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