Transport in Aqueous Solutions of Group IIB Metal Salts at 298.15 K Part 5.-Irreversible Thermodynamic Parameters for Zinc Perchlorate and Verification of Onsager's Reciprocal Relationships BY ANDREW AGNEW AND RUSSELL PATERSON* Department of Chemistry, University of Glasgow, Glasgow G12 8QQ Received 13th April, 1978 The isothermal vectorial transport properties of zinc perchlorate have been measured over the concentration range 0.1-3.0 mol dm-3 at 298.15 K. Hittorf transport numbers were shown to agree with those measured earlier using concentration cells. Conductance and salt diffusion co- efficients were also measured. These data were used to calculate mobility (&) and resistance (&) coefficients using irreversible thermodynamic theory. Analysis showed that the Onsager reciprocal relationships (O.R.R.) were obeyed.Previous papers in this series have dealt with the anomalous transport properties of cadmium iodide solutions, caused by self-c~mplexing.~-~ The anomalies in the concentration dependence of the mobility coefficients of this salt are shared by other complexed halides in the group IIB metal series in particular by cadmium and zinc chlorides.6 Studies were initiated on aqueous solutions of zinc perchlorate to allow com- parisons between complexed and uncomplexed salts of this series. It is certain that complexing, of the type observed for the halide salts, is not possible in zinc per- chlorate. Raman studies on zinc halides '-lo which have provided independent evidence of complexing have given negative results for both zinc nitrate and zinc perchlorate.ll9 l2 The activity coefficients for both salts, and transport numbers for zinc perchlorate l3 show normal behaviour and certainly no evidence for com- plexation.There is, however some doubt as to the possibility of ion-pair formation. Using conductance measurements Davies and Thomas l4 found no evidence for ion association and although Dye, Faber and Karl considered the conductance anomalous, they were unable to assign the effect to the presence of the ion pair ZnClOt. Frei and Podlahova l6 assigned to this ion pair an association constant of 4.5k0.01. It seems certain, therefore, that zinc perchlorate is completely devoid of higher complexes and that ion association, if indeed it occurs at all, is of a minor nature.Transport properties of this salt may, therefore, be used in a comparison with those of complexed zinc chloride to observe directly the effect of complexing upon the irreversible thermodynamic parameters which define the transport properties of these solutions. Excellent transport number data are available from the work of Stokes and Levien l3 on concentration cells, using zinc amalgam electrodes. In this work directly measured Hittorf transport numbers were obtained in order to verify the 28852886 TRANSPORT I N AQUEOUS SOLUTIONS Onsager reciprocal relationships, (Lik = Lki). The remaining two transport para- meters, electrical conductance and volume-fixed salt diffusion coefficients were also measured to a precision of 0.05 and 0.3 %, respectively. EXPERIMENTAL Zinc perchlorate was prepared by adding a slight excess of spectroscopically pure zine oxide (Koch Eight Laboratories) to AnalaR perchloric acid (B.D.H.) and filtering off thc unreacted oxide.This solution was made slightly more acidic by dropwise addition of perchloric acid until the pH was fir 2. On cooling to 0°C crystals of zinc perchlorate were obtained. On dissolution, these crystals gave clear solutions at all dilutions. Conductance measurements of solutions prepared from different batches of crystals prepared in this way gave reproducible conductivities. The solutions were analysed for zinc using EDTA titrations, by the methods described earlier for zinc chloride.17 The precision of the method was kO.05 %. To allow conversion between molal (mol kg-l) and molar (mol concentrations, accurate density measurements were made at 298.15+ 0.005 K.Pyknometers of capacity 30 cm3 were used and the densities obtained were reproducible to k0.02 %. Results are given in table 1, together with molality to molarity conversion equations. 