J . Chem. SOC., Faraday Trans. I , 1982, 78, 3287-3296) Velocity Correlations in Aqueous Electrolyte Solutions from Diffusion, Conductance and Transference Data Application to Concentrated Solutions of Cadmium Chloride BY REGINALD MILLS Diffusion Research Unit, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 2600, Australia A N D HERMANN GERHARD HERTZ* Institut fur Physikalische Chemie und Elektrochemie der Uni\ ersitit Karlsruhe, 7500 Karlsruhe, West Germany Received 15th January, 1982 Tracer-diffusion and intradiffusion coefficients have been measured for the Cd and C1 ion constituents and for water molecules in aqueous solutions of cadmium chloride in the concentration range 0.1-5.8 mol dm+ at 298.15 K . These measured coefficients have been combined with transport data from other sources to calculate velocity cross-correlation coefficients (VCCs).In contrast to a fully dissociated 1 : 2 salt such as magnesium chloride, the VCCs for CdC1, show much greater positivity, and in particular the anion-anion VCC is positive over the whole concentration range. An explanation for the negative transference numbers in concentrated CdCl, solutions is offered using the linear-response-theory formalism. In previous studies of this kind' velocity cross-correlation coefficients (VCCs) for aqueous electrolyte solutions have been calculated from conductance, transference- number, mutual- and tracer-diffusion and activity-coefficient data. The definition of VCCs and their mode of calculation from the primary transport data are detailed in the above papers.The electrolyte systems studied so far comprise a number of both I : 1 and 1 : 2 salts. Within the latter group have been BaCl,, CaCl,, MgCl, and NiCl,, and all of these show little or no complex formation. It is therefore of considerable interest to examine an electrolyte such as CdCl, for which there is strong evidence for complexation in aqueous solution. The evidence for complex formation in CdCl, solutions stems from a variety of observations. For example, the fact that in transference-number experiments5* the value for the cation constituent becomes negative at high concentrations has been explained in terms of its incorporation into complex negative ions. Reilly and Stokes7 have used activity-coefficient data to examine complexation in cadmium chloride solutions and have calculated stability constants for complex formation.There is also a considerable amount of spectroscopic evidence for the existence of complexes in these solutions.*. The Experimental section of this paper concerns the measurement of the tracer- diffusion coefficients of the two ion constituents and the solvent in CdCl, solutions, and is the first comprehensive diffusion study of such coefficients for a complexed system. Note that Paterson et aI.lo1 l 1 have measured tracer-diffusion coefficients for the two ion constituents (not the solvent) in aqueous CdI, solutions. However, the concentration range studied was much lower than that reported here for CdCl,, and their data were analysed in the irreversible-thermodynamics formalism.32873288 TRACER DIFFUSION I N CdC1, SOLUTIONS EXPERIMENTAL Analytical-grade CdCl, (Univar) was used without further purification. The radioactive tracers were Il5Cdrn and ,,Cl, both being obtained from the Radiochemical Centre, Amersham, U.K. The diaphragm-cell technique was used to measure tracer-diffusion coefficients for the Cd and C1 constituents; the procedures are described fully by Mills and Woolf.12 The self-diffusion coefficients of water in the solutions were determined by the n.m.r. spin-echo method and details of this technique have been given by Harris et a/.', TABLE 1 .