10.6: Mean Ionic Activity Coefficients from Osmotic Coefficients
Recall that \(\g_{\pm}\) is the mean ionic activity coefficient of a strong electrolyte, or the stoichiometric activity coefficient of an electrolyte that does not dissociate completely.
The general procedure described in this section for evaluating \(\g_{\pm}\) requires knowledge of the osmotic coefficient \(\phi_m\) as a function of molality. \(\phi_m\) is commonly evaluated by the isopiestic method (Sec. 9.6.4) or from measurements of freezing-point depression (Sec. 12.2).
The osmotic coefficient of a binary solution of an electrolyte is defined by \begin{gather} \s{ \phi_m \defn \frac{\mu\A^*-\mu\A}{RTM\A\nu m\B} } \tag{10.6.1} \cond{(binary electrolyte solution)} \end{gather} That is, for an electrolyte the sum \(\sum_{i\neq \tx{A}}m_i\) appearing in the definition of \(\phi_m\) for a nonelectrolyte solution (Eq. 9.6.11) is replaced by \(\nu m\B\), the sum of the ion molalities assuming complete dissociation. It will now be shown that \(\phi_m\) defined this way can be used to evaluate \(\g_{\pm}\).
The derivation is like that described in Sec. 9.6.3 for a binary solution of a nonelectrolyte. Solving Eq. 10.6.1 for \(\mu\A\) and taking the differential of \(\mu\A\) at constant \(T\) and \(p\), we obtain \begin{equation} \dif\mu\A = -RTM\A\nu(\phi_m\dif m\B + m\B\dif\phi_m) \tag{10.6.2} \end{equation} From Eq. 10.3.9, we obtain \begin{equation} \dif\mu\B = RT\nu\left(\dif\ln\g_{\pm} + \frac{\dif m\B}{m\B}\right) \tag{10.6.3} \end{equation} Substitution of these expressions in the Gibbs–Duhem equation \(n\A \dif\mu\A + n\B \dif\mu\B = 0\), together with the substitution \(n\A M\A = n\B/m\B\), yields \begin{equation} \dif\ln\g_{\pm} = \dif\phi_m + \frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.4} \end{equation} Then integration from \(m\B = 0\) to any desired molality \(m'\B\) gives the result \begin{equation} \ln\g_{\pm}(m'\B) = \phi_m (m'\B) - 1 + \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.5} \end{equation} The right side of this equation is the same expression as derived for \(\ln\g\mbB\) for a nonelectrolyte (Eq. 9.6.20).
The integrand of the integral on the right side of Eq. 10.6.5 approaches \(-\infty\) as \(m\B\) approaches zero, making it difficult to evaluate the integral by numerical integration starting at \(m\B = 0\). (This difficulty does not exist when the solute is a nonelectrolyte.) Instead, we can split the integral into two parts \begin{equation} \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.6} \end{equation} where the integration limit \(m''\B\) is a low molality at which the value of \(\phi_m\) is available and at which \(\g_{\pm}\) can either be measured or estimated from the Debye–Hückel equation.
We next rewrite Eq. 10.6.5 with \(m\B'\) replaced with \(m\B''\): \begin{equation} \ln\g_{\pm}(m''\B) = \phi_m (m''\B) - 1 + \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.7} \end{equation} By eliminating the integral with an upper limit of \(m\B''\) from Eqs. 10.6.6 and 10.6.7, we obtain \begin{equation} \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \ln\g_{\pm}(m''\B) - \phi_m (m''\B) + 1 + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.8} \end{equation} Equation 10.6.5 becomes \begin{equation} \ln\g_{\pm}(m'\B) = \phi_m(m'\B) - \phi_m(m''\B) + \ln\g_{\pm}(m''\B) + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.9} \end{equation} The integral on the right side of this equation can easily be evaluated by numerical integration.