly different trends in the residuals for data at 298.15 K depending on whether NaCl [Fig. 8(a)] or KC1 [Fig. 8(b)] was used as standard. However, taken together, the results are essentially random above 0.4 mol kg-[Fig. 8(e)],which confirms the accuracy of our model fit. The standard devi- ation of A4 (unweighted) at 298.15 K is 0.0024. The standard potentials of cells I, I1 and IV at 298.15 K (Table 7) agree with results obtained earlier by Pitzer et u1.l' to within 0.62 mV (cell I) and 0.063 mV (cells I1 and IV).Gardner et aZ.' ' have calculated mean activity coefficients of H2S04 from the emf data of Covington et and third-law potentials for cell I1 from 0.1 to 4.0 mol kg-' and 273.15 to 328.15 K. These mean activity coefficients agree with values calculated using the present model to within 0.001, at 298.15 K, 0.006 at 273.15 K and 0.002 at 328.15 K. Rard et uZ.,~ and later Rard,26 have evaluated osmotic coef- ficient data at 298.15 K independently of other activity mea- surements, for the use of aqueous H2S04 as an isopiestic standard. Osmotic coefficients generated by the present model agree with the most recent values presented by Rard,26 Table 6 Fitted model parameters for aqueous H2S04 (4 P'O' P'1' C'O' ti, HSO4 H, HSO4 H, HS04 91 0.227 784 933 0.372 293 409 -0.002 800 325 20 -0.025 92 -3.786 677 18 1.50 0.216200279 18.172 894 6 q3 -0.124 645 729 0.207 494 846 0.010 1500824 0.382 383 535 94 -0.002 357 478 06 0.004 485 264 92 O.OO0 208 682 230 0.002 5 WP PCO, fl"' C'O' C'" W H,so4 If, SO4 HI so4 H, so4 c 91 0.034 892 535 1 -1.06641231 0.007 647 789 5 1 0.0 0 r 92 4.972 078 03 -74.684 042 9 -0.314698817 -0.176 776 695 W00.3 17 555 182 -2.262 689 44 -0.021 192 652 5 -0.73 1035 345 q3 0.008 225 803 4 1 -0.035 296 854 7 -0.OOO 586 708 222 0.0Y4 cell I" standard error cell 11" standard error cell III"." standard error cell IV" standard error rl 1.690 998 3.68 x 10-5 0.612 357 3 3.3 x 10-5 1.077553 6.99 x 0.3527679 6.3 x 10-5 r2 8.883 153 0.18 -7.273 884 0.20 6.844 643 9 0.38 r3 0.202 140 7 2.97 x 10-3 -0.248 245 9 3.3 x 10-3 -0.26634636 6.4 x q0* standard error S1 -286.175 2.6 s2 3.677 433 0.16 s3 -0.471 039 1 0.069 and oH,s04Both oll,llsoI are equal to 2.5 mol-'" kg1I2; the coefficient c1f1,1,so4 is set to 2.0 mol-"2 kg1'2, while alf,S04was allowed to vary with temperature ~~~according to the equation L Y ~=~2 -, 1842.843(1/T -1/298.15).As is the case for earlier the mixture parameters OHS04,S04 and IC/HS04,S04,H were found not to be needed and are set to zero. " Parameters are valid for the temperature ranges: cell I, 278.15-328.15 K; cell 11, 278.15-328.15 K; cell 111, 298.15 K; cell IV, 273.15-323.15 K; C:', 283.15-328.15 K." At 298.15 K, E" (cell I) -En (cell 11) -E" (cell 111) = 1.690998 -0.6123573 -1.077553 = 0.001088 V, whereas it should be exactly zero if all of the cell potentials were internally consistent. This suggests that the standard potential of the lead sulfate/lead dioxide electrode is known only to within about 1 mV. 1888 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 7 Model parameters and standard potentials at 298.15 K" H. so4 -0.008 386 089 24 0 NaCl (298 k)0.295 903 322 b(0) 0.400 482 398 p(1)H. so4 0.314 734 575 symbol N H, so4 0.010 192 247 4 :+j +I-0.005 657 866 56 C(0) H. so4 -0.323 662 605 92-0.409 364 246 c(1) 2.0 'H, so4 2.0 x3 2.5 OH, so4 2.5 Q5 286.2 A, 0.391 475 1.69100 A,/(RT)b 0.801 844 Z 14I I A 220.612357 AcIRb 3.836 018 1.07755 t , , Idi-0.352 768 -0.0081, I I , , I + I " Units are as follows: BE) and flLt)/kg mol-'; CL:) and C:',)/kg' mol-'; A,, z,, and o,,/kgl/z mol-'I2; C$"/J K-' mol-'; E"/V; KHsoJmol kg -'.Debye-Huckel limiting slopes were calculated from the polynomial function equation given in Appendix 11. 0.004 KCI (298K) and by Rard and Platf~rd,,~ to within +0.0032/-0.0015 for svmbol N $0 04m(H2S04) < 6.0 mol kg-'. 06Comparisons are also made with thermal data. Wu and d Young's7 tabulated Lo deviate from model calculated values 0 15 by about 300 J mol-' above 0.5 mol kg-'. However, because -0.004 I 0 16 this difference is roughly constant, and arises from differences 0 between the model and Wu and Young's graphical integra- 8 0 tion of aL*/i?Jm below 0.5 mol kg-', there is little effect on the calculated differentials of activity with respect to tem- -0.008 c1 perature.The value of C$" at 298.15 K (-286.2 f2.6 J mol-' K-') determined in the fit, is consistent with -295.4 J mol-' K-' estimated by Gardner et a!.'" from integral enthalpy of solution data for Na,SO,, and -282.3 J mol-' K-adopted by Hovey and Hepler.*O At temperatures other than 298.15 K, C$O calculated from eqn. (26) agrees with the values of Gardner et d.'" to within 33 J mol-' K-' from 283.15 to 323.15 K. Reardon and Beckie12 have also fitted the Pitzer model to thermodynamic data for aqueous H,SO, (excluding C$ and using a much smaller database) over a similar range of tem-perature and molality to that used here.A comparison of their calculated values with the data yields sums of squared deviations (measured minus calculated) that exceed those obtained using the present model by factors of 5 and 2.3 for # and Adi,H, respectively. Thus, our model better represents the properties of aqueous H2S04 by a large margin, although it is more complex than that used by Reardon and Beckie. We also believe that our model equation represents the osmotic coefficient with sufficient accuracy for it to be used as an iso- piestic standard over its range 0-6.1 mol kg-' and 273.15-328.15 K. Values of activities, degrees of dissociation and thermal 0.006properties of aqueous H2S04, calculated with the present model, are listed in Tables 8-10 for 273.15, 298.15 and 323.15 K. At 298.15 K we have included calculated relative partial 0.002+ molal enthalpies [aH,O), E(H,SO,)/J mol-'1 and heat a" 40capacities [J(H,O), J(H,SO,)/J mol-' K-'1.Values of E(H2S04) calculated by the present model average about -0.002 1.25% greater than those listed in Table 5 of Wu and with cyclic deviations between the two studies of .** . tY~ung,~' the same order below 1 mol kg-', corresponding to those in -0.006 *-I 1 ' I ' '* ' I 17 the residual plot in Fig. 10. However, our model equations represent the input values of AdilH to within about 50 J mol-' in this region, which is 0.2-0.4% of C(H,SO,).There Fig. 8 Deviations between the measured and fitted stoichiometric is no systematic difference between the relative partial molal osmotic coefficients (4st)of aqueous H,SO, at 298.15 K (a)-(c), enthalpies of water calculated in the two studies, though the 273.15 and 323.14 K (4,and all temperatures (e).(a) Aqueous NaCl maximum deviations from Wu and Young ( & 9% between 0.2 used as isopiestic standard (except NaOH for N = 1); (b) aqueous KCl used as isopiestic standard ; (c)direct vapour pressure measure- and 1.5 mol kg-' H2S04) are greater on a percentage basis ment; (d) see Table 1 for isopiestic standards. (e) All data. Keys for than for L(H,SO,). We have not tabulated apparent or rela- (u)-(d): symbols are related to the dataset numbers N in Table 1.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1889 1’ I I 1 I I I I I I-’ (a1 0.4 0 (b)-!$00.2 1 icell I 0 symbol TIK cell I> EGC X 278 .-?*X --9-+ symbol TIK 283 V 293 0 H 328 0,O 298-0.2 0 D 308 -0.41, I I , I Ii -0.4 0 1.o 2.0-0.210 1.o 2.0 0.4 I I 1 1 I (cI 0.4 X cell II cell II (298 K) symbol N0.2 3 o x + svmbol TIK 0.2 > 0 278 05 E $0 + 288 0 *6 +70 308 080 U I 318 X 328 -0.2 1 v 10-0.2 O+ ‘s 11 X I --1-0.4 -0.4 tl , , , ,-J 0 1.o 2.0 0 1.o 2.0 1 I 1 1-0.4-‘ (f) symbol TIK ’ cell III (298 K) I3 cell IV 0 285symbol N 12 0 298 273 K V 310 0 323-0.2 --0.4 -1 1 I I -0.4 0 1.o 2.0 0.05 0.01 0.15 Jm/mol’ kg-’I2 Fig.9 Deviations between measured and fitted emfs of cells I-IV. Keys give cell type, with (rounded) temperature of measurement (T)and/or dataset number (N) in Table 2. (a),(b) N = 1 for all data, except filled circles (N = 2); (c) N = 5 for all data; (f) N = 13 for all data. 298 K symbol N 75 .A I I I I I,‘ 01 *2 +3-50 04I-L ’ 2525 :o symbol N TIK X 5 328 d -25 I-50 I I I I I I 0 1.o 2.0 0 1.o 2.0 Jm/mol’’2 kg-’I2 Jmlmol ’l2 kg-’l2 Fig. 10 Deviations between measured and fitted differential enth- Fig. 11 Deviations between measured and fitted apparent molal alpies of dilution at 298.15 K, plotted against the square root of the heat capacities in H,SO,. (a)298.15 K; (b) other temperatures. Keys initial molality, m,.The dataset numbers (N) of Table 3 are: (*) 1, give the (rounded) temperature of measurement (T) and dataset 3, (44.number (N) in Table 4. (0)2, (0) d 0 318 1890 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 298 K tive partial molal heat capacities at 273.15 or 323.15 K in symbol N Tables 9 and 10, as the model is unlikely to be as well con- 03 strained with respect to these quantities at the extremes of the 04 temperature range over which it has been applied. -0.04 1 +5 07 6. Conclusions I I I I I A generalised, extended formulation of the molality-based Pitzer thermodynamic model has been presented here and applied to measured properties of aqueous H,SO,, yielding a self-consistent representation of activities, apparent molal enthalpies and heat capacities of aqueous H,SO, from 0 to 6.1 mol kg-l, for the temperature range 273.15-328.15 K.The extension to the model provides a more flexible frame- work for calculating the properties of both single- and multi- -0.1 CI I I I 1 I component electrolyte solutions at low and moderate0 1.o 2.0 molalities, and its application to sulfuric acid will lead to ,/m/rnol1/’ kg-improved calculations of thermodynamic properties of acidic Fig. 12 Deviations between measured and fitted degrees of disso-sulfate mixtures. ciation of the hydrogensulfate ion. (a) 298.15 K; (b) other tem- peratures. Keys give the (rounded) temperature of measurement (T) The work of S.L.C. was supported by a grant from the Lever- and dataset number (N)in Table 5.hulme Trust. The contributions of J.A.R. and K.S.P. were Table 8 Thermodynamic properties of aqueous H2S04 at 298.15 K (0.0001) 0.9500 0.98 13 0.982 86 497.3 -267.9 -7.5689 x lo-, 917.5 -2.895 x lo-’ 34.35 (0.0002) 0.9253 0.9712 0.967 55 890.9 -252.9 -2.6763 x 1633.6 -1.006 x 10-4 61.16 (0.0005) 0.8737 0.9493 0.928 18 1874.6 -217.2 -1.3211 x lo-’ 3341.3 -4.625 x 120.3 (0.0010) 0.8152 0.9236 0.876 21 3 150.9 -174.8 -4.0382 x lo-* 5392.5 -1.270 x lop3 181.9 (0.0020) 0.7384 0.8890 0.800 41 4 997.0 -121.2 -0.1 1087 8 074.0 -2.956 x 247.0 0.0050 0.6146 0.8325 0.667 58 8 224.4 -47.4 -0.34858 12 094 -6.719 x 313.4 0.0100 0.5145 0.7867 0.555 11 10 978 -2.11 -0.7209 1 14 980 -9.873 x lo-’ 338.9 0.0200 0.41 89 0.7440 0.447 04 13 679 27.9 -1.3468 17417 -1,160 x 346.3 0.0500 0.3098 0.6987 0.328 95 16 807 46.4 -2.7538 19 864 -9.222 x lo-’ 342.8 0.1000 0.2436 0.6759 0.265 25 18 734 50.6 -4.5207 21 243 -5.837 x 340.0 0.2000 0.1916 0.6647 0.224 99 20 299 52.5 -7.2577 22 313 -1.139 x lo-’ 341.8 0.3000 0.1672 0.6647 0.212 26 21 059 54.1 -9.3826 22 795 -2.741 x lo-’ 345.3 0.4OoO 0.1525 0.6685 0.208 14 21 530 55.8 -11.088 23 069 -5.021 x lo-’ 349.0 0.5000 0.1425 0.6744 0.207 8 1 21 857 57.6 -12.528 23 248 -7.890 x low2 352.5 0.6000 0.1353 0.68 16 0.209 47 22 100 59.3 -13.856 23 382 -0.1135 356.0 0.7000 0.1300 0.6898 0.212 26 22 292 61.1 -15.216 23 498 -0.1543 359.5 0.8000 0.1259 0.6990 0.2 15 74 22 450 62.8 -16.740 23 611 -0.2016 363.0 0.9000 0.1228 0.7088 0.219 65 22 585 64.6 -18.546 23 729 -0.2552 366.5 1.m 0.1204 0.7194 0.223 86 22 706 66.3 -20.736 23 857 -0.3150 370.0 1.2000 0.1173 0.7420 0.232 78 22 922 69.8 -26.602 24 152 -0.4507 376.8 1.4000 0.1157 0.7663 0.242 08 23 122 73.3 -34.878 24 505 -0.6012 383.3 1.6000 0.1153 0.7920 0.251 57 23 320 76.6 -45.928 24913 -0.7548 389.0 1.8000 0.1157 0.8188 0.261 13 23 522 79.8 -59.993 25 372 -0.8963 393.6 2.m 0.1169 0.8464 0.270 66 23 732 82.7 -77.234 25 875 -1.0081 396.9 2.2000 0.1186 0.8746 0.280 07 23 951 85.3 -97.777 26 418 -1.0723 398.6 2.4000 0.1209 0.9034 0.289 23 24 180 87.6 -121.74 26 996 -1.0732 398.6 2.6000 0.1237 0.9327 0.297 96 24 420 89.5 -149.28 27 607 -1.001 1 397.0 2.8000 0.1269 0.9624 0.306 09 24 671 90.9 -180.56 28 250 -0.8549 394.0 3.m 0.1306 0.9926 0.3 13 39 24 93 1 91.9 -215.79 28 924 -0.6446 390.0 3.2000 0.1347 1.0232 0.319 64 25 203 92.5 -255.18 29 629 -0.3914 385.5 3.4000 0.1393 1.0542 0.324 66 25 485 92.8 -298.88 30 364 -0.1240 381.0 3.6000 0.1443 1.0856 0.328 26 25 777 92.8 -346.94 31 126 0.1264 377.0 3.8000 0.1498 1.1172 0.330 34 26 079 92.6 -399.28 31 912 0.3331 373.9 4.m 0.1556 1.1490 0.330 85 26 391 92.3 -455.64 32 714 0.4799 371.8 4.2000 0.1620 1.1807 0.329 80 26711 91.9 -5 15.62 33 526 0.5641 370.7 4.4000 0.1687 1.2123 0.327 27 27 039 91.6 -578.70 34 340 0.5963 370.2 4.6000 0.1758 1.2436 0.323 37 27 374 91.3 -644.30 35 149 0.5977 370.2 4.8000 0.1833 1.2745 0.318 25 27 715 91.0 -711.81 35 947 0.5966 370.2 5.oooo 0.1912 1.3047 0.31209 280 60 90.7 -780.66 36 727 0.624 1 369.9 5.2000 0.1994 1.3343 0.305 05 284 08 90.4 -850.37 37 485 0.7115 369.0 5.4000 0.2080 1.3630 0.297 30 287 58 90.1 -920.51 38 220 0.8878 367.1 5.6000 0.2168 1.3908 0.289 00 291 08 89.7 -990.79 38 929 1.179 364.2 5.8000 0.2259 1.4177 0.280 3 1 294 59 89.2 -1061.0 39 613 1.608 360.0 6.oooO 0.2352 1.4437 0.271 35 298 09 88.6 -1131.0 40 272 2.194 354.5 Values in parentheses are for molalities below the lower limit for which activity data (emf, #) exist.Thermal properties L*, qH20) and QH,S04) are given in J mol-’, and C:, J(H,O) and I(H,SO,) in J mol-’ K-I. * The use of a in conjunction with y* and KHSO4allows the species activity coeficients yH ,yHS04 and ysoI to be recovered. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 9 Thermodynamic properties of aqueous H,SO, at 273.15 K molality/mol kg-Yi A* za L4 QH,O) i-W,SO,) (0.0001) 0.9569 0.9845 0.991 36 202.1 -2.8804 x 10-4 362.0 (0.0002) 0.9373 0.9770 0.983 45 351.3 -1.0150 x 633.0 (0.0005) 0.8971 0.9609 0.962 23 729.6 -5.1713 x lo-' 1303.7 (0.0010) 0.85 15 0.9420 0.932 33 1244 -1.6748 x lo-, 2 173.3 (0.0020) 0.7895 0.9 1 54 0.884 81 2 044 -5.0203 x lo-, 3 437.3 0.0050 0.6827 0.8682 0.789 95 3 621 -0.183 13 5 653.8 0.0100 0.5888 0.8260 0.697 32 5 158 -0.42698 7 528.4 0.0200 0.4923 0.7828 0.59641 6 852 -0.89264 9 330.0 0.0500 0.3746 0.7324 0.469 96 9 057 -2.0532 11 337 0.1000 0.2994 0.7042 0.392 37 10531 -3.5175 12 483 0.2000 0.2382 0.6872 0.338 22 11 740 -5.4711 13 258 0.3000 0.2089 0.6836 0.319 65 12 294 -6.5342 13 503 0.4000 0.1909 0.6847 0.313 11 12 607 -6.9415 13 570 0.5000 0.1787 0.6884 0.31209 12 800 -6.8870 13 564 0.6000 0.1698 0.6939 0.31402 12 925 -6.5613 13 532 0.7000 0.1632 0.7010 0.31764 13009 -6.1489 13 496 0.8000 0.1582 0.7093 0.322 26 13 068 -5.8226 13 472 0.9000 0.1544 0.7187 0.327 50 13 113 -5.7384 13 466 1.m 0.1516 0.7291.0.333 11 13 149 -6.0342 13 484 1.2000 0.1479 0.7523 0.344 89 13 212 -8.2298 13 593 1.4000 0.1464 0.7783 0.356 92 13 280 -13.194 13 803 1.6000 0.1465 0.8064 0.368 85 13 364 -21.546 14111 1.8000 0.1478 0.8362 0.380 5 1 13 468 -33.828 14512 2.m 0.1501 0.8676 0.391 73 13 596 -50.560 15OOO 2.2000 0.1533 0.9001 0.402 34 13 749 -72.270 15 573 2.4000 0.1574 0.9337 0.412 15 13 928 -99.470 16229 2.6000 0.1623 0.9683 0.420 93 14 133 -132.61 16964 2.8000 0.1680 1.0037 0.428 45 14 364 -172.00 17 773 3.m 0.1745 1.0400 0.434 49 14 620 -217.75 18 649 3.2000 0.1818 1.0771 0.438 84 14 900 -269.72 19 579 3.4000 0.1899 1.1 148 0.441 34 15 204 -327.5 1 20 551 3.6000 0.1988 1.1530 0.441 89 15 529 -390.50 21 550 3.8000 0.2085 1.1916 0.440 47 15 872 -457.88 22 561 4.m 0.2191 1.2302 0.437 10 16 232 -528.77 23 570 4.2000 0.2305 1.2687 0.431 88 16 605 -602.26 24 565 4.4000 0.2426 1.3069 0.424 95 16 989 -677.47 25 536 4.6000 0.2556 1.3445 0.416 50 17 381 -753.61 26 475 4.8000 0.2693 1.3813 0.406 72 17 779 -830.03 27 378 5.oooo 0.2838 1.4172 0.395 84 18 180 -906.20 28 241 5.2000 0.2989 1.4521 0.384 07 18 583 -981.76 29 063 5.4000 0.3 147 1.4858 0.371 61 18 986 -1056.5 29 846 5.6000 0.33 10 1.5183 0.358 68 19 387 -1130.2 30 591 5.8000 0.3480 1.5496 0.345 44 19 786 -1203.1 31 300 6.oooO 0.3656 1.5797 0.33206 20 181 -1275.2 31 979 'See footnotes to Table 8.performed under the auspices of the Offce of Basic Energy Definitions are given below. Equations for the activity coeffi- Sciences (Geosciences and Chemical Sciences, respectively) of cient of cation M, anion X and the osmotic coefficient follow : the US Department of Energy by the Lawrence Livermore National Laboratory (contract no. W-7405-ENG-48) and the Lawrence Berkeley Laboratory (contract no. DE-AC03-76SF00098). Appendix I The modified equation for excess Gibbs energy, from which expressions for activities and thermal and volumetric proper- ties are derived, is given below for a solution containing an indefinite number of cations, c, and anions, a.For terms involving neutral species (unchanged by the extension to the model) see reviews by Pitzerg or Clegg and Whitfield.13 Gex/(nwRT) = -(4A6Z/1.2)1n(1 +1.2JZ) +11mcma(2Bca +ZCTa) ca +c1mc mc*[ 2~ccP +1ma $,,*a c<c' 1 a<a'+11ma ma,[2Qaa, +1mc +aa*c] (~11) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 10 Thermodynamic properties of aqueous H2S0, at 323.15 K” (0,0001) 0.9353 0.9741 0.962 80 1245.8 -1.9224 x 2 312.9 (0.0002) 0.9003 0.9588 0.931 45 2 227.8 -6.5493 x 4 045.5 (0.0005) 0.8273 0.9259 0.857 36 4515.6 -2.9190 x lo-’ 7 756.2 (0.00 10) 0.7492 0.8900 0.770 72 7 169.7 -7.9164 x lo-* 11564 (0.0020) 0.6547 0.8462 0.660 74 10529 -0.18784 15 743 0.5192 0.7839 0.500 02 15 454 -0.48 138 20 799 0.0050 0.4213 0.7401 0.386 34 18 987 -0.86032 23 763 0.0100 0.3346 0.703 1 0.292 17 22 009 -1.4129 25 930 0.0200 0.2416 0.6673 0.204 79 25 084 -2.5539 27 919 0.0500 0.1000 0.1875 0.6506 0.165 46 26 833 -4.0528 29 083 0.1458 0.6443 0.145 96 28 248 -6.680 1 30 102 0.2000 0.3000 0.1266 0.6468 0.142 92 28 960 -8.9699 30 619 0.4000 0.1151 0.6524 0.14440 29 418 -10.998 30 944 0.5000 0.1073 0.6595 0.147 65 29 748 -12.901 31 180 0.6000 0.1018 0.6676 0.151 67 30 003 -14.840 31 376 0.7000 0.0976 0.6764 0.156 02 30 212 -16.973 31 558 0.8000 0.0944 0.6858 0.160 48 30 392 -19.445 31 741 0.9000 0.09 19 0.6956 0.164 95 30 552 -22.373 31 932 1.m 0.0900 0.7058 0.169 36 30 700 -25.854 32 135 1.2000 0.0873 0.727 1 0.177 86 30 976 -34.738 32 583 1.4000 0.0857 0.7495 0.185 83 31 241 -46.440 33 082 1.6000 0.0850 0.7727 0.193 22 31 504 -61.072 33 623 1.8000 0.0849 0.7966 0.200 01 31 771 -78.614 34 196 2.m 0.0852 0.821 1 0.206 19 32 043 -99.004 34 791 2.2000 0.0860 0.8460 0.21 1 75 32 321 -122.21 35 404 2.4000 0.087 1 0.8714 0.216 66 32 604 -148.25 36 033 2.6000 0.0886 0.8971 0.220 90 32 892 -177.25 36 676 2.8000 0.0903 0.9232 0.224 47 33 186 -209.40 37 337 3.m 0.0923 0.9495 0.227 33 33 485 -244.95 38 018 3.2000 0.0946 0.9760 0.229 49 33 790 -284.14 38 719 3.4000 0.0971 1.W26 0.230 93 34 102 -327.13 39 442 3.6000 0.0998 1.0294 0.23 1 67 34419 -373.98 40 185 3.8000 0.1027 1.0562 0.231 71 34 742 -424.58 40944 4.m 0.1059 1.083 1 0.23 1 08 35 072 -478.63 41 714 4.2000 0.1093 1.1098 0.229 82 35 406 -535.60 42 485 4.4000 0.1129 1.1364 0.227 96 35 745 -594.77 43 249 4.6000 0.1167 1.1628 0.225 56 36 088 -655.26 43 995 4.8000 0.1207 1.1890 0.222 65 36 432 -716.04 44 713 5.m 0.1248 1.2149 0.2 19 30 36 777 -776.00 45 392 5.2000 0.1292 1.2404 0.215 56 37 121 -833.99 46 024 5.4000 0.1338 1.2655 0.21 148 37 462 -888.83 46 598 5.6000 0.1385 1.2902 0.207 12 37 797 -939.40 47 109 5.8000 0.1434 1.3144 0.202 52 38 126 -984.61 47 549 6.oooO 0.1485 1.3381 0.197 72 38 446 -1023.4 47915 a See footnotes to Table 8.The function 9in eqn. (AI2) and (AI3) is given by: where F= -A+[Jz/(I + 1.2J1) + (2/1.2)1n(1 + 1.2J1)I g(x) = 2[ 1 -(1 + x)exp(-x)]/x2 (A1 13) + 1C mc ma(K + ZC32) ca and two new functions are defined for the extended model: + c 1m,m,,wee, + 1 1 mama,was.(AI5)c<c’ aia‘ h(x) = (6 -[6 + x(6 + 3x + x2)]exp(-x))/x4 (AIM)In the above equations, Z is the molality based ionic strength, and summations c < c’ and a < a‘ are over all distinguishable h’(x)= exp(-x)/2 -2/44 (AI16) pairs of cations and anions, respectively. The symbol $,,., is a Note that g’(acaJI) is equivalent to Zd[g(a,, ,/Z)]/dI, and a ternary parameter for the interaction of an anion and two corresponding relationship applies for h’(mca41).Other func- distinct cations, and similarly for $,,., . The equations also tions for the standard model involve interactions between contain the following functions : pairs of ions (i,j)of like sign: CD, = 8, + e$) (A117) qj= O”,,Z) (AI18) CD~= e, + eE(q + ieE(I) (AI19) The value of parameter Oij is obtained by fitting, whereas the unsymmetrical mixing term OE(Z) and its derivative are obtained from theory,’13 and are equal to zero where the charges on ions i and j are equal in magnitude.These mixing J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 terms are given by: QE = (zi zj/4I)[J(xij) -1/2J(xii)-1/2J(xjj)] (AI20) 0: = a;./aI = -e8p + (zizj/s~2)[xij~~(xij) -(1/2)Xii J’(X,i) -(1/2)XjjJ’(Xjj)] (AI21) and J’(Xij)= aJ(xij)/axij (A122) The molality-dependent variable xijis given by: x..IJ = ~z.~.A,I”~1 J (AI23) The function J(xij) is an integral which has been evaluated numerically.The following approximating equation is used here to obtain values of J(xij)and J’(xij):’I3 J(X~~)xij/[4 + C1x? exp(~,x:)] (AT24)= with C, = 4.581, C, = -0.7237, C, = -0.0120 and C, = 0.528. More accurate (but complicated) methods are avail- able.’ However, in the present application the alternative Chebychev polynomial representation of J(xij) and J’(xij) (used by Harvie et a!., for example”) offers no improvement over eqn. (AI24). Appendix I1 Values of the Debye-Hiickel coefficient, A,, used here (at 1 atm, and different temperatures) are those determined by Archer and Wang,I6 calculated using a program supplied by Archer.’ l4 The Debye-Hiickel constants of Archer and Wang’ were obtained from least-squares equations for the relative permittivity of water as a function of temperature, pressure and water density.Since we are interested in the Debye-Hiickel limiting law slopes as a function of tem-perature only, at a single pressure of 1 atm, we decided to represent A, with one equation valid over the temperature range 234.15-373.15 K. We use a polynomial in Chebychev series form involving the normalised variable x [x = (2X -X,,, -Xmin)/(Xmax-Xmin)],where X is the temperature (in K). X,,, (373.15 K) and Xmin(234.15K) are the upper and lower limits of the fit, respectively. The polynomial, with 19 (N + 1) coefficients, is given by: A, = oh0 To(x)+ UITl(X) U2 T2(X) + u3 T3(x).. . + UN TN(X) (A111) where To(x)= 1, T,jx) = x, and for n 2 2: T,(x) = 2xT,-,(x) -T,-,(x) (AII2) The polynomial coefficients, ai,are listed in Table 11.Fitted values of A a ree with those calculated from the orig- @ .ginal program to within 0.5 x lo-’ below 250 K, and to within 0.95 x lop8 mol-1/2 kg1l2 at higher temperatures. Table 11 Chebychev polynomical coefficients for A,, 234.15 6 T/ K d 373.15 0.797 256 081 240 a,, -0.388 189392385 x 0.573 389 669 896 x 10-a,, 0.164245088592 x 0.977 632 177 788 x a,, -0.686031 972 567 x 0.489973732417 x lo-’ a13 0.283455806377 x -0.313 151 784342 x lo-’ a14 -0.115641433004 x 0.179 145 971 002 x lo-’ a,, 0.461489672579 x -0.920584241844 x aL6 -0.177069754948 x 0.443862726879 x al, 0.612464488231 x lo-’ -0.203 661 129991 x a18 -0.175689013085 x lo-’ 0.900924 147 948 x This level of precision was chosen to ensure accuracy in the calculated second differential of A, with respect to tem-perature, because it appears in the expression for the appar- ent molal heat capacity, obtained here by numerical differentiation (Section 2).The very low minimum tem-perature of fit was needed in order to use the same poly- nomial in a second study of the H2S0,-H20 system over a more extended range of temperature and comp~sition.~~ For program validation, the polynomial yields the follow- ing values: 273.15 K, A, = 0.376421452 mol-’/* kg’I2; 298.15 K, A, = 0.391 475 238 mol-1/2 kg’l2. 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ISSN:0956-5000    