-TRACER-DIFFUSION COEFFICIENTS OF ION CONSTITUENT SPECIES (Cd AND C1) AND H,O IN AQUEOUS CdC1, SOLUTIONS AT 298.15 K c/mol dmP3 Cd c1 H,O 0 0.057, 0.1 13, 0.204, 0.508, 0.565, 1.134 2.045 3.043 4.076 5.787 0.712 0.734 0.733 0.723 0.666 0.566, 0.404, - 0.155, 0.047, 2.030 1.420 1.299 - - 0.945 0.713 0.466, 0.170, 0.050, 2.299a 2.275 2.241 2.191 2.065 1.828 1.496 1.139 0.826 0.433, - a Value of Mills.', RESULTS The tracer-diffusion data for all constituents in the CdCl, + H,O system at 298.15 K are listed in table 1.The ion-constituent coefficients have an average precision of +0.4% and the solvent coefficients In fig. 1 we have plotted the data in table 1 and for comparison the corresponding tracer-diffusion data for the MgC1, + H,O system.15 Magnesium chloride is a strong 1 : 2 electrolyte and shows no evidence of complex formation. The difference between the two sets of results is striking. We take the C1- results first; in the CdC1, case, its tracer-diffusion coefficient decreases very steeply from the infinite-dilution value as compared with MgC1,.This rapid decrease may be attributed to the fact that even at very low concentrations, the normally mobile C1- is bound to the slower moving Cd ion constituent in complexes such as CdC1+ and CdCI,. Cd2+ has practically the same limiting diffusion coefficient as ME,+, but as the concentration increases its value rises above the latter and shows a small maximum at ca. 0.05 mol dm-3. The reason for this slight maximum in the Cd diffusion pattern probably is due to the fact that in complexes such as CdC1+ and CdCl,, the faster moving C1 constituent increases their mobility when compared to the bare Cd2+. Thereafter with increasing concentration the Cd constituent coefficients approach slowly to those of Mg2+ and coincide with them at ca.3 mol dmP3. It is also 1 %.R. MILLS AND H. G. HERTZ 3289 worth noting that the C1 ion constituent values also coincide with those of Cd and Mg2+ at ca. 5 mol dm-3. Another important difference between diffusion in the CdCl, and MgC1, systems is that in the former the mobility of water molecules is considerably greater than in the latter. A simple explanation is that Mg2+ is regarded as a strong structure-making ion with both primary and secondary water coordination shells. Cd2+ is larger than Mg2+ and even in its uncomplexed state would be less of a structure-maker than Mg2+. ... 0 1 2 3 4 5 6 c/mol dm-3 FIG. 1.-Tracer-diffusion coefficients of ion constituents and of water in CdCI, and MgCI, solutions from 0.06 to 5.8 mol dm-3 ( T = 298.15 K).However, in addition if it forms complexes with C1- such as CdCl+ and CdC1, then its structure-making capacity would be diminished even further. The diminution of structure in water, of course, leads to higher mobility of the water molecules. VELOCITY C RO S S-CO R R E LAT I ON C 0 E F F I C I EN T S GENERAL In this section the data reported above are combined with other transport and thermodynamic data to calculate generalized transport coefficients which have their basis and definition in linear response theory. One could have chosen to use the formalism of irreversible thermodynamics and calculated ionic lij coefficients, as was done in the CdI, case by Paterson et a1.l07 l1 However, it is felt that the VCC approach3290 TRACER DIFFUSION I N CdCl, SOLUTIONS is a more powerful one for complex-forming electrolytes in that the behaviour of complexes is reflected better by a dynamic variable such as the velocity.The calculation of VCCs for CdCl, solutions follows the procedures laid down in the series of papers by Hertz et previously cited. Sources of data, other than for tracer diffusion, used in these calculations are as follows : conductances, McQuillan ;6 transferences numbers, McQuillan ;6 mutual-diffusion coefficients, Rard and Miller ;16 activity coefficients, Robinson and Stokes.17 0.6 0.5 0 1 2 3 4 - cs/mol dm-3 FIG. 2.-Ionic velocity correlation coefficients faa, fa, and f,, in an aqueous solution of CdC1,.For comparison the ionic VCCs in a MgCl, uij) are given as dotted curves. The dot-dashed curves represent the predicted or standard VCCS,~’, of the constituents of CdC1, if the latter were present as a completely undissociated molecule. For further details see text ( T = 298.15 K). IONIC VELOCITY CORRELATION COEFFICIENTS The anion-anion, cation-cation and anion-cation VCCs,f,,,f,, andf,,, which have been computed from the experimental data according to the methods outlined in two previous papers3> are plotted against the concentration (on the molarity scale) in fig. 2. In the same figure we have included the corresponding coefficients of the classical fully dissociated 1 : 2 electrolyte MgCl, as dotted curves. All theLj curves for CdC1, are shifted to more positive values