11 TABLE 1.-POLYNOMIAL EXPRESSIONS OF FORM y = C ajx' X a0 C C C+ In C C C In m In m 0.997 401 1.002 753 12 1.115 7 4.833 594 1.061 359 5.650 279 0.285 227 321 1.323 017 88 X a3 C 0.00007893 C 1.728 243 84 x In C -0.006 089 6 C 8.903 89 In m 9.860 176 12 x lnm 0.171 591 501 C* -1.511 311 C -1.02225 i = O concentration range a1 a2 /mol dm-3 0.198 024 - 0.002 427 4 0.1-3.0 - 0.705 702 1.831 215 0.03-3.15 0.774 400 - 0.048 270 2 0.1-0.35 6.604 568 x 7.556 885 14 x 0.1-3.0 168.880 5 - 59.803 46 0.3-1 .O 115.267 4 -37.353 2 0.9-3.0 0.1 -2 .o 0.589 824 368 0.407 860 767 2.0-4.0 0.453 586 887 0.176 694 894 concentration range 11.4 a5 /mol dm-3 - - 0.1-3.0 1.256 318 41 x - 0.1-3.0 0.882 843 - 0.260 420 0.03-3.1 5 -0.002 091 0 - 0.1 -0.35 - - 0.3-1 .O 0.9-3.0 0.506 203 - - 6.111 581 28 x - 0.1-2.0 3.066 428 84 x - 2.0-4.0 Where p is the density of the solution at 298.15 (g CM-~), C and m are concentrations, mol dm-3 and mol kg-', respectively.Dv is the volume fixed salt diffusion coefficient (cm2 s-') and K, the conductance (In-' cm-I). Electrical conductances were measured by the methods described previously to a precision of k0.02 % and were reproducible to k0.05 %, due largely to the uncertainty in chemical analysis of k0.05 %. Results are given in table 4 where conductances at rounded concentrations in the range 0.1-3.0 mol dm-3 are given.Diffusion coefficients were measured using Rayleigh interferometry on a closed column of diffusing solution in a regular parallelopiped optical cell. The methods used were identical to those described ear1ier.l Diffusion coefficients were reproducible to k0.3 %, table 2. Data at rounded concentrations, shown in table 4, were obtained by interpolation, using computer curve fitting procedures, table 1.A . AGNEW A N D R . PATERSON 2887 Transport numbers were measured using the Hittorf method. The Hittorf cell was of a similar design to that introduced by MacInnes and Dole and used recently by Pikal and Miller.Ig* 2o The cell was constructed with 10 mm bore Pyrex glass tubing made in two sections and coupled together by a B14 " Quickfit " joint.The assembled cell could be separated into three compartments using two 10 mm bore stop-cocks. These compart- ments were the cathodic, anodic and middle portions of the cell. For the analysis of a Hittorf experiment it is essential that concentration changes are confined to the two electrode compartments. For this reason every precaution was taken to prevent thermal and gravitational convection currents within the cell. The cell was bent in several places to minimise such currents and the barrels of the stop-cocks were drilled to allow free circulation of thermostat water internally, within the stop-cock barrel. The water thermostat was maintained at 25_+0.005"C and all sources of vibration from thermostat stirrers were minimised.As with earlier designs, the electrode polarities were arranged such that the more dense electrode solution was that of the lower electrode, usually the anode. Both electrodes consisted of short lengths of spectroscopically pure zinc rod sealed into a B14/20 Quickfit cone with epoxy resin. Pure metallic electrodes were found to be unsuitable for electrolysis experiments due to side reactions, particularly hydrogen evolution at the cathode and oxide formation at the anode. These side reactions were greatly reduced by using amalgamated electrodes. These electrodes were prepared by dipping the cleaned zinc metal electrodes into dilute mercuric chloride solution for a few minutes, rinsing with distilled water and polishing gently with a clean tissue.A Solartron constant current supply (P.S.U. AS1413) was used to deliver currents of 5.8 or 15.8 mAkO.1 %. Both currents are safely below the maximum current values recommended by Pikal and Mi11er.l The current was monitored when entering and leaving the cell and no leakage of current from the cell to the thermostat was ever found. TABLE 2.