DOI:10.1039/FT9949001875

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1895-1898 Pitzer Model Parameters for Sparingly Soluble Salts from Solubility Measurements:Thallium(i) Chloride in Aqueous Solutions of Ammonium Chloride, Rubidium Chloride and Caesium Chloride at 298.15 K Kean H. Khoo,* K. Rodney Fernando and Lee-Hoon Lim Department of Chemistry, Universiti Brunei Darussalarn, Bandar Seri Bega wan 3186, Brunei Da russalam -~ Measurements are reported of the solubility of thallium(t) chloride in aqueous solutions of ammonium chloride, rubidium chloride and caesium chloride at 298.15 K at concentrations of added salt up to 4.0 mol kg-'. Pitzer model parameters for TICI, B(O) and p('), were evaluated assuming C4= 0. These parameters can account satis- factorily for the ionic interactions between TI+ and CI- without the need to assume ion association explicitly, thus simplifying the interpretation of the experimental data. Although the parameter a = 2, which is customarily used for a broad class of electrolytes, is found to be satisfactory for TICI, larger values of a generally gave better fits to the experimental data.Mixture parameters, 8 and $, relevant to the various mixed electrolytes are also reported. There is a general lack of information in the literature regard- ing Pitzer model parameters for sparingly soluble This situation is ascribed to the low solubility of these salts, which precludes evaluation of these parameters in the usual manner using pure solution activity or osmotic coefficient data. Problems arise when it is necessary to estimate the activities of such salts in concentrated brines.Pitzer model parameters for such salts can be obtained indirectly by using activity or osmotic coefficient data for simple mixtures of these salts, the simplest being a three-ion mixture. A conve-nient method for obtaining the required activity data is to study the solubility of the salt in question in the presence of another salt with a common ion, as is the case in the present study where we report the solubility of thallium(1) chloride in aqueous solutions of ammonium chloride, rubidium chloride and caesium chloride at concentrations of added salt up to 4.0 mol kg- ' at 298.15 K. Thallium (I) chloride has a number of interesting applica- tions for which some knowledge of its solubility and activity in aqueous solutions would be desirable.For example, in view of its similarity in ionic radii, Tl+ may substitute for K+ in biological cells and this makes it poisonous to man; but because it is a d"s2 ion, it can be conveniently monitored by spectroscopic, NMR or polarographic techniques so that it can serve as a useful probe for the role of K+ in biological Thallium(1) chloride also finds applications in IR spectrometry. Experimental Thallium(1) chloride was prepared by the slow and simulta- neous addition of stoichiometric amounts of 0.05 mol dm-3 thallium(1) sulfate (BDH, AnalaR) and 0.1 mol dm-' hydro-chloric acid (Ajax, UnivaR) in ca. 2 dm3 of rapidly stirred water at ca.333 K. These solutions were filtered prior to mixing to remove insoluble material. The coarse crystalline product was collected on a glass frit, washed several times with doubly distilled water and dried at 383 K. It was stored in the dark. Analysis for thallium gave the purity of the product as (99.9 & 0.2)Y0. As bef~re,'-~,~,'' analysis for thal- lium was by iodate titration in the presence of concentrated hydrochloric acid, making use of the overall reaction : 10,-+ 2Tl+ + 6H+ = I+ + 2T13++ 3H20 I+ + c1-= ICl The indicator was carbon tetrachloride, and the end-point was visually detected by the disappearance of the purple colouration (due to iodine liberated in an intermediate reaction) in the organic phase. In this work, a 10 cm' burette, calibrated and readable to 0.01 cm3, was used and the stan- dard iodate solutions were prepared so as to deliver titres close to 10 cm3.Potassium iodate (Ajax, UnivaR, >99.9%) was twice recrystallized from water. Ammonium chloride (BDH, AnalaR), rubidium chloride (Aldrich, 99.9 YO)and caesium chloride (Ajax, UnivaR, 99.9-100.1 YO)were used without further purification. Ammonium chloride solutions were prepared directly from the dried salt while the other salt solutions were prepared by weight from stock solutions which were analysed gravimetrically for chloride. Calibrated glassware was used and corrections were made for air buoy- ancy. The apparatus and further experimental details are given el~ewhere.'-~ Experiments were performed in subdued light as thallium(1) chloride is light-sensitive.However, expo- sure to light in the course of the experiments, which turned the outer layers of the salt purple, did not have a deleterious effect on the solubility measurements. Results and Discussion Table 1 lists the solublity of thallium(1) chloride in aqueous solutions of the added chlorides in the concentration range 0.005-4.0mol kg-' at 298.15 K. The solid phase in equi- librium with the saturated solutions in all cases is anhydrous TlCl and there is no evidence of mixed solid phases. As expected from the common-ion effect, the solubility decreases with increasing added chloride concentration (Fig. 1). The solubility passes through a minimum and then increases as the concentration of the added chloride increases further.This is consistent with association of ions into pairs, and possibly triplets, at higher added chloride concentrations : Tl+(aq)+ Cl-(aq)eTlCl(aq) (1) TlCl(aq) + Cl-(aq) sTlCl,-(aq) (2) Consideration of eqn. (1) and (2) complicates the analysis of the solubility data. For example, in determining the ionic strength of the solutions, the fraction of the ions forming the neutral molecule would have to be calculated. Assumptions would also have to be made about its activity coefficient.'2 1896 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Solubility and activity coefficient of thallium(1) chloride in the system TlCl (m,)-MCl (m,)-H,O at 298.15 K (solid phase TlC1) n~,/lO-~mol kg-' ?TIC1 Yc1 ?TI + mdmol kg-' NH,Cl RbCl CsCl NH,Cl RbCl CsCl NH,C1 RbCl CsCl NH,Cl RbCl CsCl 0 16.25 16.25 16.25 0.844 0.844 0.844 0.005 14.10 14.11 14.12 0.836 0.837 0.836 0.870 0.869 0.866 0.802 0.806 0.807 0.01 12.34 12.33 12.34 0.826 0.828 0.828 0.862 0.861 0.857 0.79 1 0.797 0.799 0.02 9.700 9.755 9.766 0.808 0.807 0.805 0.846 0.844 0.840 0.771 0.771 0.772 0.03 8.013 8.027 8.041 0.786 0.786 0.786 0.832 0.829 0.824 0.742 0.746 0.749 0.04 6.807 6.823 6.857 0.768 0.769 0.767 0.818 0.8 15 0.809 0.72 1 0.725 0.726 0.05 5.955 5.987 6.025 0.75 1 0.751 0.748 0.807 0.803 0.796 0.699 0.701 0.702 0.06 5.362 5.355 5.386 0.732 0.734 0.732 0.796 0.792 0.785 0.674 0.681 0.683 0.07 4.858 4.858 4.904 0.719 0.721 0.717 0.787 0.783 0.774 0.657 0.663 0.664 0.08 4.457 4.477 4.532 0.707 0.707 0.702 0.778 0.774 0.764 0.642 0.645 0.645 0.09 4.134 4.158 4.238 0.695 0.694 0.688 0.77 1 0.766 0.755 0.627 0.630 0.626 0.10 3.896 3.906 3.963 0.682 0.682 0.677 0.763 0.758 0.747 0.608 0.614 0.613 0.20 2.646 2.658 2.72 1 0.592 0.592 0.585 0.71 3 0.705 0.689 0.49 1 0.497 0.497 0.30 2.177 2.200 2.263 0.535 0.533 0.525 0.683 0.673 0.652 0.418 0.422 0.423 0.40 1.942 1.980 2.041 0.49 1 0.487 0.480 0.662 0.650 0.625 0.364 0.365 0.368 0.50 1.806 1.847 1.916 0.455 0.45 1 0.443 0.646 0.632 0.605 0.321 0.322 0.326 0.60 1.719 1.769 1.838 0.426 0.42 1 0.413 0.633 0.618 0.588 0.287 0.287 0.290 0.70 1.659 1.724 1.793 0.402 0.395 0.387 0.623 0.607 0.575 0.259 0.257 0.261 0.80 1.628 1.693 1.764 0.380 0.373 0.365 0.614 0.597 0.563 0.234 0.233 0.237 0.90 1.602 1.677 1.754 0.361 0.353 0.345 0.607 0.589 0.553 0.215 0.212 0.216 1 .oo 1.585 1.665 1.747 0.344 0.336 0.328 0.600 0.582 0.544 0.197 0.195 0.198 1 SO 1.619 1.732 1.836 0.278 0.269 0.262 0.580 0.558 0.514 0.133 0.131 0.133 2.0 1.727 1.861 2.008 0.266 0.225 0.2 17 0.568 0.545 0.495 0.096 0.093 0.095 2.5 1.874 2.099 2.240 0.200 0.190 0.184 0.562 0.538 0.484 0.07 1 0.067 0.070 3.0 2.058 2.337 2.528 0.174 0.164 0.158 0.559 0.535 0.477 0.054 0.050 0.052 3.5 2.194 2.648 2.870 0.156 0.143 0.137 0.558 0.535 0.474 0.044 0.038 0.040 4.0 2.548 2.985 3.278 0.136 0.126 0.120 0.559 0.537 0.472 0.033 0.029 0.030 Previous workers have assumed the activity coefficient of (M = NH,, Rb, Cs), neutral molecules in an ionic solution to be unity, but this is strictly valid only at low ionic strength.This problem can be TlCl(s)eTl+(aq)+ Cl-(aq) (3) circumvented by using the ionic interaction approach of andPitzer since thallium(r) chloride represents an intermediate type of weak electrolyte so that its interactions may be simply In Ks = In I" + 2 In yTICl (4)accounted for in terms of virial coefficients without having to refer to an ion association reaction. Thus, ignoring ion where Ks is the thermodynamic solubility product of TlC1, association, we have for the system TICl(mA)-MCl(m,)-H 20, yTIC, is its mean ionic activity coefficient and P' = mA(mA + mB)* Pitzer's Equations 5.0 We recapitulate the salient features of the Pitzer equations relevant to the present study.The reader is referred to Pitzer's original papersI3-' for the meaning of the various terms and symbols. The mean ionic activity coefficient of 4.5 thallium(1) chloride in the system TlCl(mA)-MCl(mB)-H20 is ch I given by the following set of equations: A0, -4.0-1 vF" cn --3.5 3.0-I I I I2.510 0.5 1.o 1.5 2.0 2 (m,/mol kg-')'/* Fig. 1 Effect of added chloride on the solubility of thallium(1) chlo- ride in aqueous solutions at 298.15 K. A, NH,Cl; 0,RbCl; 0,CsCl. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 The subscripts c and a denote cation and anion, respectively. a',",), and Cf, are single electrolyte parameters which have been extensively tabulated by Pitzer and Mayorga14 for several electrolytes, but not sparingly soluble electrolytes. It is one of the objectives of our ongoing investigation to evaluate these parameters for a sparingly soluble salt such as thallium(1) chloride.For 1 : 1 electrolytes, C4 is usually very small and it is found here and previously that C$,,, can be set equal to zero without affecting the fit of the experimental data. The parameters @TIM, @iIM pertain to the and mixtures. It is customary to set 6& equal to zero. Then @TIM and $TlMCi can be evaluated alongside @L, and /I!& by linear regression using the procedure described previou~ly.~.~ The fl('), b(') and C4 values for the added chlorides required in the calculations are taken from the compilation of Pitzer and MayorgaI4 (Table 2).It is also customary in the Pitzer for- malism to fix b = 1.2 for all solutes in aqueous solution and A@, the Debye-Hiickel coefficient, is numerically equal to 0.3915 in water at 298.15 K. The results of the calculations are shown in Table 3. These were obtained assuming Q = 2 in eqn. (8) and (9), which is the value used by Pitzer and other workers for 1 : 1 electrolytes. Since fl''', fl") and Ks are properties of the pure electrolyte, they should have the same values for thallium(1) chloride, irrespective of the mixture in which it is present. Table 3 shows that the values of these parameters are reasonably con- sistent for all the mixtures.To determine whether this will apply over a broad range of mixtures, measurements are being extended to other mixtures, such as asymmetrical mix- tures of the type TICI-MU,-H,O, in which higher-order electrostatic effects can be quite pronounced. The Pitzer approach is thus successful in describing the ionic inter- actions in the systems. A further indication of the quality of fit of the Pitzer model can be found in Fig. 2, in which differ- ences between experimental results and those calculated using a = 2 and the best-fit c1 values for each mixture are plotted as a function of rn:l2. Best-fit values of a pertaining to each of the mixtures are reported in Table 4,together with values of the associated parameters. Fig. 2 shows that whereas the fit of the experimental data is good up to ca.1 mol kg-' with x equal to two, the fit in each case can be extended to a higher concentration if a higher c1 value is used. In principle, a differ- ent value of c1 for each solute is permissible, but b must be the Table 2 Pitzer model single electrolyte parameters for aqueous solutions of added chlorides at 298.15 K" MCl 8''' p(') C@ validity range/mol kg -NH4C1 0.0522 0.19 18 -O.OO301 6 RbCl 0.0441 0.1483 -0.00101 5 CsCl 0.0300 0.0558 0.00038 5 a Ref. 14. Table 3 Pitzer model single electrolyte and mixture parameters for thallium(1) chloride in TlCl-MCI-H,O systems at 298.15 K assumingc&= 0 HCI -0.10 1.29 0.0462 0.0919 1.867 2 LiCl 0.01 1.29 -0.0777 0.0879 1.880 3 NaCl -0.07 1.26 0.0130 0.0657 1.873 2 KCI -0.06 1.35 0.0464 0.0774 1.865 3 NH,Cl 0.03 1.31 -0.0505 0.0659 1.880 this work RbCl 0.02 1.19 -0.0874 0.0590 1.888 this work CsCl 0.01 1.21 -0.0731 0.0597 1.888 this work 0 0.5 1.o 1.5 2.0 2.5 (rn ,/mol kg -' ) '/* Fig.2 Fit of the Pitzer equations shown as plots of differences. 6, between experimental solubilities and solubilities calculated using a = 2 and the best-fit value of a for each mixture. (a)NH,Cl: 0, a = 2; x, z = 5. (6) RbCl: 0,a = 2; +r = 6. (c) CsCl: A, r = 2; A, r = 4. same for all solutes. However, Q should preferably be kept the same for a broad class of electrolytes so that a systematic and meaningful relationship between the p(') and Pc') values can be obtained.The poorer fit of the experimental data at higher concentrations could also suggest that as the proportion of TlC1,-ion in solution increases, p(') may become less able to absorb the effects of triplet ion interaction, in which case a p(2)term could be added, as in the case of bivalent metal sulfates.15 Also a larger value of a may be more appropriate for an intermediate type of weak electrolyte such as thallium(1) chloride. One may also consider the ionic strength dependence of OTiM which arises from the higher-order limiting law, but this has been shown to be insignificant for activity coefficients under the present level of experimental precision.' Indeed, calculations show that the fit of the experimental data is often worse if the higher-order limiting law is included in the analysis of the data.',, The mean ionic activity coefficient of TIC1 in each of the mixtures studied in this work has been calculated using eqn.(4)and the Ks values in Table 3. The results are given in Table 1. A referee has commented that the apparent variation between the data for the different mixtures could be removed by considering single ionic activity coefficients. The concen- tration of TlCl in the solutions is so low that the activity coefficient of the C1- ion is almost entirely controlled by the NH4+, Rb+ and Cs' ions. Hence, it should be possible to calculate single ionic activity coefficients for the C1- ion using the Pitzer model by assuming pure MCl and combine Table 4 Best-fit values of a for thallium(1) chloride in the various added chloride systems added chloride a -/?(') p"' -8 t+b 104 K, NH,CI 5 0.31 1.45 0.0302 0.0460 1.881 RbCl 6 0.32 1.50 0.0777 0.0634 1.869 CsCl 4 0.25 1.25 0.0791 0.0331 1.876 these with the measured solubilities to obtain activity coefi- cients for the T1' ion in each of the mixtures.The ionic activ- ity coefficients for C1- and T1+ are given in Table 1 and, as expected, the results for T1+ are very close. The conclusion is that the Pitzer treatment in its simplest form is capable of describing the interactions between T1+ and C1- ions without the need to consider ion association explicitly. The large negative value of /?(I) reflects the weak electrolyte behaviour adequately and a value of a larger than two is necessary if the fit of the experimental data at higher concentrations is desired.A consideration of the higher-order limiting law is not necessary. Thermodynamically, it is a matter of choice how one wishes to describe the systems at the molecular level. As an associated electrolyte, part of the interactions between T1+ and C1- will have to be absorbed in an association constant for the reaction given by eqn. (1) and a less negative /3(') would then describe the interaction between the remaining free ions. But then the distinction between free and paired ions still has to be made together with a host of assumptions about the behaviour of the ion pair.This renders the ionic interaction approach using the simple Pitzer treatment far more attractive than the ion association approach. We gratefully acknowledge support of this research by Uni- versiti Brunei Darussalam under Grant UBD/T/RG.63. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 References 1 K. H. Khoo and K. R. Fernando, J. Solution Chem., 1991, 20, 1199. 2 K. H. Khoo and K. R. Fernando, J. Chem. SOC.,Faraday Trans., 1992,88,2193. 3 K. H. Khoo and K. R. Fernando, J. Chem. SOC., Faraday Trans., 1993,89, 1353. 4 R. J. P. Williams, Q. Rev. Chem. Soc., 1970, 24, 331. 5 J. P. Manners, K. G. Morallee and R. J. P. Williams, J. Chem. SOC.D,1970,965. 6 S. Krasne and G. Eisenman, in Membranes, a Series of Advances, ed. G. Eisenman, Marcel Dekker, New York, 1973, vol. 2, p. 227. 7 A. G. Lee, J. Chem. SOC.A, 1971,880; 2007. 8 C. H. Sueltor, in Metal Ions in Biological Systems, ed. H. Sigel, Marcel Dekker, New York, 1974, vol, 111, p. 201. 9 E. H. Garner and C. S. Swift, J. Am. Chem. Soc., 1936,58, 113. 10 J. B. Macaskill and M. H. Packhurst, Aust. J. Chem., 1964, 17, 522. 11 M. H. Panckhurst, Aust. J. Chem., 1962, 15, 194. 12 K. S. Pitzer, in Activity Coeflcients in Electrolyte Solutions, ed. K. S. Pitzer, CRC Press Inc., New York, 1992. 13 K. S. Pitzer, J. Phys. Chem., 1973,77, 268. 14 K. S. Pitzer and G. Mayorga, J. Phys. Chem., 1973,77,2300. 15 K. S. Pitzer and G. Mayorga, J. Solution Chem., 1974,3, 539. 16 K. S. Pitzer and J. J. Kim, J. Am. Chem. SOC.,1974,%, 5701. 17 K. S. Pitzer, J. Solution Chem., 1975,4, 249. Paper 3/04983C; Received 17th August, 1993