relative to those of MgCl,.Thus there are more positive velocity correlations between the various constituents of the solute CdCl, ; i.e. if a Cd2+ ion has a given positive (or negative) velocity at t = 0, then shortly after this instant, the C1- ions on the average still have a positive (or negative) velocity, and likewise, if a C1- at t = 0 has a given positive (or negative) velocity, shortly afterR. MILLS AND H. G . HERTZ 329 1 this instant the mean representative of all the other C1- ions has a positive (or negative) velocity. This seems to be a simple consequence of the complex formation; the constituents are (at least partly) tightly bound to each other and thus they move together, i.e. they have positive velocity correlations. In order to get a deeper and more quantitative insight it is instructive to use velocity-correlation terminology and examine the consequences of treating CdCl, as a purely molecular compound dissolved in water.1 We first divide thehi into an intramolecular and an intermolecular part, i.e. f i j = (fijlinter + (hjlintra i,.i = a, c (1) where the subscripts a, c refer to anion and cation. Then for velocity correlations within the molecule itself it can be shown29l8 for CdC1, that (fijlintra = Dc = Da (2) where the Di are self-diffusion coefficients of constituent i. for the intermolecular VCCs. as under We now formulate standard values (those predicted under certain ideal conditions) where components 1 and 2 refer to water and CdCl, and Mi and xi are the molecular masses and mole fractions of component i, respectively.Introducing eqn (2)-(4) into eqn ( I ) we obtain In fig. 2 these standard values, calculated on the assumption that CdCl, is a molecular compound, are compared with the experimentally determined values. Thus eqn (5)-(7) are shown as dot-dashed curves. First it is seen that, at high concentrations, fac andfcc are comparatively close to our predicted standard curves for the molecular species, both being more positive than the corresponding standard curves. In contrast to this faa is markedly more positive than the standard quantity fza. Moreover, for the MgC1, solution there is always a shift of thefij which is much stronger and in the negative direction relative to thef?' curves. (If computed with masses and diffusion coefficients of the MgC1, solution, the f: values for the hypothetical MgCl, molecule would come out much the same.) However, remarkably enough, the order remains the same, i.e.fcc < faa <far. To account for the deviations in the experimentalf,' from the standard molecular values, we turn now to examine the conductance, which is one of the factors in their3 2 9 2 TRACER DIFFUSION IN CdC1, SOLUTIONS computation. The ‘ reduced ’ equivalent conductance has previously been defined as where A is the equivalent conductance, z, is the charge of the cation Faraday constant. This ‘reduced’ conductance has the same units as the and velocity-correlation coefficients. In fig. 3 we show the value of A* of the concentration. This figure also gives A* for the MgC1, solution, and F is the self-diffusion as a function and the shift 0 1 2 3 ‘Es/mol dm-3 FIG.3.-Reduced equivalent conductance A* = ART/z, F 2 of CdCI, and MgCl, in water as a function of the concentration (molarity scale). For comparison the cationic self-diffusion coefficient, D,, of Cd*+ is also shown ( T = 298.15 K). of the coefficients faa,fac and fCc to more negative values as compared with those for CdCl, is accompanied by a much stronger increase in the reduced equivalent conductance A*. The reduced equivalent conductance can be separated in terms of transference numbers so that A* = t, A* -k t, A* withR. MILLS A N D H. G . HERTZ For Cs 2 3.5 mol dmP3, t, < 0, and considering eqn (9) we must have fac > D c +fee* If we assume, considering eqn (I), (5) and (7), that fcc = (fachnter then it follows from inequality (1 1) that we must have (fachntra > D c (1 1') and with eqn (9) we get (faclintra = D c - tc A*+ Application of experimental data tells us that at high concentrations (i.e.Cs = 4- 4.5 mol d m 3 ) (faJintra is only ca. 1 % larger than D,. So a very small positive deviation of (fac)intra from the cationic self-diffusion coefficient is sufficient to