-EXPERIMENTAL SALT DIFFUSION COEFFICIENTS, D, FOR ZINC PERCHLORATE Dvx 10J/cm2 s-l 1.046 1.054 1.030 1.034 1.085 1.119 1.127 1.214 C/mol ~ I r n - ~ 0.0255 0.0416 0.0823 0.0915 0.1908 0.3525 0.3632 0.6198 Dvx 105/cm2 s-' 1.222 1.493 1.507 1.574 1.564 1.539 1.347 C/mol 0.6426 1.5136 1.5677 2.1701 2.2175 2.5495 3.1705 The experimental method of determination was identical to that of Pikal and Mi1ler.l Experiments lasted from 5-15 h, dependent upon the current used and the amount of electrolysis possible before side reactions at the electrodes became troublesome.In all cases the concentration changes in the electrode compartments were restricted to a maximum of 10 % to minimise risk of concentration changes outwith the electrode compartments. The weights of solution in both electrode compartments were determined and weighed solution samples withdrawn for analysis. Two samples were also taken from the middle compartment, one close to the anodic compartment the other close to the cathodic one. Analysis of these latter solutions showed that concentration changes due to electrode reaction were confined to the electrode compartments, otherwise the experiment was discarded. Triplicate analyses of the solution samples agreed within k0.07 %, using the EDTA titration analysis method described earlier." RESULTS AND DISCUSSION CONDUCTANCE A N D DIFFUSION The electrical conductances of solutions were measured in the range 0.1-3.0 mol dm-3.Salt diffusion coefficients on a volume-fixed frame of reference, D,/cm2 s-I were obtained from light interferometry, using Rayleigh optics. Experimental details of the method have been given ear1ier.l The fringe shifts, Am, were measured between two levels at & and 3 of the total height of the diffusing column of solution, as in Equivalent conductances, given in table 4, are precise to k0.05 %.2888 TRANSPORT I N AQUEOUS SOLUTIONS Harned's conductometric method.21 Since Am is proportional to AC (the cor- responding difference in salt concentration at these two levels) only if the refractive index of the solution is a linear function of concentration, Clmol d ~ l l - ~ , independent measurements of the refractive indices of these solutions were made.The relationship was not precisely linear over the full concentration range (0.1-3.0 mol dm-3), but for small concentration differences, such as encountered in the diffusion experiments, linearity was excellent. As before diffusion coefficients were obtained from eqn (1) where I is the height of the enclosed column of diffusing solution (cni) and 8 is the time in seconds. Excellent linearity was obtained in all cases and the resulting dif- fusion coefficients were reproducible to a precision of k0.3 %.This error is due largely to uncertainty in measurement of fractional fringe shifts. Experimental data, shown in table 2, was curve-fitted as a function of ,/C to obtain diffusion coefficients at rounded concentrations, table 4. In Am = - D,(n/E)28 + constant (1) HITTORF TRANSPORT NUMBERS FOR ZINC, th+ From the definition of transport number, th,i8/2F103 moles of zinc are trans- ported by a current of i/mA, during a time, 8/s. Allowing for electrode reactions at the zinc amalgam electrodes, the change in the number of moles of zinc, An, in anodic and cathodic compartments are Ananode = (1 - th,)i8/2m03 mOl Ancathode = (th, - l ) i 8 / 2 m 3 m01. (2) (3) If the weight of solution in an electrode compartment is W/g at the end of electrolysis and the initial and final zinc concentrations are mi and mf mol kg-l (solution) respectively, then on a solvent-fixed frame of reference, where M, is the molecular weight of the salt.From eqn (2)-(4) Experimental results are summarised in table 3. The