ISSN:0956-5000    

DOI:10.1039/FT9949001895

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1899-1903 Ionic Partial Molar Volumes in Non-aqueous Solvents Yizhak Marcus,? Glenn Hefter and Teck-Siong Pang School of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA6150, Australia Standard partial molar volume data at 298 K for eight tetra-alkyl or -aryl ions and 13 smaller ions in ten solvents, including water, have been statistically related to diverse ion and solvent properties. The radius cubed is the primary ion property determining the volumes for both kinds of ions, as expected. Secondary effects are due to the polarizability of the large ions and the hydrogen bonding abilities of the smaller ones. The molar volumes of the solvents, modified by their polarizabilities as well as their solubility coefficients are the main solvent proper- ties accounting for the ionic volumes.The effects of solvent compressibility are also explored, but the dipole moment and relative permittivity do not play a role. Standard partial molar volumes, I/', of electrolytes are ther- modynamic quantities of considerable importance in a variety of applications. These include the calculation of the properties of seawater, the buoyancy of marine organisms, the properties of membranes and clays, and the behaviour of non-aqueous high-energy-density batteries. Molar volumes of electrolytes are relatively straightforward to measure by tech- niques such as densitometry and dilatometry. However, in contrast to many thermodynamic quantities, their theoretical interpretation has remained difficult.Such an interpretation is customarily made in terms of a model of the ion and its solvent environment and the interactions that take place between them. The most widely employed model has been that of Frank and Wen' who postulated the molar volume of an ion to be given by: where the subscripts int, dis, cag and el refer to the intrinsic volume, the disordered volume of the solvent associated with void-space effects, the formation of a cage of structured solvent around the ion, and the electrostriction of the solvent by the electric charge of the ions, respectively. The last term in eqn. (1) is negative. The Frank-Wen model has, however, proven to be difficult to quantify exactly and it is generally regarded as, at best, semi-quantitative.It is probably of more relevance to highly structured solvents such as water, for which it was originally developed, than for typical organic solvents [in particular the cisand I/cag terms of eqn. (l)]. More recently, Marcus2 has had some success in describing ionic partial molar volumes in water using a modified Born approach with specific allowance for the existence of a hydra- tion shell around the ion. According to this approach: V'(ion) = Kydr+ Ell + Kl2 + I/str where the subscript hydr pertains to the intrinsic volume of a hydrated but as yet unelectrostricted ion, ell to the electro- striction in the hydration shell, el2 to the electrostriction in the water surrounding this shell and str to the structuring of the water around hydrophobic ions.Despite the success of this model in accounting for ionic volumes in aqueous solutions, its application to non-aqueous solvents has not been attempted. One deterrent is the limited availability of the information (derivatives of the relative per- mittivity and the refractive index with respect to pressure) t Permanent address: Department of Inorganic and Analytical Chemistry, The Hebrew University of Jerusalem, Jerusalem 9 1904, Israel. required for the extension of this model to non-aqueous sol- vents. A more fruitful approach appears to be to attempt to identify the major factors that govern vo(ion) values in non- aqueous solvents using a statistical analysis that has been employed for other thermodynamic properties of solutes in a variety of solvents.Examples are the standard molar Gibbs energy of transfer of ions, AtrGe,3 and the (logarithms of) the equilibrium constants for the distribution of organic solutes between water and immiscible solvents, log K: .4 Data Millero5 included some 20 years ago in his extensive review of partial molar volumes of electrolytes and ions the limited amount of data then available for non-aqueous solvents. No major review of this area has since been published, although Marcus6 listed ionic volumes of transfer from water into a number of nonaqueous solvents, as has Krumgalz.' A more comprehensive review of the available vodata at 298 K has recently been completed by Pang and Hefter,* and the data used for regression analysis in the present paper have been taken from that source.The splitting of the voof electrolytes into their ionic contributions has been made according to the reference electrolyte meth~d,'.'~ specifying I/'(Ph,P') -P'(BPh,-) = 2 cm3 mol-' for all solvents. This sets a precision limit to the ionic volumes of +_2 cm3 mol-', that is considered realistic, in view of the inherent difficulties in split- ting electrolyte volumes into their ionic constituents.' Included in the database (Table 1) submitted to the statistical analysis were Poof those ions in those solvents [water, meth- anol (MeOH), ethanol (EtOH), ethane-1,2-diol (EG), N-methylformamide (NMF), propylene carbonate (PC), N,N-dimethylformamide (DMF), cyanomethane (MeCN), nitro- methane (MeNO,), and dimethyl sulfoxide (DMSO)] for which the required ionic and solvent properties are known.Some other ions, such as trifluoromethylsulfonate, for which these are unknown but vo are available, could not be included. Calculations and Results The stepwise multivariable linear least-squares regression method was applied to the ionic standard molar volumes in order to relate them to ionic and solvent properties. This method selects sequentially those properties that explain most the variance of the data, the procedure ending accord- ing to provided statistical criteria. The criteria chosen were Fisher's F = 2 for variables to enter or exit. The regression was forced through zero, i.e., no constant term was allowed, since such a term would not have a physical meaning.In principle, three approaches could have been used: (i) regress J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Ionic volume, I/', database" ion H,Ob MeOH EtOH EG NMF PC DMF MeCN MeNO, DMSO ~ Li -6.4 -18.5 -18.7 -8.7 -8.2 -8.6 -9.9 -16.2 -17.6 -5.3+ Na+ -6.7 -18.2 -9.1 -3.6 -3.5 -4.5 -1.5 -16.0 -12.6 2.9 K+ 3.5 -7.4 -0.7 6.2 4.8 2.4 6.0 -5.0 -8.3 11.2 Rb + 8.6 -3.0 5.2 11.8 (8.6) 6.8 10.1 -1.7 ( -3.4) 17.2 cs + 15.8 3.4 13.1 20.0 14.6 13.4 16.6 5.0 2.0 22.6 NH,+ 12.4 3.3 8.6 10.2 11.5 4.5 F-4.3 -0.9 -13.8 6.2 (6.0) (-5.2) (-15.1) (-26.4) 13.0 -1.3 c1-23.3 13.9 13.4 24.5 25.9 17.6 7.4 1.2 13.8 11.4 Br-30.2 22.0 15.4 31.0 31.2 23.8 9.7 6.8 21.2 16.7 1-41.7 29.2 26.3 40.4 40.9 36.4 23.6 20.6 34.1 31.1 NO,-34.5 29.5 24.4 36.8 18.7 25.5 SCN -41.2 46.0 41.9 24.7 c10, -49.6 39.8 46.3 46.0 32.1 42.8 Me,N 84.1 66.9 76.1 78.9 80.9 83.0+ Et," 143.6 125.9 138.0 137.7 139.3 142.2 140.9 134.8 140.0 Pr,N+ 208.9 196.8 206.9 206.2 210.2 21 1.4 2 12.5 206.2 211.0 Bu,N 270.2 263.6 274.5 275.3 279.3 280.2 282.1 274.5 282.0+ Pe,N+ 333.7 337.5 341.2 349.7 343.9 344.0 Ph4P+ 285.8 263.1 263.3 285.9 286.5 287.9 284.0 274.4 284.9 289.3 Ph,As+ 295.2 269.0 290.9 290.4 29 1.9 283.1 294.3 BPh,-283.1 261.1 261.3 283.9 284.5 285.9 282.0 272.4 282.9 287.3 a The data are from ref.8 and individual ionic volumes are based on the assumption that P'(Ph,P+) -ao(BPh,-) = 2 cm3 mol- l, except where noted.One decimal place is retained, although the values are known only to within k2 cm3 mol-', in order to avoid rounding-off errors. Values in parenthesis are not included in the calculations, but have been subsequently predicted. These data are from ref. 2. first with respect to the ionic properties, then regress the coef- ficients obtained with respect to the solvent properties; (ii) regress first with respect to the solvents, then regress the coef- ficients obtained with respect to the ions; (iii) first form binary products of all the ion and the solvent properties, then regress with respect to these products. In practice, only the first approach could be used, since the ionic volumes are not sufficiently sensitive to the solvent properties.Only after the data have been normalized with respect to the ionic properties could the variability with respect to solvent properties be studied. The following ionic properties were offered to the program: the radius (r),the radius cubed (r3),the reciprocal of the radius (r-'), the polarizability (ai),the softness (o),12 and the hydrogen-bonding ability [aKT for cations, /?KT for anions, collectively designated by (HB)]. The intrinsic volume is proportional to r3, the ionic field to r-l, and the subscript KT refers to the Kamlet-Taft H-bond donation and electron- pair donation abilities. The values employed and their sources are shown in Table. 2. The softness and H-bonding ability are not known for the 'large' (i.e. tetraalkylammonium and the tetraphenyl) ions, hence these were treated separately from the 'small' ions (all the other ions, cations and anions together). For the small ions, because of the limited range of the radii, r is correlated (correlation coefficient R > 0.9 for the linear relationship) with r3 and with r-', hence either r or r3 must be used in the regressions.The latter was chosen, since it can be better justified physically and its correlation with r-is not as good. Also, aiis correlated (R > 0.9) with r3 (and r) and with 0, hence it should enter only when the latter are absent. For the large ions all the variables are cor- related to some extent, and it was difficult to chose orthog- onal variables. The following solvent properties were employed in the program : the molar volume (V,), isothermal compressibility (K~),solubility parameter (a), polarizability (us), relative per- mittivity (E~), dipole moment (p), the Kirkwood dipole angular correlation parameter (g), the Kamlet-Taft H-bond donation (cIKT) and electron-pair donation (PKT) abilities.Essentially all the solvents considered are nearly equally 'hard' (except cyanomethane, which is moderately soft), so that 'softness' was not inc1uded.l6 The values employed and their sources are shown in Table 3. Among the solvent properties, only V, and a, have R > 0.9, all the other variables are less well correlated and all could be employed in the regressions. For the large ions, r3 was the first parameter chosen for all solvents, and it explained >95% of the variability.This is understandable, in view of the success of the Conway-Jolicoeur plots of vous. the relative molecular weight or the number of carbon atoms in the tetraalkylammonium ions.20*2' The rest of the variance was explained by the ion polarizability, that takes care of the differences between the Table 2 Properties of the ions ion 4 l/rb r3' aid ue (HB)f ~ Li + Na + 0.69g 1.02 1.499 0.980 0.33 1.06 0.03 0.26 -1.02 -0.60 2.07 0.83 K+ 1.38 0.725 2.63 1.07 -0.58 0.85 Rb + 1.49 0.671 3.3 1 1.63 -0.53 0.49 cs + 1.70 0.588 4.9 1 2.73 -0.54 0.47 NH,+ 1.4gg 0.676 3.24 1.20 -0.60 1.00'' F-c1- 1.33 1.81 0.752 0.552 2.35 5.93 0.88 3.42 -0.66 -0.09 2.95 1.00 Br- 1.96 0.510 7.53 4.85 0.17 0.67 1- 2.20 0.455 10.65 7.5 1 0.50 0.30 NO,- 2.00 0.500 8.00 4.13 0.03 0.09 SCN - 2.13 0.469 9.66 6.74 0.85 0.33 c10,- 2.40 0.417 13.82 5.06 -0.30 0.08 Me,N+ 2.80 0.357 22.0 9.08' Et,N+ 3.37 0.297 38.3 17.70' Pr," 3.79 0.264 54.4 24.0' Bu,N + 4.13 0.242 70.4 3 1.5' Pe,N+ 4.43 0.226 86.9 39.0' Ph4P+ 4.24 0.236 76.2 45.w Ph,As+ 4.28 0.234 78.4 45.7' BPh,- 4.21 0.238 74.6 43.1' " In A; from ref.2 and references therein. In k'.'In A3. In lo-,' m3, from Y. Marcus, unpublished compilation, based mainly on molar refractivity data from ref. 11 (with the value for Na+ = 0.65 cm3 mol-'), except as noted. From ref.12 and refer- ffences therein. From ref. 4. These values are preferred, see ref. 13. Estimated value. Ref. 14. j Ref. 15. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 3 Properties of the solvents' water 18.1 4.52 47.9 78.4 1.46 1.83 2.57 1.17 0.47 MeOH 40.7 11.95 29.3 32.1 3.26 2.87 2.82 0.98 0.66 EtOH 58.7 11.87 26.0 24.6 5.13 1.66 2.90 0.86 0.75 EG 55.9 3.82 32.4 31.7 5.73 2.28 2.08 0.90 0.52 NMF 59.1 6.27h 31.1 182.4 6.05 3.86 3.97 0.62 0.80 PC 85.2 5.90 21.8 66.1 8.56 4.98 1.23 0.00 0.40 DMF 77.4 6.22 24.1 36.1 7.90 3.86 1.03 0.00 0.69 MeCN 52.9 10.70 24.1 37.5 4.4 1 3.44 0.81 0.19 0.40 MeNO, 54.0 5.9h 25.7 35.8 4.95 3.56 0.92 0.22 0.06 DMSO 71.3 5.20 26.6 46.7 7.99 3.90 1.04 0.00 0.76 FA 39.9 4.0 39.6 111.0 4.23 3.37 2.04 0.7 1 0.48 NMA 77.0 29.4 191.3 7.85 4.39 4.2 1 0.47 0.80 AC 74.0 12.55 22.1 20.7 6.41 2.69 1.21 0.08 0.6 1 HMPT 175.7 7.9 19.1 30.0 16.03 5.54 0.86 0.00 1.05 ' From ref.6, ch. 6; unless oth erwise noted, b ulk data pertain to 25°C. In cm3 mol-'. In 10-l' Pa -'. In 5''' ~m-~". In m3. In Debye (3.335 64 x C m). From ref. 17. From ref. 18 (NMF) and ref. 19 (MeNO,). tetraphenyl ions and the tetraalkyl ions of similar size. It was with an average standard deviation of k0.30; those of (HB) rather difficult to relate the regression coefficients for the varied from -2.57 to -10.29 with an average standard devi- large ions to the properties of the solvents, but V, appeared to ation of & 1.63, both coefficients, thus, being highly signifi- play the primary role.The final results for the large ions are: cant. The former coefficients depended on the 6 and V, of the solvents, whereas for the latter a dependence on V, and a, orVo = (0.146 f0.025)r3[K -(7.1 & 1.4)~~~+ (0.44 f 0.16)6] on xT and g was found. Therefore the final results are: -(0.14 f0.03)0rj[V,-(8.6 & 2.2)aJ (3) Po = (0.08 +_ 0.01)r3[6 + (0.20 +_ 0.06)1/,] That is, the standard molar volumes for up to eight large ions -(0.66 & O.OS)(HB)[ V, -(6.0 O.5)as] (4~)in nine solvents (there are no data for MeNO,) are describ- able by a total of five parameters: two ion properties and vo= (0.08 _+ 0.01)r3[6 + (0.20 & 0.06)V,] three solvent properties.The standard deviations of the fits -(0.88 fo.ll)(HB)[~,-(0.9 k0.5)g-J (4b)vary between k3.5 for water to k18.7 for PC, the mean being k11.2 cm3 mol-' for data ranging from 67 to 350 cm3 The standard partial molar volumes of up to 13 small ions in mol-'. Notable outliers are Me,N+ and Pe,N+ in MeOH, up to 10 solvents were thus described by four parameters Ph4P+ and Ph,B- in EtOH, most ions in PC, Et,N+ in with two ion properties and three (or four, if K, is invoked) DMF, Pe,N+ and Ph,As+ in MeCN and DMSO, and solvent properties. The standard variations of the fits are Bu,N+ in DMSO. These cases were included in the regres- between f4 and k6 cm3 mol-I for Po values ranging from sions and contributed to the overall average standard devi- -20 to 50 cm3 mol-l, but MeOH (f8 cm3 mol-') and ation of the fit.MeNO, (& 12 cm3 mol-') show poorer results. Notable out- For the small ions, r3 was again the first variable chosen by liers are Na' in water, MeOH and MeCN, NH,' in MeOH the program for all solvents (except that for DMF it was on a and DMF, Rb' and Cs' in DMSO, and SCN- in PC. par with the H-bonding ability). Alone it explained from 55% For Rb+ and F-, two spherical ions with rare-gas elec- of the variability of vo(for MeNO,) up to 92.7% (for water). tronic configurations, values of Vo in several solvents are The second variable chosen was (HB): alone it explained a lacking.8 These can be predicted with the aid of the results of minor fraction of the variability of V0,but was more signifi- eqn.(4)coupled with the experimental values for other alkali cant for EG, PC and DMF. Together, r3 and (HB) could metal and halide ions. That is, differences between values explain 80%-96% of the variability of Po,except for MeNO, from the correlations and the experimental values for other (62% only). The coefficients of r3 varied from 2.32 to 4.12, ions in the series were taken into account. These systematics Table 4 Experimental and predicted values of Vo of ions in solvents for which there are no TPTB data FA NMA AC HMPT ion exptl. pred. exptl. pred. exptl. pred. exptl. pred. +Li -2.0 -9.4 -20.8 -29.1 Na -5.2 2.2 3.7 -0.4 -18 -4 -11.6 -8.0+ K+ 5.5 8.3 10.1 5.2 0.2 -2.0 NH,+ 14.5 6.6 Et,N+ 140.9 147.0 128 156 +Pr,N 209.4 208.9 20 1 226 Bu,N ' 277.4 270.3 27 1 29 1 Pe,N+ 350.8 333.7c1-22.5 16.3 -8 9 Br-33.4 28.0 28.2 23.8 21.5 20.1 I-45.0 41.2 38.6 37.0 41.5 38.0 NO,-31.7 28.5 SCN -49.0 37.2 ClO,-17 38 1902 yield for Rb+ in NMF and MeNO, predicted 'experimental' values of 8.6 and -3.4 cm3 mol- ', respectively using eqn.(4) with the data for Kf and Cs+. Similarly, for F-in NMF, PC, DMF and MeCN, eqn. (4) and the data for C1- and Br- yield predicted 'experimental' values of 6.0, -5.2, -15.1, and -26.4 cm3 mol- ',respectively. These predictions should be tested by experiment. Further predictions can be made for solvents where the TPTB approximation for splitting the voof electrolytes into the individual ionic values could not be applied directly due to the lack of data.Splitting by other means had to be resorted to, and the data were not included in the database from which eqn. (3) and (4)were derived. Such solvents are formamide (FA), N-methylacetamide (NMA), acetone (AC), and hexamethyl phosphoric triamide (HMPT).* The proper- ties of these solvents are included in Table 3, and the calcu- lated values of voresulting from the application of eqn. (3) and (4)are compared in Table 4 with the experimental data. The agreements are of the same order as for the database used for the derivation of eqn. (3) and (4), except for AC, where there exists a systematic deviation of the calculated values of +22 & 5 cm3 mol-' for both cations and anions (i.e.not due to wrong splitting). The experimental and pre- dicted values for the ions in NMA could have been brought to closer agreement (within 2 cm3 mol-l) if the value of as were 9.2 instead of 7.85 (in m3), noting that the Po data pertain to 35 "C. Discussion The thermodynamics of ion hydration has been studied by one of the authors, in a series of papers for a large number of ions of various charges, sizes, and A model, depending only on the radius r and algebraic charge z of the ions, involving a hydration shell of thickness Ar, that depends on I z I and inversely on r, is It is based on some- what similar models proposed by others previously and was shown to fit these data adequately for the Gibbs energy, enth- alpy, entropy and heat capacity of hydration. Although all these quantities, including partial molar volumes, can be understood in terms of the same general model, involving the cavity in the solvent, in which the ion is located, and the electrostatic interactions of the ion with its environment, the details can be different.The standard partial molar volumes of ions in both aqueous and non-aqueous solutions may, therefore, depend on different ion and solvent properties from those that determine other ionic thermodynamic quantities. The statistical approach adopted here reveals which of several plausible properties play the major role in determin- ing the standard partial molar volumes of ions in a variety of solvents. The present statistical evaluation shows that Po of the large ions is primarily sensitive to their intrinsic volumes, quantities proportional to r3.Because of the correlation of this quantity with r and r-' (and also r2, which is pro- portional to the surface area of the ions), nothing very defi- nite can be made of this fact. On the other hand, the linearity of the Conway-Jolicoeur20*21 plots of vo us. the relative molar weight or the number of carbon atoms per alkyl chain of the tetraalkylammonium cations is consistent with a dependence on the ionic volume. Noting that the tetra-methylammonium cations are outliers in many solvents, this means that the density of a -CH2- segment with its solva- tion shell is independent of the number of such segments per chain, provided it is 22.However, the voof the three tetra- phenyl ions, although in between those of the tetrabutyl-. and tetrapentyl-ammonium ones, fall below the straight line plots us. r3. The higher polarizability, ai, of the tetraaryl ions J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 accounts for this, eqn. (3), expressing the higher 'squeezability' of these ions. However, unquantifiable proper- ties, such as openness of the structure, which would enhance solvent penetration, thus reducing Po,could also account for the non-compliance of the tetraaryl ions with the line pre- scribed by the tetraalkyl ones. The solvent properties associated with the intrinsic volumes of the large ions in eqn.(3) indicate that these ions are 'solvated', at least as far as their voare concerned, since otherwise no solvent-property dependence would be expected. This contrasts with the finding of Kr~mgalz,~~ that these ions are non-solvated in the ordinary sense, but this was based on conductivity data. In fact, the positive depen- dence on the molar volumes of the solvents, V,, modified by a contrary dependence on the polarizability of the solvents, cc,, is due to the exclusion of the centres of the solvent molecules from an annular region around the ion. The larger the solvent molecules and the less they are able to squeeze into confined spaces, the larger is the resulting effective volume of the ion. The further, but weaker, dependence of vo on the solubility parameter, 6, indicates that the more open struc- tures of associated solvents are also reflected in the size of this exclusion shell.A similar rationalization can be made for the solvent properties associated with the polarizability of the ions, ai,in eqn. (3). It is significant to note that for the large ions, none of the solvent properties related to ion4ipole interactions, such as E, p, aKTand BKT,are of importance with respect to Vo. For the small ions, it is seen, eqn. (4), that the intrinsic volume of the ions, proportional to r3, is of importance and explains more than half of the variability of vo.Because of the mutual correlation, a dependence on r (or r2)cannot be excluded, but r-' definitely yields a statistically inferior relationship.It may be recalled that in the case of aqueous solutions, where Po data for many more univalent ions than the 13 considered here, and for ions with charges 21 are available, a dependence on r-' was found for both the intrin- sic hydrated volume, depending on (r + with Ar increas-ing with r-', and the electrostriction, depending on Ar/(r + Ar).2 In the present more restricted case, the inclusion of water in the set of solvents does not require a dependence on r-' for the statistical fit. A major part of the rest of the variability of vois explained by the hydrogen bonding or electron-pair sharing properties of the ions, (HB), i.e. aKTfor the cations and PKTfor the anions. The directional nature of such bonding is possibily responsible for this dependence, but the negative sign of this term indicates it to be related to the electrostriction. This kind of interaction should be related to the field strength of the ion, i.e.to r-', but this dependence is concealed by the r3 dependence for the limited set of ions considered here. In any case, the polarizability, xi, and the softness, 0, of the small ions do not play a significant role in making up their vo. The dependence of voon the properties of the solvents is dominated by the solubility parameter 6, i.e.,by the tendency of self-association of the solvents. This association is still manifested in the solvation shells of the ions that determine their volumes 'before' electrostriction. The molar volumes of the solvents themselves, V,, play a minor role in this respect, eqn.(4). Since solvent polarizability, a,, is fairly well corre- lated with V,, a similar statistical fit is obtained with the former replacing the latter in the first term of eqn. (4),the coefficient increasing from 0.20 to 1.8. The electrostrictive effect on the volume is related to the solvent compressibility, xT,modified slightly by the dipole orientation parameter, g, eqn. (4b). Alternatively, a quantity equal to 33-52% of the molar volume [i.e.V, -6a,, eqn. (4a)l determines the electro- strictive effect on Po. This is less easy to understand than the J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 -6 t *\ \* - I *\ \ I .L -12 ' E m EY"1.-18 4 -24 V" 0 2 4 6 8 10 12 K~/IO-'O Pa-' Fig. 1 The standard molar volumes of transfer of lithium iodide from water to non-aqueous solvents, A,r V0/cm3 mol-', us. the com- pressibilities of the solvents, ~,/10-'O Pa-' dependence on the solvent compressibility. It is noteworthy that for the small ions, as for the large ones, none of the solvent properties related to ion-dipole interactions, p,E, zKT and BKT, are required to explain the variability of vo.This contrasts strongly with the importance of these properties in explaining, for example, the solvent dependence of the Gibbs energies of transfer of ions.3 It is seen above that the standard partial molar volumes of ions in any solvent have as a major constituent the intrinsic volume of the ions, i.e.a quantity that is proportional to r3. The standard molar volumes of transfer, AtrVo,should be less dependent on this qsantity and more on the ion-solvent interactions. It is expected that the dominant ion-solvent interaction is electrostriction. If so, as shown by Hamann and Lirn," AtrV' should depend on solvent compressibility. This was tested for the four alkali-metal ions and three halide ions in the data base for which data are available (Table 4). It turned out that only for Li' among the cations and I-among the anions could linear regressions of AtrVovs. KT be obtained with reasonable correlation coefficients. This means that other solvent properties besides K~ are significantly responsible for the variability of AtrVo.For example, for C1- I903 and Br- Att Vo has some correlation with aKT.Following Hamann and Lim,28 the AtrVo of the electrolyte LiI (rather than of its individual ions) is plotted in Fig. 1 us. K~.The data for HMPT from Table 4 have been included, those of DMF are seen to be outliers, but for no apparent reason. Dr. Brian Clare is thanked for providing help with the sta- tistical calculations. References 1 H. S. Frank and W. Y. Wen, Discuss. Faraday SOC., 1957, 24, 133. 2 Y. Marcus, J. Chem. SOC., Faraday Trans., 1993,89,713. 3 Y. Marcus, M. J. Kamlet and R. W. Taft, J. Phys. Chem., 1988, 92, 3613. 4 Y. Marcus, J.Phys. Chem., 1991,95,8886. 5 F. J. Millero, Chem. Rev., 1971, 71, 147. 6 Y. Marcus, lon Solvation, Wiley, Chichester, 1985.7 B. S. Krumgalz, J. Chem. SOC.,Faraday Trans. I, 1987,83, 1887. 8 T. S. Pang and G. T. Hefter, to be published; from Ph.D. thesis of T. S. Pang, Murdoch University, 1994. 9 G. T. Hefter and Y. Marcus, J. Phys. Chem., 1994, submitted. 10 F. J. Millero, J. Phys. Chem., 1971,75, 280. 11 A. Heydweiler, Phys. Z., 1925, 26, 526. 12 Y. Marcus, Thermochim. Acta, 1986,104,389. 13 Y. Marcus, J. Chem. SOC., Faraday Trans., 1991,87,2995. 14 N. Soffer, M. Bloemendal and Y. Marcus, J. Chem. Eng. Data, 1988,32,43. 15 S. Glikberg and Y. Marcus, J. Solution Chem., 1983. 12.255. 16 Y. Marcus, J. Phys. Chem., 1987,91,4442. 17 Y. Marcus, Chem. SOC. Rev., 1993,22,409. 18 A. J. Easteal and L. A. Woolf, J. Chem. SOC., Faraday Trans. I, 1985,81,2821. 19 1. A. Brodskii and V. S. Libov, Russ. J. Phys. Chem., 1980, 54, 678. 20 B. E. Conway, R. E. Vera11 and J. E. Desnoyers, 2.Phys. Chem. (Leipzig), 1965, 230, 157; Trans. Faraday SOC.,1966,62, 2738. 21 C. Jolicoeur, P. R. Philip, G. Perron, P. A. Leduc and J. E. Des-noyers, Can. J. Chem., 1972,50, 3167. 22 M. H. Abraham and Y. Marcus, J. Chem. SOC.,Faraday Trans. I, 1986,82, 3255. 23 Y. Marcus, J. Chem. SOC., Faraday Trans. I, 1986, 82, 233; Y. Marcus and A. Loewenschuss, Annu. Rev. C, 1985,1984,61. 24 Y. Marcus, J. Chem. SOC., Faraday Trans. I, 1987,83,339. 25 Y. Marcus, Pure Appl. Chem., 1987,59, 1721. 26 Y. Marcus, Biophys. Chem., 1994, in the press. 27 B. S. Krumgalz, J. Chem. SOC.,Faraday Trans. I, 1983,79,571. 28 S. D. Hamann and S. C. Lim, Aust. J. Chem., 1954,7,329. Paper 4/00265B; Received 17th January, 1994

