Partial molar volumes of aqueous 1: 1 and 2: 1 electrolytes and a structure model

Transcription

1 Partial molar volumes of aqueous 1: 1 and 2: 1 electrolytes and a structure model LOWELL W. BAHE AND KATHRYN A. JUNG' Department of Chemistry, University of Wisconsin- Milwaukee, Milwaukee, Wisconsin Received June 3, 1975 LOWELL W. BAHE and KATHRYN A. JUNG. Can. J. Chem. 54,824 (1976). An expression for partial molar volumes is derived from the free energy equation based on field-dielectric-gradient forces, as well as coulombic forces, between ions in structured solution. Qualitative agreement between the model and experimental partial molar volume data is very good. Excellent quantitative agreement is obtained by adopting for a universal parameter of the theory a value that implies (d In k/dp) = 76 x bar-' for water at 25 "C and 1 atm. The latter value is compatible with data obtained from direct measurements. LOWELL W. BAHE et KATHRYN A. JUNG. Can. J. Chem. 54, 824 (1976). On a derive une expression pour les volumes molaires partiels a partir de I'equation d'energie libre basee sur des gradients de forces de champs ditlectriques ainsi que sur les forces coulombiques entre ions dans une solution structuree. La correlation qualitative entre les donnees experimentales de volumes molaires partiels et celles calculees a partir du modele est tres bonne; on obtient une correlation quantitative excellente si on adopte pour un parametre universe1 de la theorie une valeur qui implique que (d In k/dp) = 76 x bar-' pour l'eau a 25 "C et a 1 atm. Cette valeur est compatible avec des donnees obtenues par des mesures directes. [Traduit par le journal] Introduction The properties of strong, binary 1 : 1 and 2 : 1 electrolytes in aqueous solution have been interpreted on the basis of a model of a loose lattice structure that results from the fielddielectric-gradient interaction, in addition to the coulombic interaction, between all ion pairs (1-3). The fundamental equation which results from this analysis is (1,3) 111 log 1; = - Ac113 + Bc where.f, is the mean mole fraction ionic activity coefficient, c is the concentration in mol I-', and where A" is the Madelung constant, z+ and z- are the charges on the cation and anion, respectively, N is Avogadro's number, v = v+ + v- for the salt C,+A,_, R is the gas constant, T is the temperature, k is the dielectric constant of the solvent, a,.. is the factor for converting a particular structure into the distance of ionic separation (3), and B is a constant which depends upon the interaction between an ion and the surrounding dielectric medium (1, 3). Equation 1 agrees well with experimental data for activity coefficients (free energies) for 1 : 1 (1) and 2: 1 (3) electrolytes at 25 "C. The temperature derivative of eq. 1 predicts heats of dilution of 1 : 1 electrolytes which show excellent agreement with experimental observations (2). This paper will show that the pressure derivative of eq. 1 leads to excellent agreement with partial molar volume data for 1 : 1 and 2 : 1 electrolytes and gives a new value for the pressure derivative of the dielectric constant of water at 1 atm. Equation for Partial Molar Volumes The partial molar Gibbs free energy for an electrolyte in solution is given by [31 G, = (7,' + RTln f,"xtv where X, is the mean ionic mole fraction of electrolyte in solution. Since (ag/ap),,n.s = 8, the 'Present address: Department of Oceanography, Dalhousie University, Halifax, Nova Scotia.