first column represents zinc perchlorate molarity, C/mol dm-3. The next five columns are concentrations in units, mol kg-1 of solution. These are m i m ~ r n ~ m ~ and ma, representing the original solution concentration, final concentrations of the middle compartment, sampled at the cathodic and anodic sides and final concentrations in cathodic and anodic compartments, respectively. Wc and Wa are the weights of solution in these electrode compartments at the end of the electrolysis. Agreement between mi, mz and m;, within experimental error shows that con- centration changes were confined to the electrode compartments.Using eqn (5) zinc transport numbers (t:) were obtained from analyses of anodic and cathodic solutions, with uncertainty in the average tt of 50.01 units. These results are less accurate than might have been expected. Although amalgamation of the zinc electrodes reduced the tendency for side reactions to occur at the electrode, formation of a film of white zinc oxide or zinc hydroxide at the anode remained a problem, particularly on prolonged electrolysis. For this reasonA . AGNEW AND R . PATERSON 2889 experiments of limited duration were made and consequently concentration changes at the electrodes were much smaller than optimal ( M 10 % for dilute, reducing to 3 % for more concentrated solutions).For the most dilute solution, hydrogen gas evolution occurred at the cathode and so transport number was estimated from analysis of the anodic compartment alone. Stokes and Levien l3 have reported transport numbers from concentration cell measurements, t t , which they consider accurate to k0.002 units. These data have been interpolated to compare with the Hittorf transport numbers obtained here, table 6. The two sets, t: and th+, agree within the uncertainty of measurement. This agreement implies the validity of the Onsager reciprocal relationships (O.R.R.) which are verified by a more stringent method applied in the next section. TABLE 3 .-HITTORF TRANSPORT NUMBER DETERMINATIONS C/mol drn-3 0.097 30 0.322 25 0.460 84 0.574 31 1.015 39 1.155 71 1.463 6 1.965 6 2.424 7 C/rnol dm-3 0.097 30 0.322 25 0.460 84 0.574 31 1.015 39 1.155 71 1.463 6 1.965 6 2.424 7 mi 0.095 70 0.303 76 0.423 31 0.517 35 0.849 05 0.945 32 1.142 00 1.427 40 1.658 23 WClg - 28.430 28.912 30.074 31.701 32.788 34.190 37.146 39.095 ma "F "Z rnc m /mol kgl (ofiolution) - 0.303 43 0.423 32 0.517 39 0.848 57 0.945 36 1.142 69 1.429 00 1.658 03 wvg 26.499 27.835 28.838 29.161 - - 33.645 36.279 - 0.095 73 0.303 91 0.423 72 0.516 46 0.849 96 0.944 78 1.142 20 1.427 70 - 2Flie /mol-l x lo3 1.422 07 0.815 40 0.216 93 0.569 54 0.222 47 0.380 21 0.435 26 0.505 43 0.501 73 0.111 07 0.279 46 0.329 85 0.332 80 0.514 49 0.485 11 0.550 73 0.776 14 - 0.909 13 I 1.110 28 1.175 17 1.404 80 1.451 99 1.637 54 - ta anodic 0.406 0.3 60 0.362 0.358 - - 0.308 0.287 - zinc transport number tr cathodic - 0.384 0.359 0.360 0.336 0.325 0.320 0.307 0.279 fa (av.1 0.406 0.372 0.361 0.359 0.336 0.325 0.314 0.297 0.279 mi, m&, mt, mf and m! represent the original solution concentration, the final concentrations of the middle compartment on cathodic and anodic sides, respectively and the concentrations in the cathodic and anodic compartments at the end of the experiment.These concentrations are expressed as mol kg-I of solution. IRREVERSIBLE THERMODYNAMICS As in earlier papers in this series on group I13 metal salts zinc perchlorate will be designated as a, b, where a and b represent zinc and perchlorate ions, respectively and ra and Pb are the stoichiometric coefficients of the salt (ra = 1 and rb = 2 in this case).The theory of irreversible thermodynamics requires that, for systems close to equilibrium, there will be linear relationships between the observed flows (Ja and Jb) and their conjugate forces (Xa and Xb). In accord with earlier practice the flows J, and Jb are defined relative to stationary solvent (mol cm-2 s-l) and the forces, are defined by the local negative gradients of electrochemical potential (-grad PI) : dimensions 3 mol-' cm-l.2890 TRANSPORT I N AQUEOUS SOLUTIONS Two equivalent representations of the phenomenological equations are possible, eqn (6) and (7). In the first flows are expressed as linear functions of the forces, defining mobility coefficients, &k Ja = Laaxa+Labxb (6) Jb = Lbaxa+Lbbxb where the mobility coefficients L i k have dimensions mo12 J-1 cm-l s-l.resistance coefficients, Rik, eqn (7), In the second or inverse manner forces are expressed in terms of flows defining where resistance coefficient Rik have dimensions J cm s mol-2 and may be obtained from the mobility coefficients of eqn (6) by matrix inversion. The three isothermal transport properties, electrical conductance, transport number and salt diffusion coefficient may be expressed as functions of mobility coefficients,22 eqn (8)-( 10) where A and h: are the equivalent conductance (cm2 f2-l equiv-I) and " specific " conductance ohm-l cm-l respectively. N is the equivalent concentration (equiv dm-9 and A = k - 1 0 3 1 ~ = a ~ 2 1 0 3 p (8) is the function, Z z L a a + Z z & b + z a z b ( L a b +&a). t$ = t," = (Z:&-,afZaZ&ab)/a t: = ti = (ZzLaa+ZaZ&ba)/a.( 9 4 (9b) Superscripts h and c distinguish transport numbers determined from Hittorf experi- ments and concentration cell e.m.f. measurements, respectively. Finally, where y is the stoichiometric mean molal activity coefficient and r = ra+r,, the sum of the stoichiometric coefficients for the salt. Mobility coefficients were obtained from these three equations, assuming, on the basis of the equality of tk and ti (table 6) that L a b = Lba and so the Onsager reciprocal relationships are obeyed, table 5. The activity term of eqn (10) was obtained from osmotic coefficients, #, tabulated by Robinson and Stokes,23 using the identity (1 1) Polynomials expressing In 6 and 6 as power series of In m were used to obtain the differential in eqn (11).Coefficients for these polynomials are given in table 1, together with the concentration ranges over which they were used. The estimated uncertainties in the activity terms calculated from these data are 0.5 %. Resistance coefficients R,,, R b b and Rab were obtained from the corresponding set of mobility coefficients by matrix inversion. The additional resistance coefficients RaO and RbO relating to the interaction between the ions and water (symbol 8) were obtained from the identity 24 (I+%) = d ( 4 4 -&-. CJ,, = 0, k = 0, a, b. i = 0,a.bA . AGNEW AND R. PATBRSON 289 1 The three mobility coefficients expressed as LiR/N are shown as functions of J N in fig. 1 and 2. There are few analyses of dissociated 2 : 1 electrolytes for comparison.Only Miller’s data 22 for barium chloride has been made over a sufficient range ta make comparisons meaningful. The anomalous concentration dependences of mobility coefficients associated with self-complexing behaviour are absent.2 * Zinc perchlorate is normally considered to be a highly solvated salt. This view is sup- ported by such evidence as the abnormally large mean molal activity coefficients in concentrated solution (at 4.0 mol kg-l the activity coefficient is 37.9).23 There are however, few indications of such effects in the analysis of mobility coefficients, table 5 . The only analogy which might be made is between the concentration dependence of Lab/N for zinc perchlorate and lithium chloride.22 The latter, which is the most hydrated member of the alkali chloride series, shares with zinc per- chlorate, low values of Lab/N (designated LI2/N in Miller’s paper)22 and the fact that for both Lab/N passes through a maximum and decreases markedly thereafter with increasing concentration. 1 2 3 1/N FIG.1 .