ISSN:0956-5000    

DOI:10.1039/FT9949001899

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1905-1907 I905 Differential Scanning Microcalorimetric Study of Sodium Di-n-dodecylphosphate Vesicles in Aqueous Solution Michael J. Blandamer, Barbara Briggs and Paul M. Cullis Department of Physical Chemistry, University of Leicester, Leicester, UK LEI 7RH Jan B. F. N. Engberts Department of Organic and Molecular inorganic Chemistry, University of Groningen , Nijenborgh 4, 9747 AG Groningen, The Netherlands Dick Hoekstra Department of Physiological Chemistry, University of Groningen, Bloemsingel 10,9712 KZ Groningen, The Netherlands The scans recorded by differential scanning calorimetry are reported for aqueous solutions containing vesicles formed by sodium di-n-dodecylphosphate (DDP). The gel-to-liquid-crystal transition occurs near 35 "C,the melting temperature T,.The dependences of heat capacity on temperature near T, are analysed to yield related enthalpies of transition and patch numbers which record the number of DDP monomers which melt co-operatively. The dependences of the enthalpy of transition and patch number offer indications of vesicle size and the tightness of packing of monomers within the vesicles. In aqueous solutions, synthetic double-chain amphiphiles' aggregate to form These are interesting systems because they resemble natural membrane system^.^ The amphiphile, di-n-dodecylphosphate (DDP) as its sodium salt forms vesicles as confirmed by electron microscopy.6 Each vesicle of DDP has a diameter' less than 100 nm although the related C,,/C,, derivative can form vesicles having sig- nificantly larger diameters; e.g.1000 nm. The related di-n- hexadecylphosphate (sodium salt) can form7 vesicles using a chloroform injection method; these vesicles have a mean diameter of 270 nm. The dependence of fluorescence polariza- tion on temperature for DDP(aq) indicated' a phase trans- ition at a melting temperature T, equal to 28 "C. In the study reported here we used differential scanning microcalorimetry to study the gel-to-liquid-crystal transition for DDP(aq) in dilute solutions. The technique is sufficiently sensitive to allow the properties of DDP to be measured in aqueous solu- tions for which one can be sure that individual vesicles are comparatively far apart.Consequently, the measured param- eters are likely to describe intra-vesicular properties with only minor contribution from inter-vesicular interactions. These conclusions follow from simple calculations described pre- viously,' based on a model for solutions described by Robin- son and stoke^.^ In reviewing the patterns which emerge from these DSC studies, we use the previously discussed7 model for the vesicle structure that involves patches of monomers, the vesicle sur- faces comprising aggregates of these patches. In these terms, the gel-to-liquid-crystal transition for DDP in aqueous solu- tion is a co-operative processlo." involving over one hundred DDP molecules where [DDP] z 8.4 x mol dm-3. This transition is accompanied by a change in organ- ization of the charged head group.Experimental Materials Surfactants and other materials were prepared as described. ' Calorimetry The differential scanning microcalorimeter (MicroCal Ltd., USA) recorded12 the heat capacities of DDP solutions rela- tive to that of a corresponding solution which contained no DDP. The volume of the cell was 1.2 cm3. Temperature was increased at ca. 60 K h-'. As previously described,I2 a water-water baseline was subtracted from each scan using ORIGIN software (MicroCal Ltd.). Therefore, in the figures described below we report the dependence of the differential isobaric heat capacity SC, (sln; T)on temperature. A known weight of DDP(s) was added to 2.2 cm3 of water, heated to 55°C and held at this temperature for cu.30 min with stirring. The solution was allowed to cool to room tem- perature placed in the sample cell of the calorimeter. (For further details see the Results section.) Analysis of Heat Capacity Data In the event that an extremum is observed in the dependence of SC, on temperature, the simplest model attributes the 'bell-shaped' plot to a two-state chemical equilibrium; X (aq)=Y(aq) characterised by equilibrium contant K at temperature T and a standard enthalpy of reaction, ArHe. Hence the dependence of the molar heat capacity C,, on temperature is given by eqn. (1). C,,(T) = [~ArH~(sln))'/RT2]1y/(l+ K)' (1) Hence, by fitting the dependence of C,,(T) on temperature, we obtain the van't Hoff enthalpy term, ArHZ(sln).The maximum in C,,(T) occurs at the maximum Tm,vH. However, in the study reported here, the procedures were not straight- forward. In the case of DDP(aq), the concentration of DDP was known when expressed in (mol monomer) dm-3. However, the phenomena described below are properties of DDP vesicles. In practice, the data showed that the states X(aq) and Y(aq) describe a group of DDP monomers which change their form co-operatively. The number of monomers in each group was called the patch number. In other words, we tentatively characterise the groups by patch numbers and refer to each group as a patch. Hence, in fitting the depend- ence of SC, on T to eqn. (l),three parameters are used in the least-squares analysis ; i.e.patch number, Ar H$(sln) and Tm.vH * The equilibrium constant K in eqn. (1) describes the equilibrium between the two states, gel and liquid crystal. The curve formed by the dependence of C, on T has a maximum at Tm,intand the area defined by the bell-shaped 1906 0.02 c I Y-(D go -0.02 (e 1 I I I 20 40 60 80 TJT Fig. 1 Dependence on temperature of the differential heat capacity, JC,,, of DDP(aq) as a function of DDP concentration [DDP] = (a) mol dm- '. (b)2 x (c) 1 x lop3,(d) 5 x and (e)1 x8.4 x plot yields the integrated enthalpy term A, Hgt. Confidence in the analysis of each set of data was gained because in the cases reported below A, HEt and A, Hz are in agreement. Results As a preliminary to the results reported here, differential scans were recorded for solutions prepared using a range of protocols.We observed that scans recorded for solutions pre- pared using the ethanol-injection method' were not repro- ducible. Here we simply report details of the scans for solutions prepared using the protocol reported above. In detail, we examined the traces as a function of DDP(aq) con- centration (Fig. 1). With decrease in concentration, the intensity of the extremum declined. No extremum was observed for solutions containing <1 x (mol monomer) mol-' indicating that the concentration of vesicles was too small to be observed via a change in heat capacity as a func- tion of temperature. The scan pattern for solutions contain- ing 2.0 x mol dm-3 DDP was reproducible over four repeated scans for the same solution.Evidence of a longer- term change was shown by additional extrema after standing Y 0.041 TpC Fig. 2 Dependence on temperature of the differential heat capacity, JC,, of DDP [aq; 8.4 x (monomer mol) dm-3]. (a),(b),(c)and (d) recorded after cooling solutions to 15 "C and scanning to 90 "C. Plot (e) was recorded 10 h after scan (4. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 at 15°C for 11 h. With increase in [DDP], T, decreases slightly. Nevertheless, there. is only one important extremum in the range 15-90 "C which for a given solution was repro- ducible over a series of repeat scans; Fig. 2. Discussion The striking feature in the DSC scans for the DDP(aq) ves- icles is the extremum near 35 "C which is assigned to a gel-to- liquid-crystal transition.In other words, the n-dodecyl chains within the bilayer 'melt' and gain local freed~m.~ In the present context, use of the term 'melt' is consistent with a description of the process responsible for the extremum as a phase transition. As for other solid +liqiud phase transitions, the melting temperature for surfactants containing the same head group, depends on chain length and chain branching within the amphiphile. In these terms, the breadth of the transition recorded by DSC is attributable to a distribution of vesicle sizes associated with a given preparative meth~d.~ While the above description of the processes responsible for the DSC extrema is attractive, it is difficult to account for the reproducibility of the total scan maximum together with 'band' shape and width.In the same way, it is diflicult to account for the reproducibility of the traces through several heat-cool-heat cycles if the breadth were somehow a for- tuitous consequence of the size and shape distribution amongst vesicles in a given sample. The term melting also seems inappropriate when, according to fluorescence polari- sation studies,' some vesicle transitions occur over a range of 50°C; e.g. vesicles formed in aqueous solution from (C1,H,,O),PO,-Naf. The arguments presented above prompt us to consider other descriptions, particularly one similar to that used in the context of enzyme denaturation. In the latter case, DSC traces often show extrema at a temperature characteristic of the enzyme.13 This temperature indicates a change in the bio- macromolecule from active to inactive forms.Moreover, this change involves a change in organisation throughout large domains within the enzyme. The change in structure can often be de~cribed'~in terms of two equilibrium states leading to an equation of the form shown in eqn. (1). We suggest that this description can be used to account for the DSC scans reported here for DDP vesicles in aqueous solu- tion. Moreover, this approach builds on an intuitively attractive chemical model. In adapting eqn. (1) to the DSC scans we introduced the concept of a patch number. Thus the transition gel -+ liquid crystal does not involve the simultaneous and coupled motion of all DDP monomers in a vesicle, a number much smaller than the aggregation number, e.g.50000 for vesicles formed by dioctadecylammonium chloride (see ref. 8 for details). In other words, the patch number calculated on the basis of eqn. (1)is not the total number of DDP monomers in each vesicle. Rather this number reflects the number of mol- ecules in the DDP bilayers which, acting as a single unit, gain local mobility at T, together with increasing freedom of the phosphate head groups. The derived enthalpy parameters reflect the change in enthalpy for each group of DDP mono- mers. In Fig. 3 we compare the dependence of C,, recorded near 35°C and the best-fit dependence using eqn.(1). In the example shown, agreement between calculated and observed dependences of molar heat capacity on temperature required a patch number of 168. The derived parameters for differen- tial scans produced by solutions of different DDP(aq) concen- tration are recorded in Table 1. The agreement between ArHEt and ArH$ indicates that the analysis is on the right track. With decrease in [DDP], the patch number increases, J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Derived parameters for DDP in aqueous solution as a function of DDP concentration [DDP]/10-3 (monomer mol) dmP3 Tm,intl°C ArHZJkcal (patch mol)-' Tl."H/OC ArH:&cal (patch mo1)- n 8.42 34.8 f0.2 654 3 34.9 f0.2 653 5 168 2.0 35.1 +_ 0.1 642 f21 35.0 f0.1 660f3 250 1 .o 35.3 f0.1 811 f36 35.3 & 0.1 838 f22 500 0.5 35.7 f0.1 734 f39 35.8 f0.1 742 f18 467 patches are loosely coupled. Hence, rapid diffusion of water and other solutes across the bilayer would occur at the joins 600 of the patches.r I Y We thank the University of Leicester for a travel grant to r I 0 400 M.J.B. and SERC for their support under the Molecular -E Recognition Initiative. m 5 CIEA 200 References 1 A. Wagenaar, L. A. M. Rupert, J. B. F. N. Engberts and D. Hoekstra, J. Org. Chem., 1989,54,2638.C 2 J. H. Fendler, Acc. Chem. Res., 1980, 13, 7. 3 T. Kunitake, Angew. Chem., Znt. Ed. Engl., 1992,31, 709. TI"C 4 A. M. Carmona-Ribeiro, Chem. SOC. Rev., 1992,21,209. Fig.3 Dependence of molar heat capacity, C,, ,on temperature for 5 T. A. A. Fonteyn, J. B. F. N. Engberts and D. Hoekstra, Cell and DDP [aq; 8.4 x (monomer mol) dm-3]. Comparison between Model Membrane Interactions, ed. S. Ohki, Plenum Press, New observed dependence (-) and the dependence calculated using York, 1991, p. 215. eqn. (1) (---) together with a patch number of 168. 6 T. A. A. Fonteyn, D. Hoekstra and J. B. F. N. Engberts, J. Am. Chem. SOC., 1990,112,8870. 7 A. M. Carmona-Ribeiro and S. Hix, J. Phys. Chem., 1991, 95, tending towards a maximum near 500. Another pattern 1812. shows that as [DDP] decreases so the molar isobaric heat 8 M. J. Blandamer, B. Briggs, P. M. Cullis, J. A. Green, M. capacity at T, [eqn. (l)] decreases. The dependence of patch Waters, G.Soldi, J. B. F. N. Engberts and D. Hoekstra, J. Chem. SOC.,Faraday Trans., 1992,88,3431.number on monomer concentration indicates that with 9 R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butter-decrease in concentrations the mean vesicle size decreases. In worths, London, 2nd edn., 1959. the limit the vesicles are sufficiently small that the total 10 L. R. De Young and K. A. Dill, J. Phys. Chem., 1990,94,801. bilayer system melts in a single co-operative transition. 11 L. A. M. Rupert, J. B. F. N. Engberts and D. Hoekstra, Biochem-Expressed in terms of the enthalpy change of each istry, 1988, 27, 8232. monomer unit the changes in enthalpy are quite modest. The 12 M. J. Blandamer, B. Briggs, P. M.Cullis and G. Eaton, J. Chem. Soc., Faraday Trans., 1991,87, 1169. concept of a patch number has wider implications. Many 13 J. M. Sturtevant, Annu. Rev. Phys. Chem., 1987,38,463.models of vesicles describe a uniform bilayer in which the amphiphilic molecules (anions for DDP) are packed with hydrophobic alkyl chains in close contact. In terms of the patch model, the bilayer resembles a quilt in which the Paper 41009245 ;Received 15th February, 1994

ISSN:0956-5000    

DOI:10.1039/FT9949001905

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1909-1911 Very Large Thermal Separations for Polyelectrolytes in Salt Solutions Derek G. Leaist" and Ling Hao Department of Chemistry, University of Western Ontario , London, Ontario , Canada N6A 5B7 A conductivity method is used to measure the Soret coefficient and heat of transport of aqueous sodium poly(styrenesu1fonate) (Na,PSSA). The data are used to calculate thermal separation factors for Na,PSSA dis-solved in aqueous sodium chloride solutions. The calculations show that large steady-state concentration gra- dients in the polyelectrolyte will be established when a temperature gradient is applied to mixed Na,PSSA-NaCI solutions, ranging up to 80% change in concentration per degree as the polyelectrolyte concentration approaches zero.The Soret coefficient and heat of transport of aqueous sodium benzenesulfonate are mea- sured and compared with the corresponding values for Na,PSSA. The distribution of components in a mixture changes when a temperature gradient is applied.' In many non-isothermal salt solutions, for example, solvent migrates to the warmer regions and salt migrates to the cooler regions.2 Though thermal diffusion measurements provide fundamental infor- mation about molecular interactions, the steady-state separa- tions that can be achieved (typically 0.1-1?4 change in concentration per degree) are usually too small to be useful in practice. The purpose of the present work is to show that thermal separations for polyelectrolytes dissolved in salt solu- tions can be much larger: 10-100% change in concentration per degree.It has been known for some time that large thermal separa- tions can be achieved for high polymers dissolved in solvents of low molecular eight.',^,^ To see if polyelectrolytes show similar behaviour, thermal diffusion measurements are reported here for aqueous sodium poly(styrenesu1fonate) (Na,PSSA, n = 340) of average molecular weight 70000 g mol-'. The size of the poly(styrenesu1fonate) ion should give Na,PSSA a much larger heat of tran~port~.~ than that of a simple salt. Since the heat of transport measures the tendency of a substance to migrate in a temperature gradient, large thermal separations might be observed for aqueous Na,PSSA.In fact, the measured separations were disappointing, ca. 0.4% change in concentration of Na,PSSA per degree. (Similar results were reported in an earlier study of poly- electrolyte thermal diff~sion.~)In binary Na,PSSA-water solutions, however, the sodium and poly(styrenesu1fonate) ions must diffuse together to prevent charge separation. Even if the poly(styrenesu1fonate) ions were drawn strongly to the cold plate, they would be held back electrostatically by the sodium counterions which have much smaller heats of trans- port. By adding a supporting electrolyte, such as NaCl, it might be possible to unleash the polyions from their counterions and thereby achieve much larger thermal separations. This idea prompted us to use thermal diffusion data for Na,PSSA-water and NaC1-water to calculate thermal separations for Na,PSSA-NaC1-water solutions.Experimental The Soret coefficient,"2 defined by 0 = -(l/mKdrn/dT)steaciy state (1) is a convenient measure of thermal diffusion. It gives the frac- tional change in solute molality, rn, per degree under steady- state conditions. The Soret coefficients of aqueous Na,PSSA solutions reported here were measured with a conductivity cell' which held a 1.219 cm column of solution sandwiched between nickel-plated copper cylinders controlled at 30.0 "C (top) and 20.0"C (bottom). Thermal diffusion was followed by measuring electrical resistances Rdt) and RL(t)across upper (U) and lower (L) pairs of platinum electrodes located at and 2 of the cell height h.The quantity Y(t)= [Rdt)-R,(t)]/[Rdt) + RL(t)]decays exponentially to the steady-state value Y,: Y(t)= Y, + Y, exp(-t/8). The relaxation time is 8 = h2/n2D,where D is the mutual diffusion coefficient evaluated at the mean cell tem- perature. The method of linear least-squares was used to evaluate Y, from plots of Y(t)US. exp(-t/O). Soret coefficients were calculated from values of the initial slope', Y,, accord-ing to the relation cr = -n2Y1/2,/(3)BAT.The factor B = -(a In R/a In m)T was evaluated from plots of the log of the mean cell resistance us. the log of the mean cell molality. Mutual diffusion coefficients were measured at 25°C by the Taylor dispersion method.' Solutions were prepared by weight with distilled, deionised water and Na,PSSA (PolySciences) of average molecular weight 70000 g mol-' (n = 340). Thermal diffusion of aqueous sodium benzenesulfonate (Aldrich), which is chemi- cally and structurally analogous to the sodium styrenesulfon- ate monomer, was also measured.Results and Discussion Na,PSSA(m ,)-Water The Soret coefficient of aqueous Na,PSSA was measured at sodium ion molalities (nrnl)from 0.001 to 0.010 mol kg-'. The values of c1 at each composition were reproducible within &O.OOOl to k0.0002 K-'. Table 1 summarises the results. Despite a molecular weight of 70000 g mol-', the Table 1 Soret coefficient of aqueous Na,PSSA(rn,) at 25 "C nrnJmol kg - ~/io-~m2s-' B OlK' O.Oo0 - - 0.0052" 0.00 1 1.08 0.97 0.0045, 0.002 1.06 0.98 0.0043, 0.003 1.05 0.98 0.00418 0.005 1.04 0.98 0.0039, 0.010 1.00 0.99 0.0037, " Obtained by extrapolation of u1us.(nrn,)'". Soret coefficient of Na,PSSA is not large, only ca. 0.004 K-'. Soret coefficients of simple salts, such as aqueous NaC1, are similar in magnitude.6 To understand this result, it is helpful to use the thermody- namic identity :2*10 which relates the Soret coefficient of Na,PSSA to its chemical potential p1 and its heat of transport QT. When Na,PSSA diffuses out of an element of solution, the heat QT per mole of Na,PSSA is absorbed in order to maintain constant tem-perature. The activity of aqueous Na,PSSA is akaapSSA= (nml)nmly>+ll,and hence p1= py + RT ln[(nml)nmly>+l'] (3) where ykl is the stoichiometric mean ionic activity coefi- cient.Differentiation gives QT = (n + 1)RT201[1+ (d In yk '/d In ml)T] (4) The activity coefficient term vanishes in the limit of infinite dilution : Qfo = (n + 1)RT20y= nQi: + Qg', (5) Extrapolation of the Soret coefficient values in Table 1 gives 07 = 0.0052 (+0.0002) K-', and hence QTo = 1300 (+50) kJ mol-' for the limiting heat of transport of Na,PSSA (n = 340). Adopting the recommended value6*" Q;t;,"= 1.33 kJ mol-' leads to Qgs, = 850 f50 kJ mol-' for the limiting heat of transport of the poly(styrenesu1fonate) ion. Eqn. (5) shows that the limiting Soret coefficient of Na,PSSA is proportional to the number-weighted average of the heats of transport of n mol of sodium ions plus 1 mol of poly(styrenesu1fonate) ions: RT2ay = QTo/(n + 1) = (nQ6:+ Q$!,)/(n + 1).Consequently, thermal separations for Na,PSSA are relatively small despite its large molar heat of transport. The data for aqueous sodium benzenesulfonate in Table 2 illustrate this point nicely. The limiting heat of transport of this 1 :1 salt is Q* = 2RT200= 7.8 (k0.3) kJ mol-', about 170 (ca. 42) times smaller than that of Na,PSSA, yet the Soret coefficient sodium benzenesulfonate is nearly identical to that of Na,PSSA. The prospects for large thermal separa- tions in binary polyelectrolyte solutions are not bright. Na,PSSA(rn,)-NaCl(rn,)-Water The chemical potentials and heats of transport2," of Na,PSSA and NaCl in ternary Na,PSSA(m,)-Table 2 Soret coefficient of aqueous sodium benzenesulfonate at 25 "C m/mol kg- ' D/w9 m2 s-' B a/K - O.OO0 - - 0.0053" 0.001 1.06 0.98 0.0045, 0.002 1.05 0.97 0.0043, 0.003 1.05 0.96 0.0042 0.005 1.05 0.96 0.0040, 0.010 1.04 0.95 0.0038, " Obtained by extrapolation of D us.m"'. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 NaCl(m,)-water solutions are given by p1= p: + RT ln[(nm, + m2)"mly",+,'] (6) P2 = + RT lnC(nm1 + m2b2 7: 21 (7) QT = TuI(~PI/~In m1)T + To2(dpc,/aIn m2)T (8) Q: = Toi(d~2/8 In mi)T + To2(8~2/8In m2)T (9) Proceeding as before, the exact limiting expressions nm2 QTO/RT~ = [i + n2m1 0; (10)nm, + m2 ]rT:+ nm, + m, Q:O/RT~ = nm1 m2 nm,+ m2 oy + 11 + nm,+ m2If39 (11) are obtained.The limiting Soret coefficients of Na,PSSA(m,)-NaCl(m,) solutions are functions of the heats of transport and the ratio of the solute molalities. Limiting ionic heats of transport are strictly additive. The values QT" = 1300 kJ mol-' and Q:' = 3.99 kJ mol-' obtained for binary Na,PSSA-water and NaC1-water l1 solu-tions can therefore be used in eqn. (10) and (11) to calculate accurate limiting Soret coefficients for ternary Na,PSSA-NaC1-water solutions. Fig. 1 shows the Soret coefficient of Na,PSSA dissolved in aqueous NaCl solutions calculated according to eqn. (10) and (11). As the ratio m1/m2 drops to zero, the Soret coefficient of Na,PSSA reaches an astonishingly large value : oy(m1/m2 +0) = [Q:' -(nQzo/2)]/RT2 = 0.84 K-', imply-ing an 84% change in Na,PSSA concentration per degree.The Soret coefficient values shown in Fig. 1 are accurate only at infinite dilution where activity Coefficients, ion associ- ation and other complicating factors are unimportant. In view of the rather weak concentration dependence of ol, the calculated limiting values should nevertheless provide a rea- sonable qualitative guide to the thermal diffusion behaviour of dilute Na,PSSA-NaCl solutions. For example, the Soret coefficient for trace amounts of Na,PSSA( 1) dissolved in dilute NaCl solutions should be ca. 0.8 K-'. The thermodynamic analysis presented here shows that thermal separations for polyelectrolyte M,P dissolved in sup- porting salt solutions can be of the order of n times larger than the separations for a binary solution of the poly-electrolyte.Unfortunately, the experimental technique used in 0.9 I I I 0.8 0.7 0.6 -0.5 I Y,0-b 0.4 0.3 0.2 0.1 0.0 0.00 0.05 0.10 0.15 0.20 nm,/(nm, +m2)Fig. 1 Limiting Soret coefficient of Na,PSSA in aqueous Na,PSSA(m,)-NaCl(m,) solutions plotted against the fraction of the sodium ion molality contributed by Na,PSSA J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 191 1 the present study is not well suited to the measurement of this effect because the conductivity changes caused by thermal diffusion of the supporting electrolyte would tend to overwhelm those caused by the polyelectrolyte. Also, because polyelectrolytes diffuse very slowly in supporting salt solu- 2 3 4 J.N. Agar, in The Structure of Electrolytic Solutions, ed. W. J. Hamer, Wiley, New York, 1959, ch. 13. A. H. Emery and H. G. Drickamer, J. Chem. Phys., 1956, 26, 620. M. Schimpf and C. Giddings, J. Polym. Sci. Part B, 1989, 27, 1317. tions, polyelectrolyte gradients develop at long times where the solution columns are prone to convection. For these reasons the thermal separations calculated according to the additivity rule may be more reliable than those obtained by direct measurement, at least for dilute solutions. 5 6 7 H. J. V. Tyrrell, Chem. Commun., 1967, 456. J. L. Lin in Measurement of the Transport Properties of Fluids, ed. W. A. Wakeham, A. Nagashima and J. V. Sengers, Blackwell, Oxford, 1991, ch. 10. J. N. Agar and V. M. M. Lobo, Electrochimica Acta, 1975, 20, 319. The authors thank the Natural Sciences and Engineering 8 9 D.G. Leaist and L. Hui, J. Phys. Chem., 1990,94, 447. D. G. Leaist and L. Hao, J. Solution Chem., 1992, 21, 345. Research Council for financial support of this research. 10 K. G. Denbigh, The Thermodynamics of the Steady State, Methuen, London, 1951, p. 20. References 11 N. Takeyama and K. Nakashima, J. Solution Chem., 1988, 17, 305. 1 H. J. V. Tyrrell, Diffusion and Heat Flow in Liquids, Butter-worths, London, 1961. Paper 4/01266F; Received 18th March, 1994