explain why the transference number is negative. In the case of MgCl, the cationic transference number is positive; thus we have fa, < D c +Sac (faclintra < D c - which means that We have seen that the deviation from neutral-molecule behaviour, which determines the contribution tc A* to the reduced equivalent conductance, is very small, and now turn to the anionic contribution.We use eqn (l), and determine ( faa)inter: (faahnter = faa - (faalintra where (fa.Jintra is the same figure as derived from our t,A* values. The results for the four concentrations 3, 3.5, 4.0 and 4.5 mol dmP3 are -0.21 x lop9, -0.18 x lop9, - 0.14 x and - 1.2 x lop9 m2 s-l, respectively. However, according to eqn (3), for the molecular species we should have (faa)inter = 2fcc, the corresponding four fCc values are -0.13 x -0.12 x -0.092 x and -0.072 x m2 s-l (see fig. 2); thus the (faa)inter values are too positive. This may clearly be seen from fig. 2.T Thus we obtain the result that the electric conductivity of a concentrated CdCl, solution can be related to the fairly strong positive excess velocity correlations of the anions among themselves. VELOCITY CORRELATION COEFFICIENTS INVOLVING THE WATER MOLECULE In fig.4 we show the solute-molecule-solvent-molecule velocity correlation coefficient fsw as a function of the (analytical) CdC1, concentration. As in our previous work1* 2 * it has been calculated from the relation d In y* Dp = - 3 ~ ( l + c W , ) 3 ( ~ + ~ : czlc? M, dc,* )fsw where D,, is the mutual diffusion coefficient and y' the activity coefficient of the solute CdCl, (component 2). In this formula the molarities (c,) and molalities (c;) refer to 1 cm3 and 1 g, respectively. Fig. 4 also gives the water-water velocity correlation coefficient, fw,, which again has been calculated in the same way as described previously1* 2 v t D, = 0.160 x m2 s-l at E, = 4 mol dm-3.3294 TRACER DIFFUSION I N CdC1, SOLUTIONS The dashed curves in fig.4 represent the corresponding standard VCCs which one would predict if the solution were an ideal mixture18 (apart from an activity factor 1/3 occurring in the expression for f",) with and c , f i w I 1 I 1 I , 0 1 2 3 G 5 zs /mol dmA3 FIG. 4.-Solute-water (f,,) and water-water uww) VCCs in an aqueous solution of CdC1, as a function of the CdC1, concentration. The standard VCCs cr",) are also shown ( T = 298.15 K). It will be seen that forf,, the deviation from the standard behaviour is appreciable, which is a consequence of the marked peculiarity of the activity factor (1 + c; d In y* /dc;). DISCUSSION The most interesting point which should be discussed is the comparison of the conventional interpretation of the negative cationic transference number with the one given within the framework of linear-response theory.Let Ai be the equivalent conductance of the ith ionic species; then according to Spirolg the transference number of the ion constituent Cd is given by the expressionR. MILLS AND H. G . HERTZ 3295 where we have neglected the contributions of the neutral CdCl, molecules and of the very small amount of CdClj- ions.' With the stability constants as reported by Reilly and Stokes7 one calculates at c, = c, = 4 mol dmP3: cCp+ z 0.02 mol dm-3, cCdC,+ z 1.2 mol dm-3, cCdCl2 z 2.0 mol dmP3, ccdc,<; z 0.6 mol dm--3 and cC1- z 0.6 mol dm-3, neglecting the influence of the activity coefficients (which are also reported in the paper of Reilly and Stokes).The general physical rituation is little changed if these activity coefficients are incorporated.* If the stability constants of Reilly and Stokes, which were based o n data measured to ca. 1.2 rnol dm-3 in the CdCI, concentration, are valid up to 4 rnol dm-3 then an interesting anomaly arises. In order to obtain negative transference numbers one must assume that the conductance of a CdC1; ion is about twice as large as that of CdCI+. This seems rather unlikely. However, Lutfullah and Paterson,20 using the procedures of Reilly apd Stokes have calculated the concentrations of the various species present in CdI, solutions. At ca. 0.28 rnol dmP3, where the transference numbers become negative, the concentrations of negative and positive Cd complexes are almost equal and it becomes unnecessary to postulate