-(a) Intrinsic mobilities (Laa/N) for zinc (e) and barium ions (-) in zinc perchlorate and barium chloride solutions, respectively. (b) Coupling coefficients (&b/N) for perchlorate (e) and In an earlier paper 2 5 it was shown that solvation effects in the alkali chloride series were more apparent when the ion-to-water frictional coefficients h o / [ Z i 1 were examined. This function, defined by eqn (12), measures the friction between one equivalent of charge of the ion i and those moles of water around it per unit volume chloride (-) ions in zinc perchlorate and barium chloride solutions. (Dimensions : see table 5).(12) A 0 CiRio lzil Izit - = -- Eqn (12) allows in principle the comparison of water friction for ions of different valencies. Using the data of table 5fa0/lza\ andfbo/lZbl are compared with those of barium chloride 22 and lithium chlorideYz2 fig. 3. It is obvious that the water2892 TRANSPORT I N AQUEOUs SOLUTIOXS '. I 1 2 FIG. 2.-Intrinsic mobilities (Lbb/N) for perchlorate (0) and chloride (-) ions in zinc perchlorate and barium chloride. (Dimensions : see table 5). 0 1 2 3 - z/N FIG. 3.-Ion-to-water frictional coefficients, fio/lZil for : zinc perchlorate, @, 2:'; +, ClO? : barium chloride, -, Ba2+ ; 8, Cl- : lithium chloride, - -, L{ ; + + , C1-.A .AGNEW AND R. PATERSON 2893 friction of zinc ion is strongly concentration-dependent and rises steeply with in- creasing concentration and is larger than that of barium or even lithium ions in their respective chlorides (when compared at equal equivalent concentrations). Per- chlorate ion has a relatively low ion-to-water frictional coefficient at infinite dilution, as reflected by its high equivalent conductance, j1&,, eqn (13)26 Such anions might well be expected to continue to have low water friction even in concentrated solution. In barium chloride and lithium chloride the chloride-to- water frictional coefficient actually decreases initially from its value at infinite dilution, before rising in more concentrated solutions due to the effects of the solvent order producing cation.In zinc perchlorate, perchloratelwater friction increases from infinite dilution, with no obvious minimum and rises steeply with concentration in a manner more marked than for chloride in either lithium or barium chlorides. It appears that the dominant solvent effect is that of zinc ion which increasingly affects the bulk solvent as salt concentration is increased and consequently makes the movement of perchlorate through that solvent more dif5cult. ap = i03~2~(fi0~~zi~); i = cio,. (13) TABLE 4.-ISOTHERMAL TRANSPORT DATA FOR ZINC PERCHLORATE AT ROUNDED CONCENTRA- TIONS C m A (*) /mol dm-3 /mol kg-1 /cm2 8 - 1 equiv-1 ts Dv/cm2 s- 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o 1.5 2.0 2.5 3.0 O.OO0 0.1009 0.2032 0.3070 0.4122 0.5189 0.6271 0.7369 0.8485 0.9617 1.0766 1.6801 2.3368 3.0569 3.8512 120.16 87.57 81.65 77.65 74.52 71.66 68.99 66.45 64.03 61.73 59.52 49.21 39.67 30.83 22.78 0.439 0.409 0.389 0.377 0.368 0.360 0.354 0.348 0.342 0.337 0.332 0.331 0.294 0.280 0.271 1.1818 1.0357 1.061 7 1.0966 1.1342 1.1752 1.2104 1.2476 1.2838 1.3187 1.3521 1.4903 1.5635 1.5480 1.4198 1 ,m 0.9943 1.0825 1.1942 1.3179 1 A496 1.5878 1.7321 1.8820 2.0373 2.8999 3.9861 5.0937 6.3083 C and m axe, respectively, molar and mold concentrations.A, ta and D, are equivalent conduc- tance, zinc transport number and volume-fixed salt diffusion coefficient, respectively. The activity term (1 +d In y/d In m) was obtained from the osmotic coefficient data of Robinson and Stokes 23 using eqn (1 1) and the polynomials of table 1.ONSAGER RECIPROCAL RELATIONSHIPS The equality, within experimental error, of the transport numbers for zinc, t: and t,", has been used above as sufficient reason to assume the validity of the Onsager reciprocal relationships, using eqn (9a,b). The experimental difficulties related to electrode performance in the Hittorf experiments made the uncertainty in zinc transport number ( & 0.01) and accordingly the concentration cell data Stokes and Levien were used