ISSN:0956-5000    

DOI:10.1039/FT9949001909

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1913-1922 Redox Properties of Ubiquinone (UQ,.) adsorbed on a Mercury Electrode Gabriel J. Gordillot and David J. Schiffrin Chemistry Department, The Donnan and Robert Robinson Laboratories, P.O. Box 147, The University of Liverpool, Liverpool, UK L69 3BX The electrochemical behaviour of ubiquinone 10 (UQ,,) adsorbed on mercury in contact with aqueous electro- lytes has been investigated at coverages smaller than a monolayer. Stability regions, acid-base ionisation constants and standard potentials for redox equilibria between conjugate stable species were determined and reaction mechanisms proposed. The standard potential for the ubiquinone/ubihydroquinone couple obtained was 0.276 V vs. SCE (-0.138 at pH 7).The values of pK, obtained for the two acid-base dissociation equilibria for the ubihydroquinone, 12 and 13.6, are higher than those predicted from the Hammett equation and lower than the value obtained in a low-relative- permittivity medium (80% w/w ethanol). The ubisemiquinone ion radical has been found to be stable at pH > 13.6 with a disproportionation constant of 0.4. The corresponding constant for the protonated radical was estimated to be showing the high instability of this form. This fact, together with kinetic considerations, suggests that for pH values lower than 12 the redox chemistry proceeds via dismutation. Two-phase transitions for the reduced and oxidised forms were observed. Ubiquinone (UQ) is known to be an important mitochon- drial redox component of the electron-transport chain.The molecule contains a quinone redox centre and a long iso- prenoid chain, and it has been suggested that its lipidic char- acter is responsible for its location 0 0 ubiquinone 10 in the hydrophobic domain within the phospholipid bilayers in cell membranes.'V2 The biological function of this coenzyme is to act as an electron carrier between membrane- bound redox enzymes. It has been proposed that UQ can be present in the mem- brane in two forms, associated with electron-transport com-plexes and in a pool containing an unbound form.3 Knowledge of the redox properties of UQ is essential for the understanding of its function. In common with other quin- ones, the reduction of ubiquinone follows a two-electron, two-proton reaction.In biological membranes it is now rec- ognised that the intermediate ubisemiquinone radical is sig- nificantly stable, as shown from electron paramagnetic resonance (EPR) spectro~copy.~ However, the values of the redox potentials corresponding to the UQ/UQ'-and UQ'-/UQH2 couples are still under disp~te.~-~ It is appar- ent that the various redox potentials measured are strongly dependent on the environment in which the quinone is present. For this reason, it is important to study electron- transfer reactions under well defined conditions. Classical electrochemical investigations have relied on the use of non- aqueous solvents, such as a~etonitrile.~ It is questionable, however, if the results obtained from aprotic solvent studies can be transposed to the membrane case, where, although the t Present address : Departamento de Quimica Inorganica, Analitica y Quimica Fisica, Universidad de Buenos Aires, Ciudad Universitaria, Pab.11, 1824 Buenos Aires, Argentina. coenzyme is present in a lipidic environment, water is avail- able to participate in intermediate chemical steps. The electrochemistry of UQ in aqueous solutions has been studied by Petrova and co-workers' who used thin films of the insoluble quinone spread on carbon electrodes by evaporation of its toluene solution. Schrebler et al. attached UQ to pyrolytic graphite by absorption from a benzene solution' and Takehara and Ide" studied charge-transfer reactions to UQ attached to glassy carbon by spin coating from an acetone solution.One of the problems with the pre- vious work has been the difficulty of controlling both the amount of electroactive species attached to the surface and the state of the carbon electrode surfaces employed. For these reasons a self-assembly technique based on the transfer of UQ adsorbed at the air/solution interface to mercury was employed in the present work. This method has been suc- cessfully used by Nelson" to attach monolayers of phospho- lipids on mercury. Experimental The adsorption technique consisted of transferring an Hg drop through a layer of UQ adsorbed at the air/solution interface. A hanging mercury drop electrode (HMDE) (Kouteckjr, Poland) was used for preparing the Hg surface prior to coating.The HMDE was mounted on a vernier screw clamp which allowed the Hg drop to be lowered slowly across the surface of the cell solution containing the spread ubiquinone. The electrode entered the cell through a silicone rubber septum that allowed movement of the capillary inside the cell. A three-electrode cell was employed; the counter electrode was a Pt gauze of 1 cm2 area and the reference electrode was a saturated calomel electrode (SCE). All poten- tials are given with respect to this electrode. The cell was jacketed and the temperature of the circulating water was controlled with a Grant W14 thermostat (kO.01"C) at 25 "C. The solutions were de-aerated with high-purity nitrogen (BOC) and a constant flow of gas was kept over the surface during measurement.The spreading solution was prepared from synthetic ubi- quinone 10 (Sigma). The spreading solvent was pentane (Fluka puriss p.a.) which was distilled before use in order to remove any non-volatile impurities. Before preparing the J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 solutions, the pentane was de-aerated; the solutions were 60 kept at -20°C. The purity of the solvent used was checked by measuring the double-layer capacitance of the mercury surface after 40 being lowered through the aqueous solution surface on which only the solvent had been spread. No difference between Hg N 20 drops extruded directly in the bulk of the solution and those previously made above the surface and then transferred across the air/solution interface was observed.The solutions were prepared from triply distilled water, E O O5.‘= -20 once from alkaline permanganate followed by distillation from dilute phosphoric acid and then by a final distillation. -40 Borax (BDH, AnalaR), K,HPO, and KH,PO, (Fluka, BioChemika) were used as received. -60 A Hi-Tek (England) DT 101 potentiostat and PPRl wave- -1.2 -0.8 -0.4 0 form generator were employed. The voltammograms at high sweep rate were recorded with a Phillips PM3302 digital E/v storage oscilloscope and at low sweep rates directly with an X-Y recorder (Phillips PM8277). Capacitance measurements were carried out with a PAR 5210 lock-in amplifier by mea- suring the in-phase and out-of-phase components of the current when a sinusoidal potential of & 5 mV was applied.Results Fig. 1 shows the general voltammetric behaviour of UQ adsorbed on mercury at different values of pH. For pH 26 f2--10 - three groups of peaks are observed; the two groups at the extremes of polarisation (11, 111, IV and V) are pH indepen- dent, whereas peak I shifts with pH. A characteristic sweep- rate dependence of the voltammograms and of the peak potential is shown in Fig. 2 and 3, respectively. At low sweep rates, the behaviour expected for a surface reaction pseudo- -1.2 -0.8 -0.4 0 capacitance is observed, but the lack of linearity at high EP sweep rates shows the onset of irreversible behaviour.This can be clearly seen from the sweep-rate dependence of the peak potential (E,). The values of E, are strongly pH depen- dent as shown in Fig. 4, but the peak separation between the 2030 t ’ ’ ’ anodic and cathodic scans tends to zero at very low sweep rates. The E, values extrapolated to zero sweep rate (E’’) depend on the pH (Fig. 5). Three pH dependences are observed; for 4.1 < pH < 12, dE”/dpH = 59.6 mV decade-’; for 12 < pH 5 13.6, dE”/dpH z 30 mV decade-’ and for pH 2 13.6, dE”/dpH w 0 mV decade- ’. Discussion Adsorption Capacitance and Reorientation Processes It is striking that two different types of voltammetric peak are -30 -I I I IVC 1 observed. At the extremes of polarisation, very sharp peaks can be seen.Since the corresponding peak potentials are independent of pH it can be concluded that reorientation processes are responsible for these. Similar processes have been discussed by Nelson and Beaton for phospholipids adsorbed on Mg.” The model proposed for this behaviour12.13 is based on the decrease in hydrophobicity of the Hg surface for increasing charge densities. This results in an increased interaction between the phospholipid polar head groups and the surface, leading to an inhomogeneous layer. As the potential is increased (both for potentials positive and negative with respect to the pzc), water displaces the lipid Fig. 1 Cyclic voltammetry of ubiquinone 10 adsorbed on mercury at different values of pH for a sweep rate of 0.1 V s-:(a)pH 4.5,O.l mol dm-3, H,KPO,, (.* .) base electrolyte; (---) adsorbed QU,o; (b)pH 6, 0.1 mol dm-3 mixture of HK,PO, and H,PO,) solutions; (c) pH 8.3, 0.1 mol dm-3 mixture of HK,PO, and H,PO, solutions; (6)pH 9.2,O.l mol dm-3 borax -1.2 -0.8 -0.4 EP 0 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 0.8 0.4 Q 30 -0.4 ....... ........ ....... -0.8 . .,,.': ,. .. .. ...... -1.2, -0.5 -0.4 -0.3 4.2 -0.1 0 EP 30 20 ................................ 10Q 30 -1 0 -20 -30 -40 .... ...-50 .... I I 1 I I I I -0.7 4.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 E/v Fig. 2 Redox peaks of UQ,, adsorbed on mercury at pH 9.2 (0.1 mol dm-3 borax). A, Sweep rates from (a) to (9): 0.3, 0.2, 0.1, 0.075, 0.05, 0.01 and 0.005 V s-'.B, Sweep rates from (a) to (9): 100,75, 50, 25, 10,7.5,5 and 2.5 V s-I. from the surface to produce discontinuous bilayer structures owing to its higher dipole moment. There are two distinct processes that can be predicted from a statistical thermodynamics model describing the changes of interaction energy of the different parts of the phospholipid molecules with the surface for different states of charge.12 As the interfacial field is increased, there is a competition for the increased polarity of the surface between the polar heads and the hydrocarbon chain. It has been proposed that the first peak at -0.9 V (us. Ag/AgC1/3.5 mol dm3 KCl) corresponds to a reorientation of the film, leading to a change in position of the polar headgroup.The second peak at more negative potentials corresponds to the formation of bilayer structures on the Hg surface, with the polar headgroups pointing both to the metal and to the solution. The experimental justifica-tion for this model is that the values of the interfacial capac-itance at potentials negative of the second peak corresponded to nearly half of those obtained for a lipid-free surface. At all pH values the capacitance at potentials more nega-tive than ca. -1.15 and more positive than ca. 0.06 V us. SCE, i.e. outside the reorientation peaks, is very close to that of a ubiquinone-free surface [see, for example, Fig. l(a)]. The sharp peaks at the extremes of polarisation are related to reorientation processes and no desorption is observed at least up to a potential of ca. -1.3 V, as shown by the repro-ducibility of the voltammograms for multiple scans.In the potential range between the reorientation and the redox peaks, the low value of the capacitance indicates strong adsorption of the molecule with an orientation parallel to the 1915 I I , I c;v (a ) 0-4 1 3 Q 6 3 .-2 0 1 0 0 1 2 3 4 5 vlv s-' I'I'I'I'I'I 30 20 15. .-lo 0 v/v s-' Fig. 3 Dependence of the peak current on sweep rate at pH 9.2 (0.1 mol dm-3 borax): (a) sweep rates under 5 V s-'; (b) over the entire range of sweep rates investigated. surface. For example, at pH 11, a minimum capacitance of 5.5 pF cm-2 is found at -0.6 V for a surface concentration of quinone of 2.6 x lo-" mol crnp2.Ignoring the small diffuse-layer contribution and considering full monolayer 0.00 -0.15 $ -0.30 -0.45 -0.6C -6.0 -2.0 2.0 6.0 In(v/V s-') Fig. 4 Dependence of the peak potential on sweep rate for different value of pH: (a) 5.3,(b) 7.0, (c) 9.2,(d) 13.3.Top curves correspond to the oxidation peak. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 901916 -0.1 I\ -0.2 % -0.3Lu -0.4 -0.5 -0.6 -11111,111,1 , coverage (see below), the thickness (d)of the adsorbed layer can be calculated from: d=-EEO C where E is the relative permittivity, c0 the permittivity of free space and C is the capacitance. From eqn. (l),and consider- ing E = 2 for a hydrocarbon, d = 0.3 nm.This calculation, of course, is affected by the surface coverage and indicates that the effective relative permittivity is greater than 2. The length of the ubiquinol molecule was found to be 5.6 nm.14 The trans double bond in the isoprenoid chain introduces rigidity in the hydrophobic chain. A molecular-modelling calcu-lation" confirmed this length and indicated an average transversal thickness of ca. 0.7 nm. This would correspond to a total surface reduction charge of the quinone for a 2e- process of 8.2 pC cm-2 for a close-packed monolayer, which should be compared with a measured charge of 5.1 pC cmP2. Considering a thickness of 0.7 nm for a flat orientation at full coverage," E x 4.3. This large value of the relative permit- tivity is probably related to insufficient close packing of the isoprenoid chains leading to areas of the Hg surface exposed to aqueous electrolyte.The higher polarisability of the double bonds in the isoprenoid chain compared with that of a satu- rated hydrocarbon is unlikely to be the reason for the appar- ently high value of E, since non-polar organic compounds have relative permittivities in the range 2-3. In order to distinguish between these possibilities, experi- ments in which the coverage by UQ,, was altered were carried out. Different coverages were obtained by altering the electrode area, i.e. changing the hanging Hg drop volume for a constant amount of adsorbed quinone. Fig. 6 shows the effect of area expansion on the cyclic voltammograms. The peak height and area of the redox process remain constant, but the capacitance baseline current increases linearly with electrode area (Fig.7). The model of the interface considered is that of a surface . which is partially covered by the quinone, with areas where water molecules have free access. This is the classical two- capacitor model used extensively for analysis of the capac- itance of adsorbed organic molecules for which the capacitance is given by:'6 C = OC,, + (1 -e)Co (2) where c is the capacitance per unit area, 6 is the coverage by quinone or quinol depending on the potential range analysed, F -0.2 1#1,1,1,111, -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 E/v Fig. 6 Effect of drop expansion at pH 11 and at a constant amount of adsorbed UQ,, (2.5 x mol) on the cyclic voltammetry of UQ,, at v = 0.1 V s-'. Drop areas (a) 0.022, (b)0.003and (c) 0.039 cm2.and c,, and Coare the ubiquinone monolayer and base elec- trolyte capacitances, respectively. In terms of the area expan- sion experiments, it is covenient to express eqn. (2) as c = A,(&) -C,) + ACo (3) where C is the total measured capacitance, A is the electrode area and A,, is the surface area covered by the quinone when the amount of quinone present on the Hg surface is constant and determined by the transfer of the HMDE through the monolayer spread at the air/solution interface. From the results shown in Fig. 7 and eqn. (3), Co = 24.2 pF cm-2. The value of CuQwas calculated from capacitance measurements as a function of total surface coverage by quinone.These experiments were carried out by successive immersions of the mercury electrode through the quinone layer adsorbed at the air/solution interface and typical results are shown in Fig. 8. The capacitance reaches a nearly con- stant value of 5.1 pF cm-2 for a surface charge which is con- sidered to be due to a monolayer of quinone (QM)of ca. 5.2 pC cm-2; for increasing amounts of adsorbed quinone, the capacitance shows a small decrease with increasing coverage, probably due to the formation of surface aggregates. From eqn. (3) and the results in Fig. 7, A,, = 8.7 x lop3cm2. Con- sidering that the charge corresponding to the adsorbed 0.7 0.6 0.4 0.3 0.02 0.03 0.04 area/cm2 Fig.7 Dependence of the capacitance on area at pH 11 and E = -0.28 V on drop expansion with 2.5 x mol of UQ,, adsorbed on it J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 ~ ~~~~ 0 5 10 15 QudpCcm-2 Fig. 8 Dependence of the specific capacity on surface concentra- tions calculated from the charge involved in the redox process [reaction (I)] quinone is 0.05 pC for the experiment shown in Fig. 7, a monolayer charge of ca. 5.7 pC cm-, is obtained. This is in reasonable agreement with the value calculated from the results in Fig. 8. Furthermore, from eqn. (2), the slope dc/dQ,, for surface coverages of less than a monolayer, should be given by: (4) From the results in Fig.8, and taking QM= 5.2 pC cm-,, a value of c,, -coof -18.6 pF cm-, is calculated in good agreement with that calculated from the data in Fig. 7, of -19.1 pF ern-,. The similarity in the values obtained gives confidence in the two-parallel-capacitor model employed since different experimental approaches lead to similar calcu- lated quantities. The similarity of the capacitance values observed with those of the base electrolyte is due to the low surface area covered by the quinol molecule after reorientation. If the cathodic peak is due to a change in adsorption geometry from parallel to perpendicular to the metal surface, the area covered by the organic compound will be only small and related to the cross-sectional area of the quinol molecule. For a reorientation process for a packed monolayer from a paral- lel to a perpendicular orientation, the coverage will be given by the ratio of projected areas in the two orientations.For the quinol molecule this is approximately given by 0.7,/ 0.7 x 5.6 = 0.12. For Co x 20 pF cm-2, the measured capac- itance will be only 6% lower than that of the base electrolyte, in agreement with experimental results. Redox Peaks At low sweep rates, the UQ/UQH, reaction is reversible, as shown by the characteristic symmetry of the voltammetric waves (Fig. 1 and 2), a ratio of unity for the charge of the anodic and cathodic peaks, the linear dependence of the peak current on sweep rate (Fig. 3) and the constant value of the peak potential for different sweep rates.The peak half-width can be used for calculating the number of electrons transferred. From the theory of cyclic voltammetry for adsorbed compounds under ideal behavi~ur'~the peak current density (i,) is given by: . nFvQ 1, = -4RT 1917 Table 1 Dependence of AE,,,/mV on sweep rates for different values of the pH PH v/V s-' 6.0 7.0 8.3 9.2 10 11 12.4 13.5 0.005 50 49 51 31 51 51 --0.05 51 49 51 31 51 52 62 80 0.1 52 51 51 52 52 52 62 80 -0.5 58 60 54 56 55 54 -where n is the number of electrons, v is the sweep rate, Q is the total charge per unit area associated with the voltam- metric wave and the other symbols have their usual signifi- cance. The peak half-width (AEI,,) is given by: 90.6AE,,, = -mV; at 25°C n The redox processes will be analysed for the three pH regions shown in Fig.5. pH < 12 The limiting value of AE,,, at low sweep rates, measured to sweep rates down to 10 mV s-l or below, was 51 f2 mV between pH 11 and 6. Results for different sweep rates are given in Table 1. There are two ways to analyse these results; first, from eqn. (6) it can be considered that n = 2 and sec- ondly, that there is a non-ideal term representing lateral interactions between the adsorbed molecules to account for the slight difference with a theoretical prediction of AE,,, = 45.3 mV. The value of AE,,, was found to be independent of the degree of coverage by the quinone, from 8 = 0.2 to ca. 1. At first sight, this is an unusual result; if lateral interactions are effective in broadening the redox wave, it would be expected that AE1,, should decrease at low coverages.The voltammetric waves can be analysed following the treatment of Laviron18 and of Brown and Anson." The activity coefficients of the oxidised and reduced components of the adsorbed redox couple are given by:'9 70 = exp -l-0 + rOR rd (7) and YR = ~XP (rRR rR + PRO ro) (8)-where roo and TRR are the interaction parameters between molecules of the same type, whereas yOR and yRO are those between different species; Ti is the surface concentration of species i. The peak current is given by : (9) where rT is the total surface concentration of electroactive species (Q = t+rT), ro = roo -roR and rR = TRR -rRO.From the expression for the total current" and the surface Nernst equation : RT royoE=Eo+-In -nF (rR7a) the values of AE,,, as a function of the interaction param- eters can be derived. The expression obtained is: 2RT [In( e)AE,,, = --arT r]nF 1-a where a = -J( +) '1 1918 From eqn. (ll),the theoretical dependence of AE,,, on the product rTr can be calculated. Surprisingly, this relationship is almost linear and was found to be given by: AE,,, = 45.28 -27.426rTr (13) with a correlation coefficient of 0.9999. Following Brown and An~on,'~ the interaction parameters for quinone and quinol can be considered to be equal. The argument used for this assumption was the ifidependence of peak potential on surface concentration for a series of quin- ones.A similar situation was observed in the present work (see Fig. 6). Furthermore, the value of the interaction param- eters will be determined, to a large extent, by the long hydro- carbon unsaturated chain of the molecule, and therefore, differences in the interaction energies between the oxidised and reduced forms will be much smaller compared with unsubstituted quinones. Thus, the peak potential, given by:" E, = E'O -RTT-,-(r, -rR) 2nF should be independent of surface coverage, as is indeed observed. Also, note that the value of r = ro = rR calculated from eqn. (11) is -8 x lo9 mol-I cm2, which is five times larger than that calculated for 1,4-naphthoquinone adsorbed on graphite, reflecting the increased intermolecular inter- actions due to the isoprenoid chain present in the ubiquinone molecule.An alternative interpretation for the observed peak broadening is the existence of two well defined fast one-electron transfer reactions with close values of the standard potential and ideal surface behaviour. The analysis for this type of reaction has been carried out by Laviron." From their results it can be concluded that AE,,, should be strongly dependent on the equilibrium constant of dispro- portional, K, . This is given by: where, in the present case, E: and E: correspond to the stan- dard potentials for the formation of the ubisemiquinone and the quinol in the adsorbed state, respectively. If EY G E:, K, = 00 and the value of AE,,, is 45.3 mV.Following the analysis by Plichon and Laviron,,' the dependence of the peak half-width on K,, and hence on standard potential dif- ferences, is shown in Fig. 9. For the values of AE,,, found, of i 8o t > E..70 -%i 60 -50 - 1 I , I , I , I , I 1 -0.120 -0.080 -0.040 (f? -e)/V 0.0 0.040 Dependence of the half-width peak potential on the standard Fig.potential difference of two consecutive one-electron transfer reactions calculated from ref. 20 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 I I 0.18 1' '4 0.12 - 0.06 - 3 0.00 - -0.06 - 4.12 - -0.18 b -I -0.75 4.65 4.55 -0.45 4.35 -0.25 EP Fig. 10 Effect of drop expansion at pH 12.4 and constant amount of adsorbed UQ (4.8 x mol).Area/cm-l (a) 0.015, (b)0.024 and (c)0.031; v = 0.1 V s-'. around 51 mV (Table l), the difference between the two potentials should be E! -EY = 61 mV, with Kd = 11. pH > 12 For pH values higher than 12 the shape of the voltammetric curves depends on the surface concentration, rT.Fig. 10 shows this effect, obtained by the expansion of a mercury drop with a constant amount of adsorbed UQ (r+.As can be seen, when the area is increased, the base capacitative cur- rents increase while i, decreases. Consequently, higher values of AE,,, are found at lower coverages. In the range 12-13.5, the limiting value of AE,,, obtained by lowering rT depends on the pH. This behaviour can be explained by taking into account that at pH > pK, the products of the reduction of UQ are charged species.Repulsive interactions between the adsorbed anions should be dominant and an activity surface coefficient greater than unity should be expected. If only this interaction is considered, two effects should be found: (a) a narrowing of the peaks when r is increased, and (b) AE,!, lower than the theoretical value of 45.3 mV. The first effect is in agreement with experimental results. However, a limiting value of ca. 80 mV has been found for AE,,, at pH > 13.6. This shows that AE,,, must be related to a two-step process with very close values of the formal potentials. pH-Potential Stability Diagram In order to analyse the E,-pH diagram (Fig. 9,the reactions that need to be considered are: UQ + 2Hf + 2e-eUQH2 (1) where Ki is the corresponding acid dissociation constant.As can be seen in Fig. 5, for pH > 13.5 no dependence of E, on pH is apparent. In this pH range the proton is not involved in the overall reaction. In all this analysis it is considered that at low sweep rates the oxidised and fully reduced form of the quinone follow a Nernstian equilibrium. Therefore, the reac- J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 2 Formal potential of ubiquinone 10 in aqueous solutions at a!, + = 1 ;potentials us. SCE Table 3 Hammett constants used in the calculation of pK, of sub-stituted phenols for different positions of the substituent electrode condition Ea/V reference no.mercury 80% ethanol 0.272 7 pyrolitic carbon coated 0.267 9 glassy carbon coated 0.248 10 pyrolytic graphite coated 0.264 8 mercury coated 0.276 present work tion in this pH range is a combination of eqn. (15) and (16): UQ +2e-=UQ2-(VII) with a formal potential of E" = -0.495 V us. SCE. At pH values between 13.5 and 12, a slope dE,/dpH of 30 mV decade-' indicates that an overall 2e-, 1H' reaction occurs, which can be considered to be : UQ +H+ +2e-=UQH-(VIII) The formal potential of this reaction (for uH+ = 1) is -0.09 V us. SCE. For pH < 12, a 2e-, 2H' process is present and reaction (I) must be considered. The value of Eo (at pH 0) obtained from the results in Fig. 5 is 0.276 V us. SCE. The above results are in reasonable agreement with those obtained by other techniques, as shown in Table 2.In order to analyse the breaks in the E,pH dependence (Fig. 5), the ionisation constants of reaction (1V)-(VI) must be known. Since the changes in slope are a consequence of the change in reaction stoichiometry, the break points corre-spond to the pK of the ionisation of the reduction product, ubiquinol, from which pK,, = 12 and pK,, = 13.6. The differ- ence between the formal potentials of reaction (+) and (VII) is related to the dissociation constants through: From the E'' values (Table 2), pK,, +pK,, = 25.7, which compares favourably with the value obtained from the stabil- ity boundaries indicated above, of 25.6. Of course, no new information can be obtained from the E'' values calculated and the above discussion simply shows that the results quoted are self-consistent.It is difficult to compare the pK, values obtained with those reported in the literature. The pKs refer, in the present case, to adsorbed molecules. Morrison et uL7 obtained pK,, = 13.3 for ubiquinone dissolved in 80% ethanol-water mixtures, which was very large compared with that for other quinones. Petrova and co-workers' investigated the pH dependence of the electrochemistry of ubiquinone adsorbed on graphite and obtained pK,, =9.9. gi 0- m- P- ref. CH3 -0.13 -0.06 -0.17 21, 28 C2H5 -0.09 -0.07 -0.15 21, 28 OCH, 0 0.1 1 28 OH 0.04 0.13 -0.03 28 0- -1.1 -0.47 -0.66 28 QBr 0.68 0.70 0.37 0.39 0.23 0.27 21 21 NO* 1.24 0.7 1 0.78 21 The experimental pK, values can be compared with those predicted from linear Gibbs energy relationships, since there is no simple way of estimating independently the acid disso- ciation constants.The Hammett equation for the pK of sub-stituted phenols is:,' pK =9.92 -2.23Z0, (17) where CT~is the Hammett constant for substituent i. The values of ui used in the calculations are shown in Table 3. o for the isoprene chain was taken as 0.10 for the o-position, slightly higher than that of the ethyl group, but lower than that of -CH,. The reason for this is that the inductive effect of the first secondary -CH, group in the chain is bound to be small. The possible error due to this choice of -B is minor._~ Table 4 shows a comparison of calculations of ionisation constants of several quinols and of ubiquinol with measured quantities. The pKas of the various quinols are presented for comparison purposes only, in order to ascertain the reli- ability that can be ascribed to the Hammett equation. Most of the experimental results agree with predictions within f4%. There is a significant difference in the values of pK,, measured both in the present work and by Morrison et al. in aqueous ethan01,~ compared with those calculated from eqn. (17). In the latter case, the difference is most likely to be related to the low value of the relative permittivity of the 80% EtOH-H,O mixture used, of 32.8.22 The results obtained in the present work, although referring to an aqueous solution, are approximately two pK units greater than calculated.Since the electroactive species is adsorbed on the Hg surface, this result is not unexpected and probably reflects the reduced value of the relative permittivity in the interfacial region. The analysis of the pK is made more difi- cult by the expected n interactions with the metal surface, which would act, in this case, as equivalent to a substituent with a positive value of 6.The pKa values measured by Petrova and co-workers' are close to those predicted from eqn. (25). However, there is an uncertainty regarding the way parent compound p-benzoquinone 2-meth yl-p-benzoquinone duroquinone 2-bromo-p-benzoquinone 2-chloro-p-benzoquinone 2,6-dichloro-p-benzoquinone 2-nitro-p-benzoquinone UQ6 UQ9 UQio UQio UQio Table 4 pKa values for substituted p-benzoquinols experimental theoretical PKa, PKa 2 PKa, ~Ka2 ref.9.85 11.4 9.98 11.4 29 10.15 11.75 10.12 11.68 29 11.25 12.8 11.83 12.24 29 8.67 10.68 8.43 10.52 30 8.81 10.78 8.47 10.57 30 7.30 9.99 6.95 9.74 30 7.42 10.1 1 7.22 9.8 1 30 10.15 - 10.10 11.6 8 9.98 - 10.10 11.6 8 9.94 - 10.10 11.6 8 13.3 - 10.10 11.6 7 12.0 13.6 10.10 11.6 this work 1920 5.5 5.0 (LLa 4.5 4.0 -0.40 -0.32 -0.24 -0.16 -0.08 0.00 C%+m Fig. 11 Relationship between the pK, of substituted p-benzosemiquinones in aqueous solutions and the sum of the Ham- mett substituent constants, Coo+m.The linear correlations shown on the graph are pK, = 9.92 -2.33 (2.58 + Zoo+,,,)using a value of op of 2.58 for 0-.Parent compounds: (a)p-benzoq~inone;~~”~(b)2-methyl-p-benz~quinone;~~ 2,3-dirnethyl-p-benzoq~inone;~~(c) (6) 2,5-dimethyl-p-ben~oquinone;~~,~~*~’(e) 2,6-dimethyl-p-benzoquin-J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 and UQH’ + eeUQH-; Eiy = -0.014 (XI) Disproportionation Constants The disproportionation reaction of the radical anion is: 24‘-+ Q + Q2~ (XII) The corresponding constant, Kd2 , calculated from reaction (XII) and the difference between the formal potentials for reactions (11)and (111) is 0.4. This value shows that the radical anion is stable at very high pH.The disproportionation con- stant for the protonated semiquinone: 2QH’=Q + QH,; Kd, (XIII) can be estimated from : Kd, = Kd2 G - (18) Ka, Ka2 The value obtained, Kd, % 2 x loi4, indicates that at low pH the radical UQH’ is very unstable and that the overall process should occur via reactions (IX) and (XIII). one;23(f) 2,3,5-trimethyl-p-ben~oquinone;~~(9)dur~quinone.~~.~~ in which these results were obtained, since multilayers depos- ited on carbon were used. Therefore, it is likely that these results’ are not comparable with those obtained in the present work. The potentials for the various steps can be calculated from the estimated pK,, values and from those corresponding to the ubisemiquinone.From AE,,, obtained for pH > 13.5 and using the data in Fig. 9, E&--E&-= 24 mV. Consider- ing that i(EGQ.--E&-) = -0.495 V the following formal potentials have been estimated: E&.-= -0.483 V and EgWlP2--0.507 V us. SCE. In order to obtain a complete reaction scheme, the acid dissociation constant of the ubise- miquinone radical, KR, must be known. The pK, of semi- quinones has been found to be highly dependent on the Rate Constant and the Transfer Coefficient Deviations from the reversible behaviour become evident when the sweep rate is increased, and the cathodic and anodic peak potentials shift symmetrically from the reversible potential. The degree of reversibility depends on the pH and the reaction becomes less reversible at low pH.At higher pH, higher sweep rates are required to obtain the same peak separation. These effects are shown in Fig. 4. The theoretical analysis for linear sweep voltammetry of adsorbed layers has been developed by La~iron.~~ For the general reaction : Oads+ ne-Rads (JW and when adsorption obeys the Langmuir isotherm, the surface formal potential is defined by: polarity of the solvent. For example, Patel and Will~on~~ have determined pK, between 4.9 and 5.1 for durosemiquin- one in aqueous solution of isopropyl alcohol (1 mol dmP3) and acetone (1 mol drnp3), while in aqueous isopropyl alcohol (7 mol dm-3) and acetone (1 mol drnp3) a value of pK, = 6 was found. For UQH’, Land and Swallow24 have obtained pK, = 6.5 in methanol solutions while Patel and Will~on,~found a value of 5.9 in aqueous solutions of isopro- pyl alcohol (7 mol dm-3) and acetone (1 mol dm-3).Owing to the aromatic character of the semiquinones, it is possible to use the Hammett equation to estimate the value of pK,. Fig. 11 shows a very good correlation between predictions and measurement if a value of bp of 2.57 is assigned to the substituent 0’for the pKR determined in aqueous isopropyl alcohol (1 mol drnp3) and acetone (1 rnol dmP3). From this equation a pKR of 4.4 is obtained for UQ,, . This is very low, but shows the limiting value expected in a solution with a relative perimittivity close to that of water. As has been discussed above, the relative permittivity at the interface may be quite different when compared with the corresponding value for the bulk solution.Similar to the analyses of pK,, the pK, for the system under study should be between 6.5 and 4.4. Taking pK, as 5.5, the formal poten- tials (aH+= 1, us. SCE) for the following reactions were esti- mated : UQ + H+ + e-eUQH’; EL: = -0.157 (IX) UQH’ + H+ + e-eUQH,; EL: = 0.710 (X) where b, and b, are the adsorption coefficients of 0 and R. There is no adsorption equilibrium in the present system since ubiquinone is practically insoluble in aqueous solutions, but a similar mathematical formalism can be used to analyse the system under study. The current is given by: i = -nFAX,/at = nFAk’* x {r,exp[-af(E -EO)] -rRexpC(1 -ct)f(~ -EO)]) (20) where k“ is the rate constant, A is the area, ct the transfer coefficient,f= F/RT, To and rR are the surface concentra- tions of 0 and R, respectively, and the other symbols have their usual significance.Considering that To + rR is constant, eqn. (20) can be numerically integrated and different param- eters correlated. For reversible behaviour AEp = E,, -E,, is zero (Ep, and E,, are the cathodic and anodic peak potentials, respectively). When this difference is larger than 0.2 V the theory predicts totally irreversible behaviour. In the present reaction and for all values of the pH, a transition between these two cases is observed when the sweep rate is increased. The value of x can be determined from the potential dependence of the ratio R = (Epa-Ep)/(Epc-Ep).25R was found to be independent J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 0.20 0.15 < L' 0.102 0.05 0.00 0 10 20 30 40 50 vp s-' Fig. 12 Observed dependence of nAE, (n = 2) on sweep rate for dif- ferent values of pH: (*) 7.0, (A)8.3, (0)9.2, (e)10.5, (0)13.3 of potential and equal to one, which corresponds to a = 0.5. When nAE, is greater than 0.2 V, the reaction can be con- sidered totally irreversible. In this case, the transfer coefficient can be obtained from:25 d AE, RT -d log v 2.3n,aF where n, is the number of electrons involved up to the rate- determining step. From Fig. 4, a value of an, x 0.5 is obtained at all pH values, indicating that only one electron is transferred up to the rate-determining step.The rate constant can be determined from the dependence of BE, on the reduced sweep rate given by l/m = nFv/RTk", when nAE, c 200 mV.25 The theory of quasi-reversible surface reactions2' predicts a generalised relationship between nAE, and l/m for each value of a. This is a simple and powerful approach for the calculation of rate constants. Fig. 12 shows the dependence of nAE, as a function of the sweep rate for different values of pH. In order to put the results on a common basis, a value of a = 0.5, as previously found, was considered and a best fit was found for the results at any value of the pH by fitting to the theoretical curve.,' The result of this generalised plot is shown in Fig.13, where it can be seen that the data at different pH values are consis- tent with theory. In order to compare the results at different 0.20 0.15 a2 0.10 0.05 P 0 5 10 15 m-' Fig. 13 Experimental results (Fig. 12) fitted to the theoretical curve (solid line) for the dependence of nAE, on the reduced sweep rate, m-' = ~FV/RT~''.'~ 9.2,(0) 10.5, pH: (+) 5.3, (*) 7.0, (A)8.3, (0)(0)13.3. 3.0 F-2.0 IIn1 0 s 0, 2 1.0 0.0 5.0 7.0 9.0 11.0 13.0 PH Fig. 14 Dependence of the rate constant k" on pH. Results from the data shown in Fig. 12 and 13. pH and calculate the rate constants, values of the sweep rate for the points plotted on Fig. 12 were multiplied by a factor x at each pH, such as to obtain the best fit to the theoretical curve of nAE,,, us.m-l. Since l/m = nfv/RTk", the corre- sponding k" was obtained from x = nF/RTk". The corre- sponding pH dependence of the rate constants thus calculated is given in Fig. 14. At low pH values, a linear dependence is observed, whereas in highly alkaline solutions, the rate constant appears to reach a constant value. The pH dependence indicates the involvement of the proton prior to the rate-determining step. Furthermore, from the slope in acid solution, k" a [H']o.5. This corresponds to a le-, 1H' reaction leading to reduction to the quinol. The proposed sequence corresponds to reaction (11)followed by : UQ'-+ H+(H,O)= UQH'( +OH-) (XV) The disproportionation of the radical anion is expected to be slow owing to electrostatic repulsions.The reaction order with respect to the proton suggests that the rate-determining step is the deproportionation reaction : UQH' + UQ'-+ UQH-+ UQ (XVI) since the value of the pK, of QH' is ca. 5.5, the surface cover- age by QH' is always smaller than that of Q*-, and therefore the disproportionation reaction UQH' + UQH' + UQ + UQH, (XVII) will be much slower than reaction (XVI). In alkaline solutions, where the surface concentration of QH' becomes vanishingly small, the reaction cannot proceed according to a disproportionation pathway and the rate becomes determined by electron transfer according to reac- tion (11) followed by (111). This reaction pathway leads to a pH-independent rate constant in the alkaline range, which is indeed observed.G.J.G. gratefully acknowledges the support of CONICET (Consejo Nacional de Investigaciones Cientificas y Tecnicas de la Republica Argentina). References 1 M. Ondarroa and P. J. Quinn, Biochern. J., 1986,240, 325. 2 M. Ondarroa and P. J. Quinn, Eur. J. Biochem., 1986,155, 353. 3 C. I. Regan and C. Heron, Biochern. J., 1978,178,783. 1922 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 4 5 6 7 8 9 10 11 12 T. Ohnishi, J. C. Salerno, H.Blum, J. S. Leigh and W. I. Ingle- dew, in Bioenergetics of Membranes, ed. L. Parker, G. C. Papa- georgiou and A. Trebst, Elsevier, Amsterdam, 1977, p. 209. P. F. Urban and M. Klingenberg, Eur. J. Biochem., 1969,9,519. F. L. Crane and R. Barr, in Coenzyme Q, ed.G. Lenaz, Wiley, Chichester, 1985, p. 1. L. E. Morrison, J. E. Schelhorn, T. M. Cotton, C. L. Bering and P. A. Loach, in Function of Quinones in Energy Conserving Systems, ed. B. L. Trumpower, Academic Press, New York, 1982, 0. S. Ksenzhek, S. A. Petrova and M. V. Kolodyazhy, Bio-electrochem. Bioenerg., 1982,9, 167. R. S. Schrebler, A. Arratia, S. SLnchez, M. Haun and N. Duram, Bioelectrochem. Bioenerg., 1990,23, 81. K. Takehara and Y. Ide, Bioelectrochem. Bioenerg., 1991,26,297. A. Nelson and A. Benton, J. Electroanal. Chem., 1986,202,253. F. A. M. Leermakers and A. Nelson, J. Electroanal. Chem., 1990, p. 35. 17 18 19 20 21 22 23 24 25 26 27 28 29 A. J. Bard and L. R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, p. 521. E. Laviron, J. Electroanal. Chem., 1974, 52, 395. A. P. Brown and F. C. Anson, Anal. Chem., 1977,49, 1589. V. Plichon and E. Laviron, J. Electroanal. Chem., 1976,71, 143. G. B. Barlin and D. D. Perrin, Quart. Rev., 1966,20, 75. R. Parsons, Handbook of Electrochemical Constants, Butter-worths, London, 1959. K. B. Pate1 and R. L. Willson, J. Chem. SOC., Faraday Trans. I, 1973, 69, 814. E. J. Land and A. J. Swallow, J. Biol. Chem., 1970,245, 1890. E. Laviron, J.Electroanal. Chem., 1979, 101, 19. R. L. Willson, J. Chem. SOC.,Faraday Trans. 1, 1971,67,3020. P, S, Rao and E. Hayon, J. Chem. Phys., 1973,77,2274. L. H. M. Janssen, A. L. van Ti1 and F. B. van Duijneveldt, Bio-electrochem. Bioenerg., 1992, 27, 161. J. H. Baxendale and H. R. Hardy, Trans. Faraday SOC., 1953,49, 278, 53. 1 140. 13 A. Nelson and F. A. M. Leermakers, J. Electroanal. Chem., 1990, 30 E. P. Serejeant and B. Bempsey, Ionisation Constants of Organic 14 278, 73. B. L. Trumpower, J. Bioenerg. Biornembr., 1981,13, 1. Acids in Aqueous Solution, IUPAC Chemical Data Series No. 23, Pergamon Press, Oxford, 1979. 15 Desktop Molecular Modeller (Version 1.2), Oxford University 16 Press, Oxford, 1989. B. B. Damaskin, 0. A. Petrii and V. V. Batrakov, Adsorption of Paper 4/00807C; Received 9th February, 1994 Organic Compounds on Electrodes, Plenum Press, New York, 1971.

