J. Chem. SOC., Faraday Trans. I , 1984,80, 55-60 A Consideration of Pitzer’s Equations for Activity and Osmotic Coefficients in Mixed Electrolytes BY ROBERTO IALENTI Istituto Universitario Navale, Via Amm. Acton 38, 80133 Napoli, Italy AND RAFFAELE CARAMAZZA* Facolta di Farmacia, Universita di Napoli, Via L. Rodino 22, 80138 Napoli, Italy Received 25th February, 1983 The equations developed by Pitzer and coworkers in order to calculate the activity coefficients of two mixed electrolytes in aqueous solutions are transformed into more simple equations. It is demonstrated that these new equations do not satisfy one of the necessary requirements of thermodynamic arguments, the only exception being when there are two electrolytes that have a common ion and are of the same type. Many authors have already dealt extensively with the theoretical aspects of the ionic interactions of electrolytes in aqueous s~lutions.~-~ Much work has been done by Pitzer and coworkers,8-11 who, on the basis of statistical-mechanical considerations and accounting also for the effects of short-range forces, obtained equations which reproduce, accurately and within a rather wide composition range, experimentally measurable activity and osmotic coefficients of single electrolytes.The most interesting part of their work is the possibility of predicting the properties of mixtures of electrolytes starting from those of the single components, by means of analytic relations whose compliance with thermodynamic principles have never been challenged. However, as will be seen in this paper, when Pitzer’s equations for evaluating the activity coefficients of mixtures of two electrolytes are transformed into more simple and compact analytical expressions, a necessary condition derived by a thermodynamic argument is not always satisfied by the new relations.For simplicity we will consider here only pairs of electrolytes of the types 1 : 1, 1 : 2 and 2: 1, for which it is possible to set 2, = vx and 2, = vM. THEORY First we rearrange Pitzer’s equations8 for the activity and osmotic coefficients of a single electrolyte MX as a function of ionic strength I 5556 CONSIDERATION OF PITZER'S EQUATIONS At 25 OC and 1 atm we have (3) lny,, = -0.39212 2 I x ( +:.2rh+ 1.667 In (1 + 1.211) Btx = PMx +Ptx exp ( - 21;) Ckx = 1.5CtX where BhX, PAx and C& are characteristic parameters for any single electrolyte and are reported elsewhere.12a For mixtures of electrolytes, if we neglect the effect of interactions among ions of the same charge, the general equation for calculating the activity coefficient of one electrolyte in the mixture, written in molal terms, is 2VM lnyMX = lnyEL+- mu [BMu+@mcZc) cMul vMX a 2VX +- Z mc [Bcx + (Zmc 2,) Ccxl vMX c where MX is the electrolyte selected and the sums run over all cations (including M) denoted by the subscript c and all anions (including X) denoted by the subscript a.The coefficients Bca, BLa and C,, are given by the following expressions In the case of mixtures of two electrolytes, MX = A and NY = B, if 1 A and 1, are the respective ionic strengths, it is possible to demonstrate that, bearing in mind eqn (l), we can obtain from eqn (7) the following two equations where y i and y g are the activity coefficients of the single electrolytes in separate solutions at ionic strength I = IA+IB.The transformation is performed by means of the following devices: (a) substitution of 2, with vu and 2, with v, as noted earlier, (b) substitution of the ionic molalities in terms of the ionic strengths I A and I B using the following equations : 24, m, = - vu vcu 24, ma=- vc vc,R. IALENTI AND R. CARAMAZZA 57 where a = X when c = M and a = Y when c = N, and (c) expression of the resulting equation for the electrolyte A as a function of I and IB and that for the electrolyte B as a function of I and I A . In this way we obtain the following relationships between A , and A , and Pitzer's parameters: (BNX + B&X I ) (BMY + BMY I ) + - vA vB vY 4vx (16) where BCu, BEu and Ccu are the same as in eqn (8)-( 10).The B, and B, parameters are obtained from eqn (15) and (16) by substituting subscript A for B, M for N and X for Y and vice versa. We now demonstrate, by a thermodynamic argument, what conditions must be satisfied by the parameters Ai and Bj of eqn (1 1) and (12). Let us suppose that the activity coefficients of the two electrolytes may be expressed by equations of the following type lny, = lnyO,+C. AjIh (17) lny, = lnyL+C. BjZJ (18) i i with j = 1, 2, . . . k. First we will show that for each eqn (1 7) and (1 8) the maximum number of parameters cannot be more than two (j < 2) and then we will deduce the relationships that must hold between them.We now apply to our system the condition of cross-differentiation which must be obeyed by the chemical potentials, and so obtain, with the ionic strength as an independent variable, the following equation: vN vY (T)IA a In YA = vM vX (I) a In YB * ZB From this equation, applied to eqn (17) and (18), following a treatment reported elsewherel3 and bearing in mind that IB = I - I , and that the coefficients A, functions of I only, we can obtain the following equality: Bi are (20) This cannot be satisfied identically for all I, IA, if the left-hand side contains any terms of the form ImIz, since the right-hand side contains no such terms. This means that ( I - I*) cannot in fact take a power greater than unity, which implies that A, = 0 for all j >, 3 and (dAj/dI) = 0 for all j > 2; replacement of IA by ( I - I B ) in eqn (20) allows an exactly analogous conclusion to be drawn about the coefficient Bi.58 CONSIDERATION OF PITZER'S EQUATIONS Eqn (20) therefore reduces to where A , and B, are constants, but A , and B, may be functions of I.However, eqn (21) is satisfied identically for all IA only if which, on