for the calculation of irreversible thermodynamic coefficients (table 4). Zinc transport numbers from the Hittorf experiments, tk summarised in table 3, are compared with those of Stokes and Levien,13 t:, in table 6.2894 TRANSPORT I N AQUEOUS SOLUTIONS From eqn (8) and (9) and ~ 1 0 - 3 From table 6, it is obvious that the difference (Lab-Lba)/N is much smaller than the value of L,b/N, interpolated from the data of table 5.The ratio Lab/& is shown to be unity within the uncertainty of the experimental data. It is noteworthy (not- withstanding the inherent inaccuracies of the Hittorf determination of transport number) that zinc perchlorate is a most unfavourable system for a precise proof of the reciprocal relationships. The probable deviation of the ratio .&/Lab from unity TABLE 5.IRREVERSIBLE THERMODYNAMIC COEFFICIENTS FOR ZINC PERCHLORATE C L s a l N LablN LablN NR3a - N R m NRbb -CoRa -cORbO CO lmol dm-3 x 1012 x 1012 x 1012 x 10-12 x 10-11 X 10-11 X lO-1p x 10-11 /mol dm-3 0.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o 1.5 2.0 2.5 3.0 1.41 8 1.050 0.977 0.910 0.849 0.793 0.744 0.699 0.659 0.622 0.490 0.357 0.269 0.196 0.000 0.394 0.383 0.346 0.312 0.276 0.247 0.223 0.202 0.184 0.106 0.088 0.075 0.061 7.235 6.145 5.961 5.751 5.551 5.343 5.147 4.971 4.799 4.637 3.748 3.183 2.534 1.906 0.705 0.976 1.050 1.125 1.203 1.284 1.365 1.451 1.536 1.626 2.052 2.820 3.745 5.147 O.OO0 0.625 0.674 0.678 0.677 0.664 0.655 0.650 0.645 0.644 0.580 0.778 1.107 1.647 1.382 1.667 1.721 1.780 1.840 1.906 1.974 2.041 2.111 2.182 2.685 3.167 3.979 5.301 3.527 4.255 4.574 4.949 5.337 5.758 6.171 6.602 7.037 7.485 9.681 13.319 17.619 24.088 1.382 1.355 1.384 1 .a 1 1 SO1 1.574 1.647 1.716 1.788 1.860 2.335 2.773 3.426 4.477 55.35 54.62 54.25 53.87 53.49 53.1 1 52.73 52.34 51.95 51.56 49.56 47.51 45.40 43.24 N = 2C is the equivalent concentration, (equiv dm-3) and Co is the molar concentration of water (mol dm-9.The dimensions of Lik and Rik coefficients are molZ J-' cm-' s-' and J cm s mo1-2, respectively. TABLE 6.-TEST OF ONSAGER'S RECIPROCAL RELATIONSHIPS FOR ZINC PERCHLORATE concentration transport numbers for zinc (Lat-Lba)/N Lab X IN error limit Clmol dm-3 rt a tp b x 1012 1012 LbafLab in ratio 8LbalLb 0.0973 0.3223 0.4608 0.5743 1.01 54 1.1557 1.4636 1.9656 2.4247 0.406 0.372 0.361 0.359 0.336 0.325 0.314 0.297 0.279 0.409 0.374 0.363 0.355 0.330 0.324 0.312 0.295 0.282 0.014 0.008 0.008 - 0.015 -0.019 - 0.003 - 0.005 - 0.004 0.005 0.254 0.374 0.325 0.283 0.180 0.155 0.110 0.086 0.075 0.95 0.98 0.98 0.95 1.11 1.02 1.05 1.05 0.93 0.16 0.11 0.12 0.13 0.17 0.19 0.24 0.25 0.23 a t," are Hittorf data from table 3.b tf are interpolated data from concentration cell measure- m e n t ~ . ~ ~ = 8Lba/Lab are based on the probable uncertainty of 0.01 units in t," where G(Lba/Lab) 5% (at: + at:)K/LabZaZbFZ and K is the electrical conductance.A . AGNEW AND R. PATERSBN 2895 is almost entirely due to errors in transport number measurements, as Miller 27 has shown, eqn (16) where IC is the electrical conductance cm-l). This equation shows that the sensitivity of the test is increased by both low conductances and large coupling coefficients. For zinc perchlorate the reverse is true, conductances are large (table 4) and coupling coefficients are abnormally small, particularly in concentrated solutions, table 5. Zinc perchlorate is, therefore, a most unfavourable system for a precise proof of the Onsager reciprocal relationships.Self complexed salts, such as cadmium and zinc chlorides,6 which are typified by low electrical conductances and large coupling coefficients provide much more sensitive tests of the reciprocal relationships ; the latter is discussed in the succeeding paper. a(L,a/Lab) 53 (dt;+dt,") "/LabZaZbF2 (16) R. Paterson, J. Anderson and S. S. Anderson, J.C.S. Furaduy I, 1977,73, 1763. R. Paterson, J. Anderson, S. S. 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