How is the formula of mean activity coefficient derived?

Question

The mean activity coefficient is defiend as follows:

$$\gamma_\pm = (\gamma_+\gamma_-)^{1/2}.\tag{1}$$

If Debye-Hückel equation

$$-\log\gamma_i = 0.5z_i^2\mu^{1/2}\tag{2}$$

is used, then the mean activity coefficient has the form

$$-\log\gamma_\pm = 0.5|z_+z_-|\mu^{1/2},\label{eqn:3}\tag{3}$$

where $\mu$ is the ionic strength, $z_i$ is the charge of species $i,$ $\gamma_i$ is the activity coefficient.

But I'm confused about the equation \eqref{eqn:3}. It can be found as equation \eqref{eqn:3-38} in Snoeyink and Jenkins' Water Chemistry [1]:

[…] developed for ionic strengths of less than approximately $\pu{5E-3}$ and can be stated as

$$-\log\gamma_i = 0.5Z_i^2\mu^{1/2}.\tag{3-34}$$

Because anions cannot be added to a solution without an equivalent number of cations (and vice versa). it is impossible to determine experimentally the activity coefficient of a single ion. Therefore, Eqs. 3-34, 3-35, and 3-36 cannot be verified directly. However, it is possible to define, and measure experimentally, a mean activity coefficient, $\gamma_\pm,$ as,

$$\gamma_\pm = (\gamma_+\gamma_-)^{1/2}.\tag{3-37}$$

The Debye-Hückel and Güntelberg relationships can be extended to the mean activity coefficient thus:

$$-\log\gamma_\pm = 0.5|Z_+Z_-|\mu^{1/2},\label{eqn:3-38}\tag{3-38}$$

Below is my derivation process:

$$ \begin{align} -\log\gamma_+ &= 0.5z_+^2\mu^{1/2};\tag{4.1}\\ -\log\gamma_- &= 0.5z_-^2\mu^{1/2},\tag{4.2} \end{align} $$

so

$$ \begin{align} -\log\gamma_\pm &= -\log[10^{-0.5z_+^2\mu^{1/2}}\times 10^{-0.5z_-^2\mu^{1/2}}]^{1/2} \tag{5.1}\\ & = -\log[10^{-0.5\mu^{1/2}(z_+^2+z_-^2)}]^{1/2} \tag{5.2}\\ & = 0.25\mu^{1/2}(z_+^2+z_-^2). \tag{5.3} \end{align} $$

This result is different from $0.5|z_+z_-|\mu^{1/2}.$ Can someone tell me which step is wrong in my process?

Reference

1. Snoeyink, V. L.; Jenkins, D. Water Chemistry; Wiley: New York, 1980. ISBN 978-0-471-05196-1.

Answer 1

The problem is you were starting with an expression for the mean activity coefficient for salts where both the cation and anion were monovalent, and then attempting to derive, from this, a general expression for salts of all empirical formulas. To correct this, you need to start with the general expression for the mean activity coefficient of salts of any emprical formula.

If the empirical formula of the salt is of the form $\ce{A_pB_q}$, where A is the cation and B is the anion, then the general formula for the mean activity coefficient is:

$$\gamma_{\pm} = \sqrt[^{p+q}]{\gamma^p_+\gamma^q_-}$$

[Adapted from: https://en.wikipedia.org/wiki/Activity_coefficient, where I've taken the Wikipedia expression and substituted $\gamma_+$ for $\gamma_A$, and $\gamma_-$ for $\gamma_B$, to correspond to the nomenclature you are using.]

Charge balance dictates that $|z_+| = q$ and $|z_-| = p$. [E.g., in $\ce{A_2B_3}$, the charge on A must be 3+, and that on B must be 2- (or some integer multiple of those).]

Substituting, we have:

$$\gamma_{\pm} = \sqrt[^{|z_-|+|z_+|}]{\gamma^{|z_-|}_+\gamma^{|z_+|}_-}$$

If you use that in place of eqn. (3-37) in your post, and work through your calculations again, you will get the result show in eqn. (3-38), which is a general result for any values of p and q.