a large mobility difference.Further e.m.f. measurements of the type performed by Reilly and Stokes to 4 mol dm-3 in CdC1, solutions are obviously necessary to clarify this matter. In conclusion, we wish to describe the particular behaviour of CdCl, (and similar solutions) in terms of single-particle (or distinct) velocity correlation integrals (u~')(O)c~')(t))dt k , l = c,a, i,j= 1,2. c: We recall that om VCCs are defined f k l = N , lx (~(1') ( 0 ) ~ f ) ( t ) ) dr then for the case of a molecular solute according to eqn (4), (6) and (7) we derive 0 1 " 1 M , P( 1 -x, 41) D2 0 M c;c = { ( u p (0) UP) ( t ) ) o dt = -- V (14) 1 " 1 M , V ' ( 1 - ~ , 8 2 ~ ) D, D2+- 0 M N2 czc, = (da)(0) d c ) ( t ) ) , dt = -- V where V' = V/NO is the mean particle molecular volume and M = x, M , +x, M,; N , = N,,n,, n, is the number of moles of solute added to the system.Next we write the corresponding integrals for the case of a fully dissociated electrolyte such as MgCl,. (15) The result is18 * Since this paper was written Prof. Stokes was kind enough to compute the molarities of the various complex species present for given stoichiometric CdC1, concentrations. The computed data indicate that this statement is correct.3296 TRACER DIFFUSION I N CdC1, SOLUTIONS The constants Bi and Ci (i = 1-3) are specific quantities determined by the system; the constants Ai (i = 1-3) are probably of a more general nature but at this stage not exactly known. We may summarize the statements given in eqn (12)-(17) in the following way.Consider a set of N, particles of kind C and 2N, particles of kind A (e.g. elements) which are dissolved in a certain solvent; the volume is fixed and it has the value V. Then if the N, dependence of the single particles' (distinct particles) velocity correlation integrals is described by the limiting behaviour for N2 -, 0 as given by eqn (12)-(14), the solute compound is a molecular one; if, on the other hand, the limiting behaviour as N, -+ 0 is described by eqn (15)-(17), the solute compound is present in the ionized state. Now it will be seen that CdCl, (and similar substances) behave in a particular manner which lies in a characteristic way between the two standard types just mentioned. At low concentrations (small values of N,) c,, lies between the patterns given by eqn (1 3) and (16), that means that c,, is approximately proportional to N;:, whereas according to eqn (13) and (16) it should be proportional to N i and N;Z, respectively. At high solute concentrations c,, follows fairly well the prediction of eqn (1 3), as shown in fig.2. In the same way at small concentrations la, lies between the behaviour of expressions (14) and (1 7). Whereas according to the latter equation c,, again should have a proportionality to N;$ and according to eqn (14) should be K N;l, experimentally one finds something like c,, oc N;;; again the behaviour of la, at high concentrations is fairly well described by eqn (14). Finally, the most marked deviation from the typical standard behaviour occurs for caa.According to the ionic-type behaviour at small concentrations caa should be negative [eqn ( 1 5 ) ] . However, within the range we were able to study, c,, was always found to be positive; this is the sign predicted by eqn (1 2). However, at higher concentrations Caa remains more positive than predicted by eqn (12). So the strongly positive (distinct) particle velocity correlations between the chloride ions are the typical and very specific physical effect which can be related to the interesting anomalies in this type of electrolyte solution. It is the velocity analogy of what from another point of view is described as auto-complex formation, i.e. as a strong crowding of anions around most of the cations. H. G. Hertz, Ber. Bunsenges. Phys. Chem., 1977, 81, 660. H. G. Hertz and R. Mills, J . Phys. Chem., 1978, 82, 952. A. Geiger and H. G. Hertz, J . Chem. SOC., Faraday Trans. I , 1980, 76, 135. A. 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Hertz, in Diflusion and Conductance in Ionic Liquids, 2. Phys., Chem. N.F. Suppl., in press. l9 M. Spiro, J . Chem. Ed., 1956, 33, 464. *O Lutfullah and R. Paterson, J . Chem. SOC., Faraday Trans. I , 1978, 74, 484. (PAPER 2/076)