ISSN:0956-5000    

DOI:10.1039/FT9949001913

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1923-1929 Studies of Silver Electronucleation onto Carbon Microelectrodes J. P. Sousa* Departamento de Engenharia Quimica , FEUP/Universidade do Porto, Rua dos Bragas, 4099 Porto Codex, Portugal S. Pons Department of Chemistry, University of Utah, Salt Lake City, UT 84112,USA M. Fleischmann Department of Chemistry, University of Southampton, Southampton, UK SO9 5NH It is shown that it is possible to use microelectrodes to develop a number of strategies for the measurement of silver electronucleation kinetics at the molecular level. The direct measurement of individual events and there-fore stochastic deterministic growth of the centres allows determination of an accurate value of the arrival time of the first nucleus.The distribution of these arrival times follows the simplest model of a pure birth process with identical rate constants for all the aggregation steps. The electrodeposition and electrocrystallisation of several metals have been widely studied and the mechanisms inten- sively investigated. 1-3 The early development of the theory of crystal growth stemmed from the ideas of Gibbs, the main contributors being Erdey-Gruz, Volmer, Becker and D~ring.~ The model arrived at was that crystals would grow until the object (substrate surface) was bounded by perfect low-index faces so that the crystal minimises its surface energy. Further growth would then require two-dimensional nucleation of layer planes and the formation of each nucleus would lead to the deposition of unit lattice repeat or, possibly, a subunit of lattice. On the basis of this model, crystal growth takes place at the edges of layer planes.In the 1950s a new model was proposed by Burton et according to which an emerging screw dislocation generates a self-perpetuating step, i.e. crystal growth leads to the winding up of these steps into spiral growth forms (which have been frequently observed by electron microscopy). The edges are 'rough' and lattice formation occurs at these self-perpetuating edges. The electrocrystallization of silver was investigated repeat- edly during the 1960s using a variety of electrochemical tech- niques (ac impedance, galvanostatic steps and double potential step methods).6-' In these investigations no special steps were taken to ensure the formation of smooth surfaces, i.e.these measurements can be related essentially to those in the present study. A general conclusion drawn from those investigations was that there is a relatively high concentra- tion of adatoms at the substrate surface and that the exchange current associated with the formation of these adatoms is comparable to that of the Hg2+/Hg reaction, which is one of the fastest electrochemical processes known (ky = 1-10 cm s-').'' During the past decade, special attention has been devoted to the study of the initial stages leading to the formation of metallic centres. However, several difficulties in characterising the nucleation process have been reported.'2*' The nucleation of silver on conventional carbon electrodes (diameter in the cm range) under potentiostatic conditions has been studied by Hills and co-~orkers.~~-~~ It was found that the growth of metallic crystallites is a three-dimensional process and is diffusion controlled.Attempts have been made to establish a relationship between the applied overpotential and the kinetics of the process. The main difficulty in estab- lishing such a relation was the overlap of the metallic centres undergoing deposition. Therefore, the only kinetic parameter obtainable was the number density of nuclei generated during a certain potentiostatic pulse by an indirect process of rela- ting the observed current transient to the area of the cluster formed.The rate constants and the kinetics of formation for the growth of single, three-dimensional centres can be estimated by using small devices such as microelectrodes (diameter in the pm range) as substrates. The importance of microelec- trodes is widely recognised and interest in their application to several areas of research has increased dramatically over the past decade.17-22 These devices have made an impact in the electrocrystallization domain, both fundamental and applied, unequalled by almost any other technique in modern times. Microelectrode properties contributing to improvements in the quality of the experimental data in the electrodeposition field include:23 (i) the ability to define the formation of an even smaller microelectrode (area in the A range) on the sub- strate surface; (ii) high sensitivity, which produces a high signal when a single molecular event takes place at the sub- strate surface; (iii) the individual molecular events are sto- chastic and therefore constitute a deterministic process.The above properties, among others, allow study of the nucleation and growth of an extremely small crystal, i.e. a single centre. The essentially deterministic growth of the first nucleus (defined as a set of metal atoms large enough that they are thermodynamically stable at a given potential) allows one to determine an accurate value of the arrival times. A large collection of these arrival times for each over- potential gives a distribution of first passage times.These dis- tributions are determined by the kinetics of the formation of the first nucleus and therefore can be conveniently exploited in order to determine the kinetic parameters that characterise the metallic nucleation process. In the early 1970s, nucleation started to be regarded as a random phen~rnenon;~~ the random birth of nuclei within a given time interval was assumed to follow a simple Poisson distribution law: P, = N" exp(-N) m! where N is the average number of nuclei to be expected in the average time interval. When using small devices such as microelectrodes in metal- lic electrodeposition studies one can observe from analysis of the ensembles of the experimental crystal growth transients that the process of forming a single nucleus can be described by the Chapman-Kolmogorov relations : The difficulties in obtaining general solutions to such sets of equations are well known.Both Aj and pj are complicated functions of the nucleus size. One cannot therefore derive any sensible master equation. Algebraic solutions can be attained only for values of K < 3.25 Recently, Fleischmann et have proposed a model based on the facts that under severe experimental conditions (high overpotentials and highly inert substrate surfaces) the growth rates were much higher than the dissolution rates and that the birth rates were equal (Ao = A1 = = Ij) through- out the nucleation process, i.e. nucleation occurs by a pure birthprocess. The probability of forming a cluster of size j is governed by the system of equations: with initial conditions P0--1; Pj+l=0; t=O (4) The expressions comprising eqn.(3) can be integrated one at a time to give the general Poisson distribution law (At)jPj(t)= -exp(-It)I! The probability of forming a cluster of size greater than K (the size of the critical cluster), in a time interval 0 to t is then given by m (It)j1Pj(t) = 1 -C 7exp(-IIt) K+l o J! and this is illustrated in Fig. 1 as plots of PXt) us. the dimensionless variable T(=At) for K = 1 to 10, while in Fig. 2 the natural logarithms of these parameters are plotted. It can be anticipated that the first major detectable devi- ation from this behaviour (pure birth process) would be the influence of low rates of death (which occurs accordingly with the classical and atomistic approach for the nucleation process) in a general birth and death process.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.oo 0.80 0.60 P 0.40 0.20 0.00 T Fig. 1 Model of the pure birth process for different K values, using T = At. Plots of Cz+ Pj(T)US. T.k = 1 (a),.. . ,10 (j). (7) Closed-form solutions of eqn. (7), can be derived only if K < 3. However, by solving the Laplace transforms of these equations one can arrive at the general expression m * (1”t)j1pj(t) = 1-C Iexp(-k) + K(AK+ l)ptK(+2) exp(-At)K+ 1 j=o 1-(K + 2)! Eqn. (8) has been used to construct Fig. 3-5. 0.00 -2.00 -4.00 k c--6.00 -8.00 -1 0.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 In T Fig.2 Plots of In Cz+l Pj(T)us. In T for the pure birth model; (a)-(j) as Fig. 1 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T Fig. 3 Plots of cr (T)0s. T for the birth and death model for dif- ferent values of /3, where /3 = p/2 and A. = A, = . ..ik= A. Arrow indicates p increasing from 0.1 to 1.0 in increments of 0.1. /3 = 5 (a), 10 (b),50 (c),100 (d)and 500 (e). As can be seen, the inclusion of death rates accounts for a slight broadening of the theoretical plots only for small values of K. This leads to the conclusion that, under forced experimental conditions, p assumes very low values which can be ignored and the nucleation process can be regarded as being a pure birth process.The pure birth model is here applied to study the kinetics of the nucleation and growth of single centres of silver on highly inert substrate surfaces (carbon fibres) under poten- tiostatic conditions. These substrates are characterised as having a very low energy of adhesion between the substrate and the new phase. This leads to a low rate of nucleation and hence causes the deposition of only one or, at most, a few centres, which is followed by their subsequent growth. A sta-tistical treatment of the arrival time ensembles is presented, showing that silver nucleation can be regarded as a random process, i.e. obeying the Poisson distribution law. 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T Fig. 4 Plots of cg+(T) us.T showing broadening due to the inclusion of small values of fl (fl = p/A) [eqn. (8)]. Arrow indicates increasing fl, as in Fig. 3. k = 1 (a), 5 (b),10 (c). 1925 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T 0.00 3.00 6.00 9.00 12.00 15;OO T Fig. 5 Plots of IF+(T)vs. T for the birth and death model [eqn. (8)]; using A, fl = 0.1, B, fl = 1.0.(a)+) as for Fig. 1. Experimental The measurements were carried out using the potentiostatic technique for the cathodic deposition of silver from an aqueous solution of composition 0.1mol dm-3 KN0,-5.0 mmol dm- AgNO, . All chemicals were purchased from J. T.Baker Chemical Co. and the water used to prepare the solutions was triply distilled from a Corning MP-1system. The working electrode, Fig. qa), consisted of a carbon fibre microdisk of 5 pm diameter sealed into glass, as has been reported in the literature.18*’ 1,27*28These electrodes were treated with 1 :1 HNO,-H,O solution and mechanically pol- ished with fine-grade alumina powder (1.0to 0.05 pm) after several measurement cycles. Between each potentiostatic measurement the working electrode was cleaned by anodic stripping (g = 180mV). As a secondary electrode, Fig. qb),a silver wire (99.9% purity), of dimensions much larger than the working electrode, was used. All of the potentiostatic mea- surements were carried out using a two-electrode system on a single-compartment cell.A waveform generator, Hi-Tek PPR1,was employed to apply the chosen overpotentials to the investigated system. The currents were measured using a Keithley Model 617pro-grammable electrometer and recorded on a Houston Instru- ments Model 200 X-Y recorder. Low-noise coaxial cables were used to make the electrical connections and the electro- 1926 (a) copper /'' lead epoxyc./A 2F epoxy g-resin resin seal seal ~'glass 1 ,glass tube tube i . ~ lead seal carbon microelectrode Fig. 6 Side view of the electrodes: (a) carbon disc microelectrode (WE); (b)silver wire electrode (AE) chemical cell was placed inside a Faraday cage to avoid external interferences.Dry nitrogen was bubbled through the electrochemical solution for 20 min before each experiment and for each set of potentiostatic measurements a fresh solution was prepared. All measurements were carried out at laboratory pressure and temperature. Results and Discussion The linear sweep voltammograms, such as that shown in Fig. 7, display the characteristic features of nucleation and phase growth, namely the large peak separation and crossover on the cathodic branch." The shapes of these voltammograms are a function of the induction times, chosen limits of poten- tial sweep, sweep rates and surface reactions. For the case of metal electrodeposition on microelectrodes the current-potential (Aq = vt) transients present character- istic features: on the forward sweep, the current rises owing to the formation of a metallic centre at the substrate surface and this rise continues until the cathodic potential limit is reached.On the return sweep, these transients can assume a variety of responses depending on the complexity of the J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 / I/\ /I I ,f i,' 11 200 -2 00 overpotential/mV Fig. 7 Linear sweep voltammogram for the deposition of silver onto a carbon microdisc electrode from 5.0 mmol dm-, AgNO, in 0.1 mol dmP3 KNO, aqueous solution; v = 150 mV s-'; I = 165 pA cm-system undergoing investigation.' 3,2 ',23 A commonly observed pattern is that the current continues to increase for the whole of the cathodic region of the return sweep owing to the continued expansion of the growth centre(s).The different shapes observed within the anodic branch can be regarded as being due to surface passivation processes, i.e. film formation. Metallic silver centres were occasionally formed on the carbon fibre surface following the cathodic reduction of Ag', according to the following reaction : Ag+ + 1 e- -+Ago (1) A large number of current-time transients were recorded over a large overpotential range. At least 200 measurements were made for each value of overpotential applied to the system (-200, -190, -180 and -170 mV) in order to obtain a statistically accurate distribution of the induction times as shown in Table 1for the highest overpotential used.The data presented in this table show that the Poisson dis- tribution is indeed appropriate to describe the nucleation phenomena. Considering that under the experimental poten- tiostatic conditions, the observed induction times are distrib- uted randomly (Fig. 8), one has to assume that the measurements fluctuate from observation to observation, i.e. statistical fluctuations are observed. Therefore, the method of Table 1 Distribution of experimental induction times for silver electronucleation onto a carbon microdisc electrode at q = -200 mV 0.00-0.25 0.26-0.50 0.51-0.75 0.76-1.10 1.10-1.25 1.26-1.50 1.51-1.75 1.76-2.00 2.10-2.25 2.26-2.50 xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxx xxx xx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx 52 110 35 8 5 210 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 2 Experimental nucleation times and probabilities -200 mV -190 mV In t In P(t) In t In P(t) -1.386 -3.057 -0.693 -1.820 -0.693 -1.398 O.OO0 -0.814 -0.278 -0.629 0.405 -0.454 O.OO0 -0.260 0.693 -0.283 0.223 -0.1 16 0.9 16 -0.171 0.405 -0.064 1.098 -0.115 0.559 -0.039 1.252 -0.056 0.693 -0.024 1.386 -0.020 0.810 -0.010 1.504 -0.005 0.9 16 O.OO0 1.609 O.OO0 maximum likelihood must be used in order to determine the best statistical mean of the di~tribution.~'-~' Also, all of the measurements are affected by different uncertainties.Thus, for the present case, the mean is given by (9) The variance is equal to the mean, but the mean is affected by an uncertainty estimated according to Eqn. (9) and (10) have been applied to determine the sta- tistical parameters that characterise the data presented in Table 1. The results obtained were: p = 79.97 counts s-', CJ = 8.94 counts s-and o,, = 0.22 counts s-'. Once the statistical parameters that characterise the experi- mental data (p, CJ and CJ,,)have been calculated, one can analyse the parent distribution, which describes the experi- mental data in terms of its shape. This can be carried out by performing the x2 test of the distribution.For the Poisson distribution, x2 is defined according to the following equation where NP(xj)for the Poisson distribution is simply the mean. However, when dealing with binned data the method of least squares3' provides a more convenient expression for x2, without significant loss of information. For a very large ensemble, the N events can be sorted into bins of width W, where each bin contains nj events. The total x2, summed over i /-/ time/s Experimental current-time transients of silver growth onto a carbon microdisc electrode from 5.0 mmol dm-3 AgNO, in 0.1 mol dmP3 KNO, aqueous solution: q = -190 mV; t = 0.5 cm s-'; I = 25 pA cm-' -' -180 mV -170 mV In t In P(t) In t In P(t) O.Oo0 -2.145 O.Oo0 -4.074 0.693 -1.136 0.693 -1.966 1.098 -0.614 1.098 -1.366 1.386 -0.384 1.386 -0.823 1.609 -0.248 1.609 -0.623 1.791 -0.187 1.791 -0.462 1.945 -0.147 1.945 -0.360 2.079 -0.116 2.079 -0.283 2.179 -0.092 2.179 -0.191 2.303 -0.067 2.302 -0.141 2.397 -0.054 2.397 -0.108 2.484 -0.03 1 2.484 -0.080 all bins, is given by the following equation: j=1 Jj whereL represents the ideal number of expected events for each bin and is given by fj = NWP(xj;p) (13) The results obtained for the analysis of Table 1 using both eqn.(12) and (13) were x = 1.97, v = 4 and P = 0.75. These values indicate that the determined probability (P)is >0.5 which leads to the conclusion that the Poisson distribution fits reasonably the random sets of experimental data.The fea- sibility of performing such an analysis to gather information on the nature of the statistical distribution of arrival times when dealing with metal nucleation using high overpotentials and small devices, such as substrate surfaces, is demonstrated. The time dependences of the distribution Pj(t) are obtained directly from Nt/NtOtal,where N, is the number of nuclei formed within a period of time, t, for any given ensem- ble of first passage times and Ntota,represents the total number of events for each overpotential. The data obtained for the system under investigation are given in Table 2. Through an iteration method and a fit with the theoretical data for the pure birth model, it appears possible to deter- mine the most appropriate values of the rates of birth, i,for each overpotential, and thus obtain an experimental value of T.This procedure also allows the experimental sizes of the initial clusters to be obtained. The data obtained for the silver studies herein reported are given in Table 3 for the four overpotentials for which the experimental investigation was carried out. In Fig. 9 the experimental values of the dimensionless vari- able T are plotted against I;+ Pj(T).From Fig. 9 we can see that there is excellent agreement between the theoretical plots and experimental data when using the determined A and K values (Table 3). The applicability of such a simple model to describe the nucleation of silver might appear to be a matter of coin- cidence, essentially because A is assumed to be constant for successive steps in the process and there is no dissolution of Table 3 Experimental values for 1,K and ky overpotential/mv critical cluster size, K birth rate /s-' k,mo1 cmW2 s-' ~~~~~~ ~ -200 1 3.64 9.03 -190 3 2.21 5.52 -180 5 0.94 2.36 -170 6 0.71 1.78 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T Fig.9 Fit of the experimental data obtained at q = -200, -190, -180 and -170 mV (using the values of I and K + 1 listed in Table 3) to Egtl Pj(T) us. T plots calculated for the pure birth model. (a)+) as for Fig. 1. the centres formed. Nevertheless, the most likely effect of the increase in the surface energy of the subcritical nuclei with cluster size will be a decrease of the I values rather than an increase of the dissolution rates, p, for such high over-potentials. At these potentials the electrical work term, neq, is Imucn larger man any conceivame sunace-energy term, so that one would expect the condition 2 % p to be satisfied.Therefore, the model of the pure birth will apply as a limiting case at high overpotentials. As for the data presented in Table 3, it is clear that there is a relationship between the applied overpotential and the kinetic parameters. In the table the rate constants for silver deposition onto unit area of surface (1.0 cm’) derived from the rates of birth are presented. The molar volume of Ag is 10.28 cm3 giving a volume per atom of 1.71 x cm3 (considering N = 6.023 x atoms mol- ’).Taking into consideration the face-centred cubic (fcc) structure of silver, then the area of any face of the cube (assuming that any edge has a length of 2.58 x lo-* cm) is 6.64 x cm2. This can be considered as the area onto which deposition of silver takes place giving the rates 5.42 x 1015, 3.31 x lo1’, 1.41 x lo’’ and 1.07 x lOI5 atoms cm-’. Division by N gives the rates of deposition in mol cmP2s-l listed in Table 3. These are much lower than the rates one would calculate from literature values of k; appro-priate to the experimental conditions (5.0 mmol dmP3 Ag+) and overpotentials (the literature values are 4.34 x 3.72 x 3.19 x lop6 and 2.74 x mol cm-2 sP1).lJ2 Needless to say, the rates of deposition to form the nuclei must be much smaller than the rates of deposition on the growth centres, otherwise the phenomenon of nucleation would not be observed.On the other hand, 2 changes very rapidly with potential. This variation is faster than can be accounted for by any simple electrochemical rate law,’ 1,23,25926 even faster than which might be expected to hold for a rate-determining trans- formation of an adatom of silver. However, such a mecha- nism could explain the lack of dependence of the rates of birth on the cluster size, since the rates of these steps could then not depend on the number of lattice-forming sites for sufficiently small clusters. J. CHEM. SOC. FARADAY TRANS., 1994, VOL.90 The simplicity of the results obtained here, namely the applicability of a simple pure birth model and the marked increase of the rates of nucleation with overpotential, may be contrasted with the predictions based on the classical theory of crystal gr~wth.‘~,~~ However, the results attained upon using microelectrodes show a marked change in the sizes of the critical clusters with overpotential, which does not in any way fit the predictions based on the classical theory of crystal growth. With regard to the sizes of the clusters, Table 3 also reveals a progressive decrease in the critical cluster size dimensions with increasing overpotential, as described in the literature for other system^.'^,^^,^^ The sizes of these critical clusters can be interpreted in terms of a face-centred cubic (fcc) struc- ture of silver.21 Therefore, one can expect the deposition of a stable single entity (K = 1) at the origin of a plane, entities of size three and six (K = 3, 6) for the formation of sections of the (111) plane and an entity of size five (K = 5) for an element of the (100) plane of the unit cell.A similar behaviour has been observed for the nucleation of other systems on carbon rnicroele~trodes.~~~~~*~~In the present study, tran- sitions between the various cluster sizes (or types) appear to take place in a very narrow range of potential. This indicates the necessity for further research in this area in order to establish whether such transitions can be detected experimen- tally and whether these transitions are theoretically feasible.However, considering that we are dealing with an elementary system under special experimental conditions, it is reasonable to expect to observe the formation of very small critical clus- ters. Conclusions It has been shown that the use of microelectrodes enables the study of silver nucleation as well as determination of the kinetic parameters controlling the crystal growth. The main reasons why microelectrodes are so useful compared with conventional electrodes for these kind of studies are (i) they restrict nucleation to single centres, and (ii) they allow direct measurement of the initial stages of growth, leading to mea- surement of the arrival time of the first nucleus.The distribu- tion of these arrival times is, in turn, defined by the kinetics of formation of the critical nuclei, i.e. by the kinetics at the molecular level. Silver nucleation under limiting conditions is satisfactorily explained by the pure birth model which, in turn, is a limiting case of the depositiondissolution model. Under these condi- tions the sizes of the nucleus can be explained in terms of the known structure of silver. Glossary Ii Birth rate/s- pi Dissolution rate/s- t Time/s Arrival time/s Probability of forming a centre Dimensionless time Size of the critical nuclei Overpotential/V Potential sweep rate/V s-l Heterogeneous rate constant/mol cm-’ s-Standard deviation Mean References 1 M.Fleischmann and H. K. Thirsk, in Advances in Electrochem-istry and Electrochemical Engineering, ed. P. Delahay, Inter- science, New York, 1963, vol. 3, ch. 3. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1929 2 R. Kaischev and B. Mutaftschiew, Electrochim. Acta, 1965, 10, 643. 20 R. M. Wightman and D. 0. Wipf, J. Electroanal. Chem., 1989, 15, 267. 3 T. Titanov, A. Popov and Budevski, J. Electrochem. Soc., 1974, 121, 207. 21 Microelectrodes: Theory and Applications, ed. M. I. Montenegro, M. A. Queiros and J. L. Daschabach, NATOIASI, Kluwer, Dor- 4 M. Volmer, Die Kinetik der Phasenbildung, Th. Steinkopff- drecht, 1991. 5 Verlag, Leipzig und Dresden, 1939. W. K. Burton, N. Cabrera and C. F. Frank, Philos. Trans. R. 22 23 J. 0.Howell and R.M. Wightman, Anal. Chem., 1984,56,524. Ultramicroelectrodes, ed. M. Fleischmann, S. Pons, D. R. Rolin- 6 7 SOC. London, Ser. A, 1951,123,299. H. Gerischer, 2. Elektrochem., 1958,62,256. W. Mehl and J. OM. Bockris, Can. J. Chem., 1959,37, 190. 24 son and P. Schmidt, Datatech Science, Morganton, NC, 1987. S. Toschev, A. Milchev and S. Stoyanov, J. Cryst. Growth, 1972, 1S14, 128. 8 W. Lorenz, Z. Phys. Chem. NF, 1959,19,377. 25 J. P. Sousa, Ph.D. Dissertation, University of Utah, 1991. 9 M. Fleischmann and J. Harrison, Electrochim. Acta, 1966, 11, 26 M. Fleischmann, L. J. Li and L. M. Peter, Electrochim. Acta, 749. 1989,34,475. 10 11 J. O'M. Bockris and W. Mehl, J. Chem. Phys., 1957,27,818. R. D. Armstrong, M. Fleischmann and H. R.Thirsk, J. Electro- 27 28 T. E. Edmonds, Anal. Chim. Acta, 1985, 175, 122. G. Schulze and W. Frenzel, Anal. Chim. Acta, 1984, 159,95. anal. Chem., 1966,11,205. 29 P. R. Bevington, Data and Error Analysisfor the Physical Sci- 12 M. Brady and R. C. Ball, Nature (London), 1984,309,225. ences, McGraw-Hill, New York, 1989. 13 Instrumental Methods in Electrochemistry: Southampton Electro- 30 R. J. Barlow, Statistics: A Guide to the Use of Statistical chemistry Group, ed. R. Greef, R. Peat, L. M. Peter, D. Pletcher Methods in the Physical Sciences, Wiley, New York, 1989. 14 and J. Robinson, Wiley, New York, 1985. G. Gunawardena, G. Hills and I. Montenegro, J. Electroanal. Chem., 1982,138,241. 31 32 R. A. Fischer, Contributions to Mathematical Statistics, Wiley, New York, 1950. 0. Grinder, Dissertation, Royal Institute of Technology, Stock- 15 A. Milchev, B. Scharifker and G. Hills, J. Electroanal. Chem., holm, 1977. 1982,132,277. 33 L. J. Li, M. Fleischmann and L. M. Peter, Electrochim. Acta, 16 G. Gunawardena. G. Hills and I. Montenegro, Electrochim. Acta, 1978, 23,693. 34 1989,34,459. L. J. Li, S. Pons and M. Fleischmann, unpublished results. 17 R. M. Wightman, Anal. Chem., 1981,53, 1125A. 18 19 M. I. Montenegro, Port. Electrochim. Acta, 1985,3, 165. S. Pons and M. Fleischmann, Anal. Chem., 1987,59, 1391. Paper 3/07212F; Received 6th December, 1993

