Thermodynamically Consistent Equations for the Accurate Description of the Logarithm of the Solvent Activity and Related Properties of Electrolyte Solutions with a Unique Set of Parameters: Critical Analysis of the Mean Activity Coefficient Evaluation

Abstract

Thermodynamically consistent equations for the accurate description of the dependence on concentration of the properties of electrolyte solutions are proposed, based on a polynomial of order i/4 and a unique set of adjustable coefficients. The equation for the logarithm of the solvent activity ln a₁(m) was derived first, followed by analogous expressions for the osmotic coefficient ϕ(m), the solvent activity a₁(m) and activity coefficient γ₁(m) and finally for the mean ionic activity coefficient $\gamma_{ \pm }^{o} (m)$. The descriptive capability of these equations was verified and favorably compares with the extended Pitzer equation. Finally, it was demonstrated that the evaluation of $\ln \gamma_{ \pm }^{o} (m)$ is strongly influenced by the fitting capability of the expression used in the correlation of ϕ(m) in the low molality region.

Appendix

The explicit form of Eq. 8 is,

$$\ln a_{1} = \left( {\ln a_{1} } \right)_{z = 0} + \left( {\frac{{\partial \ln a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)_{z = 0} z + \frac{1}{2!}\left( {\frac{{\partial^{2} \ln a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{2} }}} \right)_{z = 0} z^{2} + \cdots + \frac{1}{n!}\left( {\frac{{\partial^{n} \ln a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{n} }}} \right)_{z = 0} z^{n}$$

(A1)

Comparing Eq. A1 with Eq. 8, Eq. 9 is obtained,

$$A_{i} = \frac{1}{i!}\left( {\frac{{\partial^{i} \ln a_{1} }}{{\partial z^{i} }}} \right)_{z = 0}$$

(9)

In the same way, the explicit form of Eq. 18 is,

$$a_{1} = \left( {a_{1} } \right)_{z = 0} + \left( {\frac{{\partial a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)_{z = 0} z + \frac{1}{2!}\left( {\frac{{\partial^{2} a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{2} }}} \right)_{z = 0} z^{2} + \cdots + \frac{1}{n!}\left( {\frac{{\partial^{n} a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{n} }}} \right)_{z = 0} z^{n}$$

(A2)

Comparing Eq. A2 with Eq. 18, Eq. 19 is obtained,

$$B_{i} = \frac{1}{i!}\left( {\frac{{\partial^{i} a_{1} }}{{\partial {\kern 1pt} z^{i} }}} \right)_{z = 0}$$

(19)

The relationship between A_i and B_i is obtained from the successive differentiation of ln a₁(m). The first derivative is,

$$\frac{{\partial \ln a_{1} }}{\partial z} = \frac{1}{{a_{1} }}\frac{{\partial a_{1} }}{\partial z}$$

(A3)

Taking the limit for z → 0 and bearing in mind that $\mathop {\lim }\limits_{z \to 0} a_{1} = 1$,

$$\mathop {\lim }\limits_{z \to 0} \frac{{\partial \ln a_{1} }}{\partial z} = A_{1} = \mathop {\lim }\limits_{z \to 0} \left[ {\frac{1}{{a_{1} }}\frac{{\partial a_{1} }}{\partial z}} \right] = B_{1}$$

(A4)

Differentiating Eq. A3,

$$\frac{{\partial^{2} \ln a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{2} }} = \frac{{a_{1} \frac{{\partial^{2} a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{2} }} - \left( {\frac{{\partial a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }}{{a_{1}^{2} }}$$

(A5)

Taking the limit for z → 0 and considering Eqs. 9 and 19,

$$\mathop {\lim }\limits_{z = 0} \frac{{\partial^{2} \ln a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{2} }} = 2A_{2} = \mathop {\lim }\limits_{z = 0} \left[ {\frac{{a_{1} \frac{{\partial^{2} a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z^{2} }} - \left( {\frac{{\partial a_{1} }}{{\partial {\kern 1pt} {\kern 1pt} z}}} \right)^{2} }}{{a_{1}^{2} }}} \right] = 2B_{2} - A_{1}^{2}$$

(A6)

Repeating the differentiating process (until the tenth derivative), as well as the application of the limiting condition, taking into account Eqs. 11–13, as well as Eqs. 20–23, and applying the relationships $A_{j}^{\text{o}}$ = A_i−4 and $B_{j}^{\text{o}}$ = B_i−4, Eqs. 25–33 are finally obtained.