2 ~(g)~,~'~ BAHE AND JUNG 825 pressure derivative of eq. 3 at constant temperature and constant composition gives The pressure derivative of eq. 1 gives 1 dlnf, acl/3 -- [5 I In 10 ( ap )T,n.s = -A(F)~,~.~ - + C"~(&)T,~?~ Since k is the only pressure sensitive variable in eq. 2, we obtain while the pressure derivative of the concentration, c, gives + c (g)t,n.s where p (= -(l/v)(dv/dp)t.n.,) is the isothermal compressibility. Combining eqs. 4, 5, 6, and 7, we obtain [9 1 = VZ0 + APc1I3 + Bpc where and B, = vrt (In 10) /?B + [ - (::)T,".l We can now compare the predictions of eq. 9 with actual experimental determinations of partial molar volumes. Coefficient of c1i3 it is now claimed that both procedures give A, in eqs. 9 and 10 should be determinable the same results, at least for the interpretation since p has been measured repeatedly, and (d In on the basis of the Debye-Huckel theory (10). k/dp),,,., has been reported from direct measure- The direct measurement of (d In k/dp), at ments on pure water (4-7). However, it has 1 atm is difficult. One reported value for also been suggested that (a In k/dp),,,., be found (d In k/dp),, c,=, is x lop6 bar-', which from partial molar volume data rather than gives a value for (BklaP),, c,=, of about direct measurement since the value of (a In 3.7 x lop3 bar-' (4). To find measurable changes k/dp),.,., from direct measurement is difficult in k, the accuracy of which is no better than to obtain and much variation has existed f 0.01, the increments in P must be fairly large; among reported values (8). On the other hand, the increments reported range around it has also been noted that a thorough and care- bars, and k changes by a few tenths with this ful analysis on the basis of a complete theory, pressure change (5). A common procedure insuch as that of Debye and Huckel, IS necessary volves measuring k at 1, 100, 200 bars, etc. to account for the variation of V or 4, with c and then curve fitting the results SO that a to avoid mistaken conclusions (9). And lastly, function of k equals a function of P; the

3 826 CAN. J. CHEM. VOL. 54, 1976 derivative is then found from this functional relationship. In our case we need the value of (8 In k/dp), at P = 1. Finding the slope of a fitted curve at the very end of the curve with data whose measured increments are very large makes one apprehensive. One could feel less apprehensive if the data could be collected at increments of perhaps 1 atm or less, but inaccuracies of method will not allow such measurements at the present time. In view of this difficulty we have chosen to follow a procedure similar to that of Redlich and Meyer (8), namely, to find A, from P2 data rather than use the values reported for the direct measurement of (d In k/dp)t,,=l. The important criterion that must be met is that a single value for [(d In k/dp), - 8/31 (eq. 8) must be found that will apply to all strong, binary salts for which partial molar volume data are available. Experimental Partial Molar Volumes Commonly results are reported in terms of 4,, apparent molar volumes, rather than in terms of P, since 4, is obtained easily from density data while Pis more involved. The relationships are where V is the volume of the solution, n, is the number of moles of solvent, P10 is the molar volume of pure solvent, and n, is the number of moles of solute. From eq. 12 it follows that where m is the molality of the salt. At very low concentrations the last term on the right of eq. 9 can be disregarded and c can be replaced by porn, where po is the density of pure water, so that eqs. 9 and 13 can be combined to give FIG. 1. Apparent molar volume, 4,, plotted against cl" for LiCl(12). which can be integrated to yield Equation 15 predicts that, at low concentrations, a plot of 4, us. c1l3 should give a straight line. Data for LiCl with four points from c = 0.02 to c = 0.16 (1 1) are shown in Fig. 1 and a straight line with slope 1.50 is obtained. The value for the slope was obtained by linear regression analysis. As was the case with heats of dilution, data of very high accuracy at very low concentrations are needed for a plot such as that in Fig. 1. Even then, the linear term in c, as will be seen below, cannot be disregarded for many salts. Fortunately, LiCl has a small B, (see below) so that the analysis of Fig. 1 is quite good. Two approaches are available for testing the data over a greater concentration range. Since 4, is the experimental quantity and V is the quantity most directly related to the theory one can either convert the 4, data to P quantities or one can rearrange eq. 9 into the 4, form. The latter, as can be seen from eqs. 9 and 13, requires that In principle, the integrations of eq. 16 can be carried out either graphically or numerically after which V20 and the two coefficients, A,