ISSN:0956-5000    

DOI:10.1039/FT9949001923

出版商:RSC    

年代:1994

数据来源: RSC

摘要:

J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1931-1940 Self-diff usion and Viscoelasticity of Dense Hard-sphere Colloids David M, Heyes and Paul J. Mitchell Department of Chemistry, University of Surrey, Guildford, UK GU2 5XH Brownian dynamics (BD) simulation is used to calculate the viscoelasticity of model near-hard-sphere colloidal liquids using a continuous potential r-“ interaction between the model colloidal particles. The exponent n was varied between 6 and 72. The real and reciprocal components of the complex shear viscosity, q’ and q”, were computed via time-correlation functions under no-shear conditions using a Green-Kubo formula. Also, oscil-latory shear non-equilibrium BD was employed at finite strain amplitude in the linear-response regime. We find that the normalised stress autocorrelation function can be approximated very well by a two-parameter stretched exponential over the complete volume-fraction range.The parameters used to specify the stretched exponential and also the viscosities and long-time self-diffusion coefficients are quite sensitive to the value of n at a chosen volume fraction. The Newtonian viscosity decreases and the long-time self-diffusion coefficient increases with the softness of the interaction, in agreement with experiment. The value n = 36 gives best agreement with the experimental data, and therefore appears to be a good ‘effective’ interaction which we suggest includes the time-averaged effects of the many-body hydrodynamics to some extent. The state dependence of the derived spectrum of relaxation times is determined.As for experimental systems, the complex viscosity scales with the dimensionless (‘ longest’) relaxation time, Doz,/a2, where a is the radius of the particle and Do is the self-diffusion coefficient in the zero-density limit. Also in the intermediate frequency regime 20 <a2w/Do < 200 we find that both the real and imaginary parts of the complex shear vis- cosity decay as ca. o-’l2in agreement with experiment. The rheology of stabilised dispersions has been widely studied by experiment, simulation and statistical mechanics theory. An area of particular interest recently is their visco- elastic behaviour which is conveniently characterised by a complex viscosity composed of real and imaginary parts, q’ the in-phase and q” the out-of-phase response, respectively. The linear viscoelasticity for stabilised dispersions has been measured experimentally by van der Werff et al.’ in the volume-fraction range 0.10 < 4 < 0.60.They discovered an inverse-square root dependence of both q’ -q’(c0) [where q’(c0) is the infinite oscillation frequency viscosity] and q” in an intermediate frequency regime ca. 20 < a20/D, < 200 where a is the radius of the particle and Do is the self- diffusion coefficient in the zero-density limit. Cichocki and Felderhof2 and de Schepper et aL3 derived analytic expres- sions for the complex viscosities which also have a high-frequency ca. o-’/* limiting behaviour. Here we report the results of BD simulations of model sta- bilked dispersions that explore the viscoelastic response of model colloidal liquids consisting of spherical particles, as a function of the volume fraction of solids and the extent of repulsiveness of the interaction potential.This is the first time that this has been performed for a simple generic colloidal interaction using a well established representation of colloidal dynamics (the Langevin equations of motion without many- body hydrodynamics) which is often used as a reference rep- resentation of the colloidal liquid state. Wide ranges of solids fraction and particle softness are considered. The motivation is to discover how this simple model system (with a minimum of parameters and assumptions) behaves, before extending the model in the future to include hydrodynamic forces.On the technical side we propose two new methods to cal- culate the linear dynamic viscosities for model colloidal systems. In the first method, the temporal evolution of the shear stress fluctuations in an unsheared sample are used to calculate a time-correlation function, C,(t), which is numeri- cally identical to the shear stress relaxation function mea- sured in linear stepstrain rheology experiments. The Fourier transform of C,(t) gives the complex moduli and shear vis- cosities. This approach has the advantage of applying no shear rate or strain to the system, so the response function is guaranteed to be in the linear-response regime, which can be time consuming to establish by the direct application of a shear strain profile to the system in experiment or simulation.The second route to the dynamic viscosities is to use non- equilibrium BD. The model dispersion is subjected to a homogeneous oscillating shear flow field with an exp( -iot) time variation, in analogous fashion to the operation of oscil- latory shear rheometers where the liquid is excited into a non-equilibrium state by an externally applied strain field. Brownian Dynamics Simulations The BD method is the same as we have employed in previous work4 and calculates the trajectories of N model colloidal particles from N coupled irreversible equations of motion. If each colloidal particle has mass m, index i and is at position denoted by ri within the (cubic) BD cell, then the position update algorithm is based on, mL’i = Fi + Ri -5i.i (1) where F is an effective interaction force calculated from Fi = -v c V(Jri-rjl) (2) j# i where V is an effective non-bonding chemical interaction between colloidal particles i and j, which is assumed to be pair-wise additive.R is the Langevin random force and 5 is the friction coeficient. The timescale for momentum relax- ation of the colloidal particle, called the Brownian relaxation time, is zB= mt-’ = m/3nrsqs, where qs is the viscosity of the suspeqding medium. A finite difference integration of eqn. (1) is used to evolve the assembly of particles through time and space, r,(t + At) = r,(t) + [F,(t) + R,(t, At)] x At/( + P(t)Atr, (3) where At is the time-step.The last term in eqn. (3) allows for the inclusion of a time-dependent linear shear-flow field in 1932 the suspending medium. In the present model we have omitted many-body hydrodynamics, to discover the predic- tive ability of a simple 'reference' model of a colloidal system. It is interesting to discover the properties of this (still much used) description of a colloidal system for the volume-fraction and interaction-potential dependence of the complex vis- cosity. The colloidal particles interact through an inverse- power potential, V(r)= E(CT/T)" (4) where o is the equivalent hard-core diameter of the model colloid molecule and r is the separation between the centres of two model particles.We set E = k, T and 6 < n < 72. This interaction would represent a stabilised colloidal particle and for n greater than ca. 36 is sufficiently hard to be equivalent to a hard-sphere system for many purposes. The reduced number density of particles, p = No3/V and the solids volume fraction, 4 = 7rp/6. The frequency, a,is made dimensionless by multiplying by a characteristic structural relaxation time, z,, which is the time it takes a colloidal particle at infinite dilution to diffuse a distance a = 012, Z, = 3TCCT3z7s/4kBT = a2/D, (5) For colloidal particles of diameter in excess of 0.1 pm we have, tB< t,, so we can choose a time-step h such that T, 4 h 4 t,. The time-step in the simulation, h, is chosen with h = 622D,, where 6, is the standard deviation of the random displacement.The value of 6, is chosen as large as possible within the bounds of algorithm stability and accuracy (determined empirically from a trial series of simulations), The values of z, and h depend on the choice of Z, (an input parameter), although the ratio of h/z, (the efficiency of trav- elling through configuration phase space) is independent of the value of zB. We typically chose the value zB = 0.316 x 10-3~(rn/~)1/2and 6, = 0.0090. The value of hz, < in these potential-based reduced units is comparable to the values chosen by other workers, (e.g.ref. 6 and 7). For the inverse-power potentials, the interaction energy, pressure and mechanical properties are simply related.The average interaction energy per particle, u, is given by rN where rij= ri -rj. Then the osmotic pressure is given by, P = np(u)/3 (7) and the infinite-frequency shear rigidity modulus in the zero strain amplitude limit, G, = G'(w .+ m), is given by G, = (n2 -3n)p(u)/15 (8) making use of the formula of Zwanzig and Mountain8 Simi- larly for the infinite-frequency bulk rigidity modulus in the zero-strain amplitude limit, K, = K'(o-, co),is given by, K, = (n2 + 3n)p(u)/9 (9) The components of the stress tensor, CT,are needed to compute the dynamic rheology uaS= (raijrpij/rij)Vij (10)N i=l j=i+' Eqn. (10)only includes the contribution to the stress from the direct interactions between the colloidal particles. It does not include the solventsolloidal particle or solvent-mediated hydrodynamic forces between the colloidal particles.The stress in this free-draining level of approximation is treated at J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 a pair-wise additive level. For a more realistic representation including hydrodynamic interactions, the relationship between particle stress and macroscopic properties is much more complex (see e.g. ref. 9). In eqn. (lo), the total stress has a kinetic term pk, TI, where I is the identity matrix. On the timescale of the motion of the Brownian particle, they have reached thermal equilibrium owing to the large number of collisions with the solvent molecules. The magnitude of the kinetic component of the stress is negligible compared with the potential term [eqn.(lo)]. Therefore we do not need to consider this component. Also, as we are mainly interested in the off-diagonal elements of the stress tensor, we do not need to include it because the kinetic contribution to the stress is a diagonal matrix on the rheological timescale. Self-diffusion We also compute the self-diffusion coefficient from the 'local' slope of the mean-square displacement, W(t). For an arbi- trary particle, index 1, we have, W)= ml(t) -r1(0)l2> which is averaged over all particles and dWD(t) = -dt The short-time diffusion coefficient, Ds is D(t)as D(t + 0) and the long-time diffusion coefficient, DL is D(t + m). Let us define the relaxation function, CD(t), D(t) -DL cD(t) = Ds -DL (14) It follows from the Smoluchowski equation that the dimen- sionless function CD(t)can be expressed as CD(t)= rP,(u)exp( -ut) du (15) with a spectral density normalised to, [PD(u) du = rPD(u) d ln(u) = 1 (16) A two-pole approximation to the spectral density valid as p --+ 0 derived by Cichocki and Felderhof" if we neglect hydrodynamic interactions is (17) We make a comparison between this analytic [i.e.eqn. (17) substituted in eqn. (15)] and the BD simulated CD(t)below. Shear-stress Time Correlation Function A linear-response expansion of the position Langevin equa- tion used in this work'' leads to a Green-Kubo expression for the linear shear viscosity in terms of the shear-stress time autocorrelation function, C,(t)defined as where indicates an average over time origins in eqn.(-0.) (18). This method was first used by Levesque et who aplied it to molecular liquids. The infinite-frequency linear shear modulus is given by G, = C,(O). In an unsheared system, the stress fluctuations of all the off-diagonal elements of the stress tensor are equivalent. Therefore, we improved the statistics of C,(t) by considering ox,, ox2,and oy2separa-tely in eqn. (18); and then averaging the three at the end of J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 the simulation. (The stress tensor is symmetric so gap= osol.) The function, C,(t),is exactly the same function as the stress- relaxation function derived from step-in-strain experiments taken in the linear strain limit.The time correlation functions have to extend typically for ca. 20000 time-steps to ensure decay of the function to zero. In order to reduce the com- puter memory requirements, the correlation function was constructed in a piece-wise fashion from three separate corre- lation functions with time origins started (and with a resolution of) every 1, 10 and 100 time-steps. These three correlation functions extended for progressively longer in time, and non-overlapping pieces were merged for the purpose of subsequent analysis and presentation. The number of entries in the histogram used to calculate the time correlation function decreases as time increases. Nevertheless the statistics are reasonable for the ca.500000 time-steps covered for each production simulation. The present BD model’ only incorporates the thermody- namic interactions between the colloidal particles and ignores the many-body hydrodynamic solvent-mediated forces. A measure of the contribution of many-body hydrodynamics to the total viscosity is given by the experimental value of the viscosity in the second Newtonian plateau, qm, which is entirely hydrodynamic in origin. If we assume that the hydro- dynamic contribution to the viscosity is equal to q, at all shear rates, then the Green-Kubo formula in the present model gives the difference between the Newtonian viscosity, qo (the zero-shear-rate limit) and qm. qo is related to C,(r) through vo = dt (19)v, + p) where we need to take a value for q,,, from another source.This is not as serious as it may first appear because often the main interest, certainly for dynamic rheology, is the deviation in the behaviour of the sample from the qocvalue. To be rig- orous, adding the high-frequency or high Peclet limiting vis- cosity in this fashion should be accompanied by a correction to the Stokes drag coefficient as the short-time diffusion coef- ficent is strongly affected by hydrodynamic interactions.’ It is convenient to render the colloid liquid’s viscosity in dimen- sionless units, by dividing them through by the host liquid’s viscosity, to form the relative viscosity, qro = qo/qsand qrL = q,/qs. The complex dynamic viscosity, q*(u)= ((0)+ iq”(o) (20) is related to the dynamic shear modulus, G*(o) through G*(o) = wq*(w).We have = q‘(c0) + FI i C,(t)exp(-icot) dt 1 (21)~’(0) r and q”(w)= 9I i rCs(t)exp(-iot) dt I (22) where q‘(m)= qm which is hydrodynamic in origin.Cs(t)can be represented by a superposition of exponential relaxation functions, C,(t*) = G, J:’P,(~)exp( -t*/z) dt (23) with the normalisation condition j2 P,(z) = 1. Then ~*(c;o)= f(m) + G, where H(z)= zP,(z). Cichocki and Felderhof” give a limiting form for P, as p +0 in the absence of hydrodynamic inter- actions, 1 9U3l2 n (1 -4~)~Ps(u)= -+ 9u(l -u)’ where t = 2t*U2/9D0. If we substitute eqn. (25) in eqn. (23) we have an analytic prediction for C,(t). Applied Oscillatory Shear The second method we use to calculate the viscoelasticity is to apply an oscillating shear strain field to the colloidal par- ticles.The contents of the BD cell are sheared homoge-neously with a time-dependent strain, y(t) over n cycles, y(t) = ’Jo cos ot (26) where yo is the strain amplitude. The analytic expressions for the dynamic viscosities are q’ = -i r2naIw a,,(t‘) dt’ cos(ot’) nT0‘J 0 and ql = -a,,(t’) dt’ sin(wt’) n=yo‘J0 The number of cycles in a simulation of a specified number of time-steps and oscillatory frequency was adjusted to be a whole number by scaling down the time-step. The number of cycles varied with the value of oz,.The number was adjusted empirically to produce reasonable statistics for the average quantities. For example, for COT, > 100 we chose n = 200 and for oz,< 0.1 we used n = 10.Limitations on the availability of computer time necessitated a smaller number of cycles at low frequency, as each cycle takes an increasing number of time-steps as the frequency decreases. Therefore, the sta-tistical uncertainty of the cycle averages was larger as the fre- quency was lowered. In the following section we use the time-correlation method and direct application of a shear flow to explore the linear viscoelasicity of these model colloidal liquids over a wide volume-fraction range. We focus on the effect of the softness of the interaction potential, which is adjusted via the value of the potential exponent, n, in eqn. (4).The softness in experimental systems can be adjusted by varying the relative diameter of the (hard) core and the stabilising barrier (shell).13 In order to cover as much of parameter space as possible, we investigated 4 over essentially the whole liquid volume fraction range for n = 36.Also at the higher 4 values we explored the effects of varying the value of n between n = 6 and n = 72. Results and Discussion A series of equilibrium BD simulations was carried out at a range of volume fractions using N = 108, 256 and 500 model colloidal particles in the simulation cell. Simulation details and the computed properties of the system are given in Table 1. Above a volume fraction of 4 x 0.4 there is an increasing system-size dependence, as expected because configurational phase space becomes more structured with increasing 4.Only those configurations consistent with a finite periodic system are allowed, which produces an increasingly more unrepresentative average with increasing 4. Generally the internal energy (and other derived properties) and the vis- cosities decrease with increasing N. The self-diffusion coefi- cients increase with N. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Thermodynamic, mechanical and Newtonian transport coefficients by Green-Kubo n 4 tsiml'r u KD ~ 108 36 0.075 455 0.0322 27.98 0.880 0.365 0.016 0.038 108 36 0.115 143 0.0563 48.97 0.799 0.979 0.044 0.073 108 36 0.150 444 0.08 19 71.35 0.749 1.857 0.069 0.115 108 36 0.250 113 0.1892 166.3 0.581 7.155 0.295 0.366 108 36 0.350 63 0.3893 346.3 0.402 20.6 1 0.995 1.17 256 36 0.400 330 0.5443 485.7 0.29 1 32.93 1.90 2.26 108 18 0.427 195 1.3359 272.5 0.279 19.61 1.90 3.34 -108 36 0.427 397 0.6572 590.1 0.253 42.45 3.05 -256 36 0.427 377 0.6574 589.9 0.235 42.46 2.40 -500 36 0.427 170 0.6599 592.3 0.230 42.62 2.95 500 6 0.450 74 3.7603 71.6 0.455 3.88 0.54 4.79 -500 12 0.450 180 2.1908 192.9 0.285 13.56 2.16 -500 18 0.450 202 1.5434 3 17.3 0.227 23.88 2.75 -500 36 0.450 207 0.7750 700.0 0.193 52.75 3.97 -500 72 0.450 1228 0.37 18 1323 0.177 105.83 >2.70 500 6 0.472 159 4.0706 78.84 0.43 1 4.40 0.61 7.00 -500 12 0.472 175 2.4609 218.91 0.254 15.97 2.54 -108 18 0.472 60 1.7518 362.33 0.195 28.43 3.40 -500 18 0.472 164 1.7631 365.00 0.196 28.61 3.6 -108 36 0.472 153 0.8939 81 1.7 0.154 63.82 7.10 -256 36 0.472 151 0.8945 811.8 0.1 50 63.86 7.05 500 36 0.472 217 0.90 16 819.1 0.157 64.37 5.7 --500 72 0.472 46 0.4392 1567.1 <0.136 131.12 >6.69 ~___ KD is the value of qro -q,, from the Krieger-Dougherty form~lae.'~ The mean-square force in the a-Cartesian direction is (F:).The thermodynamic and mechanical quantities are accurate to ca. 1%, while the viscosity and self-diffusion coefficients have a larger error which, at the highest solids fractions, is ca. 10%. Only the interaction component of the thermodynamic and mechanical quantities are given. tsim is the length of the production simulation. We first consider the variation of the self-diffusion coeffi- increases, the particles do no have as far to diffuse before they cient with volume fraction and interaction softness.In Fig. 1 interact strongly with their first coordination shell, which is shown the volume-fraction dependence of the diffusion gives rise to the start of the 'long-time' diffusion regime. In coefficient relaxation function, CD(t)for the n = 36 potential. Fig. 2 we explore the effect of softness of the interactions on The function decays more rapidly with time with increasing the form of CD(t),for the q5 = 0.450 state and varying volume fraction. At the lowest volume fraction considered, 6 d n d 36. Although broadly similar, there is an indication 4 = 0.075, there is very good agreement between the pre- of a more rapid decay with time for the higher n.The Iong- diction of eqn. (15) and the simulated curve up to times time self-diffusion coefficient becomes larger with increasing <0.2a2/D, at least. The agreement is not so good at longer softness of the potential (decreasing n) as shown in Fig. 3 for times, although it must be borne in mind that there is greater 4 = 0.472 states. The soft potential facilitates cooperative statistical uncertainty in the simulation results at low volume motion of the particles and consequently enhances the long- fractions. The simulated curve is a little above the theoretical time self-diffusion coefficient. The slower decay of the softer prediction, which we note is itself an approximation. The (smaller n) potentials especially for r > 0 leads to greater simulation CD(t)decays more rapidly with time as the solids coupling between the trajectories of the particles and hence a fraction increases, which we attribute to the decreasing dis- larger mobility.The particle self-diffusion coefficient tance between nearest neighbours. As number density decreases as the particles aproach closer to the hard-sphere limit (i.e.as n + co). 1.4 1.2 1.o 1.0 c 0.8 n 0.8Bcv 0 0.6 c, 0.4 0.2 0.0 -0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 tD,a-2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Fig. 1 CD(t)defined in eqn. (14) as a function of volume fraction, C$ tD,a-2for n = 36. 4, from top to bottom, 0.075, 0.115, 0.150, 0.250, 0.350, 0.427, 0.450, 0.472, 0.490 and 0.527. (-) is the C$ -0 prediction Fig.2 As for Fig. 1, except that CD(t) as a function of n for given by eqn. (15). 4 = 0.450 are shown. n, from top to bottom, 6, 12, 18 and 36 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.1 1 I I I I I 1.0 c 4 0.9 0.8 0.7 5c 0.6 a‘ 0.5 0.4 0.3 0.2 ’ 0.1 4 1 I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 tD,ir2 Fig. 3 D(t)/D, for 4 = 0.472 for n, from top to bottom, 6, 12, 18, 36 and 72 We now turn to the stress relaxation function C,(t),which unlike CD(t)represents a property of the whole system rather than being a single-particle function. Fig. 4 shows the C,(t) for n = 36 at a range of volume fractions. The function becomes more long-lived with increasing 4, in contrast to cD(t)which decays more rapidly with increasing volume frac- tion.Therefore there is a qualitatively different density depen- dence between cD(t) and C,(t).In general, and particularly at low 4, CJt)decays significantly more rapidly with time than the corresponding CD(t).This is reasonable as the shear-stress pair interaction decays rapidly with pair separation. There- fore the particles do not have to move as far to cause a decay of the same magnitude for C,(t) when compared with cD(t). The function C&) essentially measures the extent of displace- ment of the particles, whereas C,(t) also measures this dis- placement but, in some sense, multiplied by the rapidly decaying pair-interaction function, V(r). The displacement trend inherent in C,(t) is ‘scaled’ by the interaction potential to form C,(t).The great difference in timescales between cD(t)and C,(t), produced by the simulations at low volume fractions, is not reflected in the corresponding analytic approximations to these two decay functions based on eqn. (15) and eqn. (23, respectively. The distinct difference in timescales does not 1.0 I I 1 I I I 0.8 i\ 0.6 h ;c 0.4 Ll 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 tD,a-2 Fig. 4 C,(t) for 4, from top to bottom, 0.075, 0.115, 0.150, 0.250, 0.350, 0.427 and 0.500. N = 108 in each case. (-) is the 4 -+ 0 pre- diction given by eqn. (25) used in eqn. (23). 