1.11.2: Gibbs-Duhem Equation- Salt Solutions- Osmotic and Activity Coefficients
We consider an aqueous solution prepared using \(1 \mathrm{~kg}\) of solvent, water, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (\(\cong p^{0}\)). For an aqueous salt solution, the chemical potential \(\mu_{j}(\mathrm{aq})\) for salt \(j\) at molality \(\mathrm{m}_{j}\) is given by equation (a) where \(\gamma_{\pm}\) is the mean ionic activity coefficient of the salt.
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right) \nonumber \]
By definition, at all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1 \nonumber \]
For the solvent, water,
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{\star}(\ell)-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
By definition, at all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0 \nonumber \]
Chemical potentials \(\mu_{j}(\mathrm{aq})\) and \(\mu_{1}(\mathrm{aq})\) are linked by the Gibbs-Duhem equation. This,
\[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+v \, m_{j} \, d \mu_{j}(a q)=0 \nonumber \]
\[\begin{aligned}
&\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\
&\quad+\mathrm{v} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]=0\right.
\end{aligned} \nonumber \]
\[\begin{aligned}
&-v \, R \, T \, d\left[\phi \, m_{j}\right] \\
&+v \, m_{j} \, v \, R \, T \, d\left[\ln (Q)+\ln \left(m_{j}\right)+\ln \left(\gamma_{\pm}\right)-\ln \left(m^{0}\right)\right]=0
\end{aligned} \nonumber \]
\[-\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{v} \, \mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)+\mathrm{d} \ln \left(\gamma_{\pm}\right)\right\}=0 \nonumber \]
\[-\phi \, d m_{j}-m_{j} \, d \phi+v \, m_{j} \, d \ln \left(m_{j}\right)+v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=0 \nonumber \]
Then,
\[-\phi \, d m_{j}-m_{j} \, d \phi+v \, d m_{j}+v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=0 \nonumber \]
\[v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=\phi \, d m_{j}-v \, d m_{j}+m_{j} \, d \phi \nonumber \]
Or,
\[\mathrm{d} \ln \left(\gamma_{\pm}\right)=(\phi-v) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{v} \, \mathrm{m}_{\mathrm{j}}} + \frac{\mathrm{d} \phi}{\mathrm{v}} \nonumber \]
For a solute where one mole of pure solute forms one mole of solute in solution,
\[\mathrm{d} \ln \left(\gamma\right)=(\phi-1) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}} + \mathrm{d} \phi \nonumber \]
Then,
\[\ln (\gamma)=(\phi-1) \,+\int_{0}^{\mathrm{m}(\mathrm{j})}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right) \nonumber \]