integration, yields where K is an integration constant. However, there is another relationship between parameters of eqn (1 7) and ( 1 8) that is obtained if the Gibbs-Duhem equation is applied to our ternary system. It is possible to demonstrate that if (a) I = IA + IB is held constant, (b) two limiting cases are considered, that is IA = 0 and IB = 0, and (c) the corresponding activities of the solvent are expressed as a function of osmotic coefficients, then we can obtain the following relationship 2 vM vX (B1+$B2 I ) - v N vY I ) = 7 i V N v Y ( 4 A - l)-vM vX(dB- l)1 (24) where 4A and 4B are the osmotic coefficients in the two limiting cases corresponding to the single electrolytes separately dissolved at ionic strength I.We may summarize the discussion thus far by stating that when activity coefficients of binary mixed electrolytes obey eqn (1 7) and (1 8), it is possible to conclude : (a) there are not more than two non-zero parameters A , B in each sum, (b) only A , and B, are functions of I, while A , and B, must be constant and (c) the parameters are inter-related by eqn (23) and (24).(All these conditions are in fact satisfied in the systems NaCl + CoCl, and CaCl, + CoCl, studied by Dowries.'*) Returning to eqn (1 5) and (16), it is easy to see that the parameters A , and A , are both functions of I, as Pitzer's Hca values, which according to eqn (9), are functions of I, appear in eqn (16). The same is also valid for B, and B,. So, there is evident disagreement between Pitzer's equations, rewritten in the form of eqn (1 1 ) and (12), and the thermodynamic requirement derived from the cross- differentiation condition, eqn (19), which implies the constancy of A , and B,. These parameters are zero only if the mixtures are formed by two electrolytes of the same type and having one common ion, i.e. a system for which Hunde's rule is valid.It is possible to verify this result bearing in mind that, in eqn (15) and (16), if the electrolytes have a common anion, MX = MY = A and NY = NX = B, or, if they have a common cation, MX = NX = A and NY = MY = B. For such mixtures it also follows that A , = -B,. Therefore only in such 'common-ion' cases is the above- mentioned requirement satisfied. As to the two relationships that must hold between the parameters Aj and Bj, expressed by eqn (23) and (24), it is possible to verify that the second one is satisfied,R. IALENTI AND R. CARAMAZZA 59 but the first one is not. In fact, bearing in mind eqn (15) and (16), it can be shown that the left-hand side of eqn (23) is equal to ( Bb I - BB) 8v v v 8vA vM vX Y ( ~ ; ~ - ~ A ) + V A VB + 8vM vy(BMy - B d y I ) + 8vN vX(BNX -B&XI).(25) On the basis of the eqn (8) and (9) this sum must be a function of Z and so is not constant, as required by eqn (23), except for mixtures of electrolytes of the same type and with a common ion, because in this case it is equal to zero. Eqn (24) is, however, satisfied by Pitzer’s equations. Its left-hand side may be rewritten as Using eqn (8) and (9) again and by comparison with eqn (2), ( 4 ) and (9, it is possible to show that for any type of mixture this expression is indeed identical with the right-hand side of eqn (24). CONCLUSION It follows from the preceding analysis that the equations of Pitzer and coworkers may be considered valid for evaluating the activity and osmotic coefficients of single electrolytes. However, they are in disagreement with thermodynamic theory for mixed electrolytes, except for the above-mentioned special cases which are related to Hunde’s rule.The same conclusions are obtained if additional parameters such as 8, 8‘ and v/, which are related to the differences in the interactions between ions of the same sign, are considered in our previous treatment. According to the Bronsted principle of specific interactions3 these terms should be equal to zero, and Pitzer12b affirms that ‘ they have only a small effect, if any, on mixing electrolytes, the principal effects arising from differences in the pure-electrolyte parameters $, B1 and 0. Nevertheless, if the 8, 8’ and ly parameters are introduced into eqn (7), as rarely occurs, they will appear in eqn (17) or (16) as additional terms for A , and A,.These last two coefficients will remain as functions of I, in contrast with the thermodynamic requirement previously demonstrated. Moreover, we need to remember that it is common practice to obtain the 8 and v/ values for a mixture of two electrolytes by calculating the differences between the experimental and theoretical values of 4 or lny calculated using the appropriate Pitzer’s equations with zero values for 8 and v / . Therefore, such terms seems to be more useful for an accurate reproduction of the experimental data than for the extension of the ionic interaction theory to mixed electrolytes. S. R. Milner, Philos. Mag., 1912, 23, 551 ; 1913, 25, 742. * G. N. Lewis and G. A. Linhart, J. Am. Chem. SOC., 1919,41, 1952. J. N. Bronsted, J. Am. Chem. Soc., 1922,44, 877, 938. G. N. Lewis and M. Randall, J. Am. Chem. Soc., 1921,43, 1 112. P. Debye and E. Huckel, Phys. Z., 1923, 24, 185; 344; 1924,25, 97. E. A. Guggenheim, Philos. Mag., 1935, 19, 588. K. S. Pitzer, J. Phys. Chem., 1973, 77, 268. K. S. Pitzer and G. Mayorga, J. Phys. Chem., 1973,77, 2300. ’ E. A. Guggenheim and J. C. Tuegeon, Trans. Faraday SOC., 1955,51, 747.60 CONSIDERATION OF PITZER’S EQUATIONS lo K. S. Pitzer and G. Mayorga, J. Solution Chem., 1974, 3, 539. l1 K. S. Pitzer and G. Mayorga, J. Am. Chem. SOC., 1974,%, 5701. l2 M. R. Pytkowicz, Activity Coeficients in Electrolyte Solutions (C.R.C. Press, Boca Raton, Florida, l3 H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold, New York, l4 C. J. Downes, J . Solution Chem., 1979, 3, 191. 1979), chap. 7, (a) p. 157, (b) p. 187. 1958), chap. 14, p. 620. (PAPER 3/302)