1.5.11: Chemical Potentials- Solutes
A given aqueous solution is prepared using \(1 \mathrm{~kg}\) of water at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The molality of solute \(j\) is \(\mathrm{m}_{j}\). The chemical potential of solute \(j\), \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is related to \(\mathrm{m}_{j}\) using equation (a).
\[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{\mathrm{a}}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp} \nonumber \]
\[\text { By definition, } \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 \nonumber \]
\(\mu_{j}^{0}(\mathrm{aq} ; \mathrm{T})\) is the chemical potential of solute \(j\) in an ideal solution (where \(\gamma_{j} =1\)) at temperature \(\mathrm{T}\) and standard pressure \(\mathrm{p}^{0}\left[=10^{5} \mathrm{~Pa}\right]\).
For solutions at ambient pressure which is close to \(\mathrm{p}^{0}\), \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is related to \(\mathrm{m}_{j}\) using equation (c).
\[\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Henry’s law forms the basis of equations (a) and (c).