4 BAHE AND JUNG 827 TABLE 1. Parameters of eq. 17 determined by multiple linear regression analysis Concentration Salt range u' a' b' Reference CsBr BaC1, , CaCI, RbF , , KF , RbBr RbCl , KC CsCl NaCl O NaBr , KI NaI , LiCl KBr LiBr , LiI and B,, can be found by suitable analysis. In practice, this procedure worked poorly; the values obtained for the three constants, V20, A,, and B,, were inconsistent and not reproducible even when multiple regression analysis was used. The alternative method requires that 4, be converted into V2. The raw data can be fitted to the empirical relation [ 171 4" = V' + + btrn by multiple regression analysis. The form of eq. 17 was chosen since, as eq. 16 indicates, this form would result if c = rn. The values obtained for v', a', and 6' were not intended to be significant other than as curve fitting results. The derivative of eq. 17 gives and eqs. 17 and 18 can be combined according to eq. 13 to give Equation 19 allows the evaluation of V2 at various rn and should be most valid between the highest and lowest values of rn. Thus v' is an empirical constant not necessarily the same as Po. Also, the a' and 6' are to be considered empirical. Because the experimental error increases very rapidly as c becomes small (9), values below about 0.1 rn were not included in finding the empirical constants in eq. 17. In addition, since previous results have indicated that deviations occur at fairly high concentrations (1-3) and because the equating of rn to c becomes worse at high concentrations, experimental results above about 1 M were not used. The values of v', a', and 6' obtained for several salts are shown in Table 1. Values of V2 obtained from a set of data should be independent of the method used to evaluate V2. In the ORNL report (16) covering his NaCl data, Vaslow determined values for V2 using an equation based on a4,/d,/e as suggested by the Debye-Hiickel theory. Values of V2 determined from eq. 19 were either identical to or slightly lower (never more than 0.02 ml mol-i or 0.1%) than the NaCl values recorded by Vaslow up to concentrations of 1 M. Before proceeding with the calculations, an analysis of the possible errors which may be encountered should be considered. Equation 9 indicates that the error in A, can be estimated from SV2 + SV20 + S(Bpc) + A,SC~'~ [20] SAP= c1 13 The values of V2 will be found from 4, and (84,/arn) (eq. 10). Even though the experimental values of 4, may be accurate to perhaps k0.002 (depending on the method and the

5 828 CAN. J. CHEM. VOL. 54, 1976 TABLE 2. Slopes, Bp, and intercepts, A,, of eq. 21 for the electrolytes shown in Fig. 2 Salt Bp Ap NaCl KC CaCI, FIG. 2. The left-hand-side of eq. 21 plotted against (c, - c,)/(c,'~~ - c,'~~). Top curve, CaCI,; middle curve, KCI; bottom curve, NaCl. The data were taken from Vaslow (12) and Dunn (15). concentration range where measured), the numerical evaluation of a derivative increases the error appreciably (17). Not only this, but V20 is obtained by extrapolation outside the range of experimental values which will introduce additional error. Contributions of several hundredths to as much as 0.1 to the error in A, might be anticipated from V2 and V20. Since the error in the concentration should normally be limited to something near f the error contributions of the last two terms in the numerator of eq. 20 should be small. The c1i3 term in the denominator of eq. 20 could increase the error by as much as a factor of 4 or 5 when c is as low as 0.01 (O.Ol1I ): Equation 9 can be recast as to eliminate the V20 obtained by extrapolation, and a similar error analysis would lead to an error estimate of several hundredths to one- tenth in the value of A,. Even to achieve this accuracy, very accurate data on 4, over a considerable concentration range must be available. By eliminating one of the constants, eq. 21 proved useful for testing experimental data and for evaluating A, and B,. The left-hand-side of eq. 21 is plotted against (c, - c,)/(~,'/~ - cl1i3) in Fig. 2 for the salts NaC1, KC1, and CaC1,. Very nice straight lines are obtained. The values for the slopes and intercepts, determined by linear regression analysis, are given in Table 2. A statistical error treatment (18) of the data in Fig. 2 indicated that the error in the slope and intercept might be as small as However, from our discussion of the errors in A,, above, we doubt that the accuracy is as good as the statistical analysis indicates. The curve fitting has apparently made a particular set of data internally more satisfactory than is truly allowable. Analysis of data for other salts, though also internally consistent, confirmed that the error in A, is probably larger than It was pointed out above that the value for ([(a In k)/(dp)],,,., - P/3) must be the same for all strong binary salts. Table 2 shows that NaCl and KC1 have identical A, within the experimental error. In addition, eq. 10 indicates that the ratio of A, for 1: 1 and 2: 1 salts must be given by (1, 3) and the ratio from Table 2 is = In addition to these numbers, we should also remember, as indicated by eq. 15, that the slope of 4, us. c1i3 at very low concentrations should be 314 of A,; the results in Fig. 1 for LiCl indicated a slope of 1.50 which gives A, = 2.00