1935 appear to be accounted for in the analytic approximations. The analytic model is essentially a zero 4 expansion.Using eqn. (25), the analytic C,(t) decays significantly more slowly than the lowest volume fraction BD computed function (for 4 = 0.075). The discrepancy is somewhat reduced as volume fraction increases but the volume fraction is not a feature of the theory. The C,(t) can be represented very well by a so-called ‘fractional’ or ‘stretched’ exponential [except at very short times t/z, < 0.2 x where this analytic form underesti- mates the simulation C,(t)]. The stretched exponential has the analytic form Two examples of least-squares fits to the BD C,(t) using the analytic form of eqn. (29) are given in Fig. 5 for two extreme values of n at 4 = 0.472. The figure shows the BD and fitted C,(t) for n = 6, the softest potential considered and n = 72, the most steeply repulsive interaction used.Bearing in mind that there are only two disposable parameters the level of agreement between the fitted and BD function is very good in both cases. This stretched exponential function is useful for post-simulation analysis of the viscoelastic behaviour of the system, because of its comparative analytic simplicity. In Table 2 we present the values of z’ and for the different systems. We also give in the table the mean relaxation time z = z‘T(l/p)/p for the normalised function The simulations show that the mean relaxation time increases with volume fraction in agreement with the experimental data on sterically stabilised silica particles by van der Werff et a/.’ In Fig.6 we show the function C,(t) for a range of softness parameters (n) at 4 = 0.450. The function decays more rapidly as n increases i.e. the potential becomes more repulsive. The interaction becomes more short-ranged with increasing n so that the particles do not have to move apart as far before their contribution to the stress is significantly reduced. This causes C,(t) to decay more rapidly for the high n, steeper potentials. Both C,(t) (Fig. 2) and C,(t) (Fig. 6) manifest the same qualitative variation with n. In the 4 +0 limit the stress relaxation function C,(t) appears to be relatively insensitive to the value of 4 (Fig. 4 1.o 0.8 0.6 h%0.4 c, 0.2 0.0 I I I I I-0.2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 tD,a-2 Fig.5 Normalised stress relaxation autocorrelation function from simulation, (-) and (---), and the stretched exponential least- squares fit, (-----) and (. . .), for the states (a) n = 6, 4 = 0.472, = 0.653 and T’ = 0.0414, and (b) n = 72, 4 = 0.472, 0= 0.238 and Z‘ = 0.0009066. N = 500. 1936 Table 2 Parameters of the stretched exponential least-squares fit to the simulation C,(t)from eqn. (29) 256 36 0.400 0.331 0.00372 0.0228 108 18 0.427 0.45 1 0.01 7 194 0.0423 108 36 0.427 0.330 0.00420 0.0292 500 6 0.450 0.632 0.0427 0.0602 500 12 0.450 0.473 0.0258 0.0578 500 18 0.450 0.433 0.0186 0.0502 500 36 0.450 0.315 0.00465 0.0341 500 72 0.450 0.29 1 0.000830 0.00841 500 6 0.472 0.653 0.04 14 0.0563 500 12 0.472 0.485 0.0306 0.0648 108 18 0.472 0.425 0.0195 0.0553 500 18 0.472 0.416 0.0191 0.0568 108 36 0.472 0.287 0.00369 0.0398 256 36 0.472 0.283 O.Oo409 0.0465 500 36 0.472 0.309 0.00472 0.0373 500 72 0.472 0.238 0.0009 1 0.0198 The mean relaxation time is defined in eqn. (30).and Table 2). In this region the parameters for the stretched exponential representation of C,(t) have values, = 0.360 & 0.005 and z’ = 0.0035 f0.001 for 4 < 0.25 and n = 36. There is a gradual decrease in the value of p and increase in z’ with increasing volume fraction. The functional form of eqn. (29) can be written alternatively in the relaxation time spectrum form of eqn. (23). The distribution of relax- ation times Ps(~)or equivalently H(z)for several representa- tive choices of p and z’ are presented in Fig.7 using the numerical procedure developed in ref. 15 specifically for the stretched exponential. The figure shows that the stretched exponential produces a relaxation-time distribution function which is broad at the low z end but has a relatively sharp cutoff at the high z end, which becomes sharper as j?+ 1. (At fl = 1 we recover a process with a single relaxation time, so the relaxation spectrum takes the form of a S function.) There is a slower decay of H(z) in the z +0 limit. The spectrum of relaxation times becomes broader as /I decreases with increasing 4. If each of these relaxation times is identified with a particular microscopic mechanism for stress release, then this trend indicates that there are many more significant dynamical processes that come into existence at the higher volume fractions.The growth in population of relaxation times is especially at the lower z end. The rapid decrease in C,(t)occurs in the time domain of the transition between the short-time and long-time diffusion coefficients. 1.0 I I 1 I I I I 0.8 0.6 hc % 0.4 u 0.2 0.0 -0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 rD,a-2 Fig. 6 Normalised stress relaxation autocorrelation function from simulation for n, from top to bottom, 6, 12, 18 and 36 at 4 = 0.450 and N = 500 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 -1 n h c) v3 -7 -6 -5 -4 -3 -2 -1 log(7&la2) Fig.7 Stretched exponential relaxation time spectrum distribution function for the 4 = 0.450 state values: (a) = 0.291 and z‘ = 0.000840, (b) p = 0.473 and 7’= 0.00258 and (c) fi = 0.632 and Z’= 0.04266 In Table 1 we give the integrated shear viscosity from eqn. (19). The states at # = 0.472 and above manifest a strong N dependence with systems in excess of N = 500 considered essential. The # = 0.472 state for example has an N depen-dence with the N = 500 state point qro -qrm value being 5.7 as opposed to 7.0 for N = 256. These high volume fractions and relatively small system sizes have a noticeable N depen-dence for the viscosity. The trend is for the shear viscosity to increase for the smaller N systems. This is reasonable as the trajectories of the particles through phase space are more frustrated by the constraints imposed by the periodic bound- ary conditions.Pathways to lower stressed states have higher activation energies than for larger N systems, and this causes the viscosity to be higher. The relaxation mechanisms are slower, and they have longer stress relaxation times and therefore larger viscosities. The N dependence of the viscosity decreases with volume fraction. For example, Table 1 reveals that for n = 36 the 108 to 500 values of qrO-q,, at $ = 0.427 are 3.05 to 2.95, respectively, compared with 7.10 to 5.7 at 4 = 0.472. Therefore there is a small N dependence at $ = 0.427 but a more significant effect at 4 = 0.472. Over the complete volume-fraction range the following analytic expressions fit the experimental relative viscosity data of near hard-sphere dispersions (summarised in the Krieger-Dougherty expression^'^) quite well.The expressions are, and, qroo= (1 -4/0.71)-2 Table 1 shows that the Krieger-Dougherty expression for qrO -qrm, give values that are quite close to the simulation results for n = 36 (obtained by numerical integration of the time-correlation functions). This is noteworthy as the model does not include many-body hydrodynamics in the equations of motion, which is often considered to be important. Clearly these interactions are present in the real system. Nevertheless, these data indicate that at these high volume fractions, the simple Langevin dynamics embodied in the computer algo- rithm used here, coupled with the continuous interaction V(r),in some ‘mean-field’ sense reproduces well the rheology of the real systems.More significantly, perhaps, is that it sug- gests that the contribution of the many-body hydrodynamics to the non-Newtonian viscosity is relatively insensitive to J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 shear rate. At high volume fraction there is probably appre- ciable screening of the hydrodynamic interactions, so that their effective range is likely to be only several particle diam- eters at most. At lower volume fractions, this screening is likely to be less effective, so that the current hydrodynamics- free approach should be a poorer representation of the system and a mean-field hydrodynamics treatment less ade- quate.This is corroborated by the viscosities given in Table 1. Best percentage agreement with experiment using n = 36 is for the high 4 states, and it becomes less satisfactory as #) +0. In Table 1 we see that the interaction energy, mean-square force, elastic moduli and viscosity increase with n. This trend for thermodynamic and mechanical properties is a feature of inverse power potentials, for which it has recently been shown that u cc n-’.16 The behaviour of the viscosity is con- sistent with the experimental evidence of Mewis et all3 who showed that for the same effective hard-sphere particle diam- eter the Newtonian viscosity (and also the so-called infinite shear rate viscosity) increased with the particle ‘hardness ’.These experimental results are consistent with the trends exhibited by our model systems. As n increases, then the New- tonian viscosity increases. The experimental colloid particles had a varying stabilising layer thickness. Larger colloidal particles with the same stabilising shell thickness are harder when distance is scaled by the its mean diameter. Mewis et al. noticed that larger particles at the same mean volume frac- tion exhibited higher relative Newtonian viscosities.’ The simulated viscosities decrease with decreasing n. We now consider the oscillatory shear simulations carried out using the non-equilibrium BD method. We are interested in the linear response region here, so it is important to deter- mine the maximum strain amplitude that can be used (which will depend on frequency) to remain, to a good approx- imation, within the linear response regime.In Fig. 8 we explore the strain amplitude dependence of the storage modulus, G’, for a 4 = 0.472 state. Simulations were carried out over a range of yo between 0.01 and 1.5 at a series of fixed frequencies, ma2/Do= 10100 and 1000 which covers the practical range of interest. Fig. 8 shows that the response is linear for yo < 0.1 at all frequencies, and there is a reduction in the dynamic modulus or shear strain ‘softening’ for higher amplitudes. There does not appear to be a significant fre- quency dependence to the value of yo above which there is a significant departure from linear response.A value of yo = 50 I -kl 30 --0” 25 0+ + + ++++ ++ +20:; 1 15 -+ O0 0 10 t 1 Fig. 8 Strain amplitude, yo, dependence of the storage modulus of an N = 108 and 4 = 0.472 system at the frequencies, oaZ/D,: 0,10;+, 100 and 0,lOOO 1937 0.02 was chosen for the oscillatory shear simulations reported below to ensure linear response. It is interesting to explore the time-average response of a range of physical properties to the applied oscillatory shear as the strain amplitude increase into the non-linear response region. Fig. 9 shows the amplitude dependence of the average interaction energy per particle, u, for the 4 = 0.472 and n = 36 state point. We note that in the strain amplitude regime where there is shear softening of G’ (ie.yo > 0.1) there is an associated dramatic increase in u, which becomes larger in magnitude with higher frequency. This is reasonable, as an increase in strain amplitude and frequency will both indepen- dently cause a departure from the equilibrium state. We expect their combined effect to be roughly additive. The larger yo and o then the further from equilibrium on average is the state, which causes an increase in the internal energy u. In Fig. 1qa) we show the corresponding plot of the first normal pressure difference, P,, -P,, where P = --b. This manifests a quite distinct dependence on yo when compared with u. At the two highest frequencies considered oa2/Do= 10 and 100 there is a negative dip in the function between 0.5 < yo < 1.0 before going positive and rising steeply with yo at yo x 1 (more steeply than u).We note that the region of rapid ascent is delayed to larger amplitudes than in the case of internal energy. The second normal pressure difference P,, -P,, and its dependence with yo at three fixed frequencies is given in Fig. lqb). These data follow very much the same trend as the first normal pressure differences of Fig. lqa). The complex viscosities obtained from the stretched expo- nential fits to the time-correlation functions using eqn. (21) and eqn. (22) are statistically indistinguishable from those obtained using the direct application of an oscillating shear flow field for oa2/Do< 1OOO. At higher frequencies systematic differences appear which arise from the failure at very short times of the fitted function to agree with the simulated C,(t).The frequency domain of oa2/Do> loo0 is, however, not of much practical interest. The real and imaginary parts of the relative complex viscosity for two widely separated volume fractions are given in Fig. 11. The qf and qf’ for n = 36 shift to lower frequencies as 4 increases, in agreement with the experimental results of van der Werff et al.’ In Fig. 12 we present the real and imaginary parts of the relative complex viscosity for two values of n at the same volume fraction, 4 = 0.472. As the interaction becomes softer (i.e. n decreases) the peak in the imaginary part of the viscosity shifts to higher frequency and becomes sharper (associated with an increase in p, shown in Table 2). The corresponding real part also I I3.0 1 2.8 2.6 2.4 2.2 2.0 U 1.8 1.6 1.4 1.2 -2 -1 0 1 log Yo Fig.9 Variation of the time average value of u with strain ampli- tude at frequencies as in Fig. 8 for the 4 = 0.472, n = 36 states 1938 12 10 8 26 I h kk 4 0 o..., 1-2-‘ -2 -1 0 1 log Yo 35 I I I I 0 30 (b)25 t i0 0 0 0 I I-10 ‘ -2 -1 0 1 log Yo Fig. 10 Variation of (a)the time-average value of P, -P,, and (b) P,, -P,, with strain amplitude at frequencies as in Fig. 8 moves to higher frequency. Although we are unaware of any experimental studies that would confirm or deny this trend, it is intuitively reasonable in the same sense as we rationalised the softness dependence of the Newtonian viscosity.The softer particles (but maintaining the same effective hard- 1.o 0.9 > z 0.8b.$ 0.7 0.6 c -Y x 0.5 8E-v-‘e10.150.4 0 E* 0.3 .-> -x 0.2 ?2 0.1 0.0 -3 -2 -1 0 1 23 4 0.00 0.05 0.10 0.15 0.20 log oa2/Do (w a2/D,)-’I2 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 00 0 0 0 0 0 ~ O+++&+0.4 +, 1 -3 -2 -1 0 1 23 4 log oa2/Do Fig. 12 As for Fig. 11, except we compare two systems with differ- ent n (0, +, n = 6; 17, x, n = 36) at the same solids fraction, 4 = 0.472 sphere diameter, a) possess interactions that facilitate coop- erative motion.Therefore it is necessary to go to a higher frequency to cause the same extent of viscoelastic response as for harder (i.e. higher n) particles. Workstation movies of the simulation cell under shear distortion in the linear response regime do not reveal any restructuring (e.g. long-range order) during oscillatory shear, if the affine shear distortion is extracted. The dynamic viscosities are presumably therefore primarily influenced by the competing relative timescales of the mobility of the particles within their essentially equi- librium structures, and that of the oscillatory shear (ca. 2n/o). Between a20/D, = 20 and 200 Fig. 13 shows that q’ decays as o-”’.Both q’ and q” can be approximated in this regime by the following analytic forms q; = q:, + A’(a*m/Do)-”2 (33) and 7:’ = A”(a2m/Do)-l’* (34) This behaviour is in agreement with the results of van der Werff et al.’ who measured the viscoelastic behaviour of 1 1 1 n 8 0.20 Fig.11 Real (upper curves) and imaginary (lower curves) part of the Fig. 13 Normalised real part of the relative complex shear viscosity relative complex shear viscosity as a function of frequency for the as a function of (aZw/D,)-’/2for n = 36. 0,4 = 0.350, calculated by n = 36 model colloidal systems at the volume fractions: 4 = 0.150 Fourier transformation of the correlation function. +, 4 = 0.350; 0, (0,+), N = 108 and q5 = 0.450 (0,x), N = 500 0.427 and x ,0.472, calculated by the non-equilibrium BD method. J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 sterically stabilised silica spheres. It is important to note that at very high frequency, there is always a departure from this relationship. This is essential in practice as a q' decaying as at higher frequency implies a G' that grows as 01/2 which is physically unreasonable as this is equivalent to G, -,co for a -,00. This is understandable, at least for a system without hydrodynamics, because this limit represents an affine deformation of the microstructure by shear, which is impossible for rigid spheres at contact. The dimensionless constants A' and A" in eqn. (33) and eqn. (34), respectively, have values 1.8 & 0.1, 3.6 & 0.2 and 5.5 & 0.5 at the volume fractions # = 0.35, 0.427 and 0.472, respectively, for n = 36.The experimental values at these volume fractions are approximately double in magnitude. For 4 = 0.35, 0.427 and 0.472 these parameters have the values 3.5 & 0.5, 9.0 f1.0 and 30 f5. The slope of the linear regime of the complex viscosity with LO-'/'is sensitive to the softness parameter n. The dependence of the imaginary component of the complex viscosity with n is presented in Fig. 14. As the value of n diminishes the dimensionless constants A' and A" decrease. In this case, at 4 = 0.472 there are A" = 1.3, 3.7 and 7.8 for n = 6, 12 and 72, respectively. In Fig. 15 we show the imagin-ary component of the complex viscosity with 4 for n = 36 8T 0.20 + :-I0 ox 0 8"$ 0.15 0 +A 0 X r-+ O O *exe: 0.10 0.05 10.00 0.20 0.00 0.05 0.10 0.15 (w aZ/D,)-!2 Fig.14 Normalised imaginary part of the relative complex shear viscosity for the model colloidal systems as a function of (u~~/D,,-''~at 4 = 0.472 for n: 0,6; +, 12; 0,18; x, 36 and A, 72, using the correlation function method 0.35 , 1 a 0.30 -a + A 0.25 h 8 0.20 n ++ h v 3 r-U v z .' 0.15 h 0 z +0 E-0.10 X 0.05 0.00 0.00 0.05 0.10 0.15 0.20 (w a2/D,)-'Iz Fig. 15 As for Fig. 14 except the dependence on 4 for n = 36 is presented. 4: 0,0.075; +, 0.115; 0,0.150; x, 0.250; A, 0.350; *, 0.400;0,0.427; +, 0.450; 0,0.472 and x, 0.490. 1939 using the correlation function route. We note that the linear regime for o-'I2 is compressed into a higher frequency band as volume fraction decreases. Van der Werff et a/.also show that the complex viscosity scales with a reduced frequency, oa2zl/Do where tl is the 'longest' relaxation time in their fit to the data. If we use the van der Werff values for a2zl/D, then the present simulation data also scales onto the same curve which is also statistically indistinguishable from their experimental data (see Fig. 16). The particle size and volume fraction therefore can be incorp-orated in this dimensionless scaling parameter. The values of a2z,/Doare approximately an order of magnitude larger (but roughly proportional to) the values of a2z/Do which we obtained from our simulations. The Stokes-Einstein relationship is for this system D,q,(O)/D, = 1.The ratio DLqr(O)/D0is essentially linearly increasing with volume fraction up to 6 z 0.45, before lev-elling off. This ratio also shows an approximately linear dependence with n-l, as shown in Fig. 17 for two of the higher solid fractions. This indicates that as the softness 1.o ' """""""d, ' 1 1 0.9 t >..z 0.8 8 F.p 0.7 W B+0.6 t 0 0.5 0 -fi 0.4 0 0.3 .-5 -t;; 0.2 ?? 0.1 0.0 -3 -2 -1 0 1 2 3 log wrl Fig. 16 Normalised (a) real and (b) imaginary part of the relative complex shear viscosity as a function of frequency for three volume fractions 4: 0, x, 0.150; +, A, 0.427 and 0,*, 0.490 us. m1,where Dot,/u2 = 0.08, 0.32 and 0.45, respectively. The 'bars' on the figure represent the limits of the band of experimental points from van der Werff et al.' a" 3.4--.- h 3.2 - ca'' 3.0 2.8 - 0 2.6 - 2.4 - + + 0 2.2 I ' L 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 1/n Fig. 17 DLq,(0)/Do11s. n for the simulation data. 4: 0,0.450 and +,0.472. increases (i.e. n-increases) the self-diffusion coefficient increases more rapidly than the relative viscosity decreases. Conclusions We have shown that BD computer simulation can be used to investigate the viscoelastic behaviour of model near-hard- sphere colloidal liquids up to volume fractions of ca. 0.50 using two complementary techniques (equilibrium and non- equilibrium). We have explored the influence of solids volume fraction and extent of softness of the interparticle interactions on the self-diffusion and complex viscosity.The linear stress relaxation function (stress autocorrelation function) is approximated well by a stretched exponential for all states considered. The values of the parameters in the stretched exponential depend on solid fraction and softness of the interaction potential between the particles. Despite the formal absence of many-body hydrodynamics, the simple model adopted accounts well for the experimentally observed trends in viscoelastic behaviour as solids fraction and particle soft- ness are varied. Newtonian and complex shear viscosities are also numerically quite close to the experimental values at the same volume fractions if we choose a potential index, n, close to 36.This suggests that, for the future, it could be possible to develop an essentially ‘mean-field’ treatment of many-body hydrodynamics which is suitable for simulation at high solids fraction, and which is not much more computationally inten- sive than the current model (in marked contrast with existing approximate schemes for including many-body hydrody- namics e.g. ref. 17). P.J.M. thanks the SERC and ECC International Ltd for a research fellowship (grant number GR/H80644). Computa-J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 tions were carried out on the CONVEX C3 at the University of London Computer Centre, Personal DEC Stations and a Silicon Graphics Indigo XZ workstation in the Chemistry Department at the University of Surrey. The referees are thanked for helpful comments. References 1 J. C. van der Werff, C. G. de Kruif, C. Blom and J. Mellema, Phys. Rev. A, 1989,39,795. 2 B. Cichocki and B. U. Felderhof, Phys. Rev. A, 1991,43,5405. 3 I. M. de Schepper, H. E. Smorenburg and E. G. D. Cohen, Phys. Rev. Lett., 1993,70, 2178. 4 J. R. Melrose and D. M. Heyes, J. Chem. Phys., 1993,98,5873. 5 D. M. Heyes and J. R. Melrose, J. Non-Newtonian Fluid Mech., 1993,46,1. 6 K. J. Gaylor, I. K. Snook, W. van Megen and R. 0. Watts, Chem. Phys., 1979,43,233. 7 H. Lowen, J-P. Hansen and J-N. ROUX,Phys. Rev. A, 1991, 44, 1169. 8 R. Zwanzig and R. W. Mountain, J. Chem. Phys., 1965,43,4464. 9 J. F. Brady, J. Chem. Phys., 1993,99,567. 10 B. Cichocki and B. U. Felderhof, Physicu A, 1993,98,423. 11 W. Hess and R. Klein, Adv. Phys., 1983,32, 173. 12 D. Levesque, L. Verlet and J. Kurkijarvi, Phys. Rev. A, 1973, 7, 1690. 13 J. Mewis, W. J. Frith, T. A. Strivens and W. B. Russel, AIChE J., 1989,35,415. 14 W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dis-persions, Cambridge University Press, Cambridge, 1989, p. 466. 15 C. P. Lindsey and G. D. Patterson, J. Chem. Phys., 1980, 73, 3348. 16 D. M. Heyes and P. J. Aston, J. Chem. Phys., 1994,100,2149. 17 G. Bossis and J. F. Brady, J. Chem. Phys., 1989,91,1866. Paper 4/00386A; Received 21st January, 1994

ISSN:0956-5000    

DOI:10.1039/FT9949001931

出版商:RSC    

年代:1994

数据来源: RSC