6 BAHE AND JUNG 829 TABLE 3. Slopes, Bp, and intercepts, V,', for the 1 : 1 and 2: 1 electrolytes shown in Fig. 3. Salt BP VI0 CsBr BaC1, CaCI, RbF KF RbBr RbCl KC1 CsCl NaCl NaBr KI Nal LiCl K Br LiBr LiI to compare with the 2.01 obtained from NaCl and KCl. We therefore have three 1: 1 salts and one 2: 1 salt which give the same value for ([(a In k)/(dp)it,,., - p/3). We adopt a value of 2.0 f 0.1 and 6.7 i 0.3 as the value for A, in eq. 9 for 1 : 1 and 2: 1 salts, respectively. More will be said below about the resulting value of (8 In k/ap),,=,=, for water. Many other salts have been examined according to the procedure just outlined for eq. 21 but not all behave as consistently as the three cited. Extremely accurate experimental results are necessary to use eq Vaslow's data have been used preferentially because of their great accuracy and because the concentration range is large. Data on NaCl from other investigators have been examined, and, though none were apparently as accurate as those of Vaslow, the results confirmed the values obtained from Vaslow's data. Smith (19) measured 4, for NaCl at 25 "C over the concentration range of 0.002m to 0.09m. At these small concentrations where c = pom, eq. 16 can be cast in the integrated form as Using B, determined above from Vaslow's results this equation gave a value of 1.97 for A, by linear regression analysis, using Smith's data from 0.01m to 0.09m. Millero (20) also reported FIG. 3. The left-hand-side of eq. 24 plotted against c. The curves are arranged in the same order, from top to bottom, as the salts in Table 3 with CsBr on top and LiI on the bottom. Each increment on the ordinate represents 1 ml mol-'. The number to the left of each right angle represents the value of that intersection in rnl mol-'. 4, as a function of m for NaCl at 25 "C. Again using eq. 16 in the integrated form, which is valid at low concentrations, where c = pom, we evaluated 4: = 4, ~'~~ ~ for Millero's seven values between 0.01m and 0.20m and found an average of with a standard deviation of 0.05 with no discernable trend in the values. The universality of the A,'s can be demonstrated for many salts. Equation 9 can be rearranged to give so that the left-hand-side of eq. 24 plotted against c should give a straight line. The results of doing this using the universal values found above for A, are shown in Fig. 3. Very nice straight lines are obtained for fifteen 1 : 1 and two 2 : 1 salts. The intercepts, Pz0, and slopes, B,, are shown in Table 3. A further check on the consistency of the results may be obtained by testing the additivity of the TZ0's. A partial list of the possible comparisons is shown in Table 4. Only results from

7 TABLE 4. Comparison of A VZo for several I : I electrolytes Salts CAN. J. CHEM. VOL. 54, 1976 A VZ0. _._/ ' -/---a- NaCI-NaI LiCI-LiI NaCI-LiCl NaBr-LiBr NaI-LiI LiBr-LiC1 NaBr-NaC1 NaI-NaBr LiI -LiBr the data of Vaslow (1 1, 12) have been included since our previous work indicated that his data were very good. Comparisons of values which include data from other investigators give somewhat larger, but adequate, differences of differences. Allowing an error of about ml mol-' in T20 indicates that the V20 are additive. Discussion The partial molar volume data agree well with the model of structured ions in solution for a number of 1 : 1 and 2: 1 electrolytes. These results are further evidence supporting this model, and they show additional evidence for the validity of the field-dielectric-gradient force operating between ions in solution. All strong 1 : 1 electrolytes apparently assume a loose face-centered-cubic structure in solution while strong 2: 1 electrolytes assume a loose fluorite structure in solution. The value 2.0 for A, for 1 : 1 electrolytes allows the determination of the value for the pressure derivative of the dielectric constant of water. From eq. 10 we see that and if we use the value p = x 10" bar-' (21), we obtain for water at 1 bar. This value is considerably larger than the recent value of x IK) IX)LX) P bor FIG. 4. The dielectric constant of water as a function of pressure. 0, ref. 6: 0, ref. 5. The Owen el al. value at 100 bars was multiplied by the ratio of dielectric constants at P = I to put it on the same scale. The solid line is an arbitrary line drawn through thedata points with the slope at P = I predicted from the V, treatment. The dashed line is the slope drawn through k at 1 atm. bar-' (4) found by directly measuring k as a function of P. The discussion above pointed out some of the difficulties inherent in the direct measurement of k at various P to determine the value of this derivative. We have drawn in Fig. 4 a plot of the three data points reported by Harris el al. (5) at 25.6 "C along with the value from Owen el al. (4) at 100 bar corrected by the ratio of dielectric constants at 1 atm to put it on the same temperature scale. We have drawn an arbitrary line through the experimental k's with the slope our results indicate at 1 atm pressure. Until better data have been collected, or until a theory for the expected behavior of the dielectric constant as a function of pressure has been developed, we conclude that the slope generated by our T2 treatment is not impossible and may even be preferable to that obtained by direct measurement. The results obtained here also give a value for B, in eq. 9. The values for this coefficient, as expected, are specific to each salt, and have both positive and negative values as shown in Table 3. Since B can be determined from activity coefficient data and p for water is available, the pressure derivative of B, (db/ dp),,,.,, can also be evaluated for each salt, as shown by eq The values for (db/dp)t,n.s, shown in Table 5 for those salts for which B is available (1, 3), display both positive and negative values. Though the correlation is not perfect, in general for the same cation the value increases as the ionic weight of the anion decreases, and for the same anion the value

8 BAHE AND JUNG 83 1 TABLE 5. Values for the pressure derivative of B Salt (I m o ) (ml mol-' atm-i) KC1 NaCl KBr NaBr KI NaI LiBr LiCl CaCI, BaCI, *References 1 and 3. increases as the ionic weight of the cation increases. One of the referees has pointed out that eq. 23 of ref. 3 implies that the B's of eq. 1 should be additive, i.e., that BNacl - BKcl leads to &,+ - BK+, and a similar additive result should therefore be expected for (db/i?p)i. The values of B and of db/dp are not additive as can be seen by examining the various tables. The nonadditivity of the B's apparently resides in the inability to determine individual-ion activity coefficients. Writing is a mathematical convenience without physical significance; only the mean ionic activity coefficients can be measured (22). Let us cast the function If eq. 27 were physically valid then the lefthand-side of eq. 28 should be zero (all activity coefficients referring to the same concentration). The left-hand-side of eq. 28 is a thermodynamic relationship; the values are independent of any theory of behavior in solution. Experimental values of activity coefficients substituted into the left-hand-side of eq. 28 do not give a value of zero which confirms the physical invalidity of eq. 28. Correspondingly eq. 23 of ref. 3 substituted into the right-hand-side of eq. 28 implies that AB = 0, but the measured values of B do not give a value of zero. Thus the nonadditivity of the B's apparently resides in the impossibility of determining individual-ion activity coefficients. Guggenheirn (23) has considered the limitations on relationships between single-ion activity coefficients. His "specific salt" effects come into play in the B's and B,'s and therefore two salts which differ in one or both ions, even at the same concentration, are not in the same "media" as Guggenheim has defined "media." Comparison of the additivity of the V201s, however, as was done in Table 4, is legitimate since at zero concentration the effects of electrical interactions and specific salt effects has been removed. Finding V20's as was done above does not require that the model and the resulting equations be valid at all concentrations down to zero concentration. V20 has the properties of an integration constant which can be evaluated over the domain of the function. Operationally V20 can be evaluated by an extrapolation to zero concentration, but it can also be evaluated without such an extrapolation. The properties of the function (eq. 9) are such that V2O measures the partial molar volume of a salt when the contributions of the coulombic and field-dielectric-gradient electrical interactions to the volume have been removed from the structured model. Values of V20 evaluated over the domains of other functions, such as that of Debye and Hiickel, may also be obtained and compared. The values of V20 obtained from different functions will probably not be the same, yet each set can have a physical interpretation that can be valid and significant. The values of V20 obtained from the structured model are very close to those obtained from a square-root, linear function, the values in Table 3 often being smaller by a few hundredths of a ml mol-' (24). Acknowledgment It is a pleasure to acknowlege the support of the University of Wisconsin-Milwaukee Graduate School for one of us (L.W.B.), and the National Science Foundation through its Undergraduate Research Participation Program for the support for the other (K.A.J.). 1. L. W. BAHE. J. Phys. Chem. 76, 1062 (1972). 2. L. W. BAHE. J. Phys. Chem. 76, 1608 (1972). 3. L. W. BAHE and D. PARKER. J. Am. Chem. Soc. 97, 5664 (1975). 4. B. B. OWEN, R. C. MILLER, C. E. MILNER, and H. L. COGAN. J. Phys. Chem. 65, 2065 (1961).

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