1.1.10: Activity Coefficient- Two Neutral Solutes- Solute + Trace Solute i
A given solution is prepared using n1 moles of water(\(\lambda\)) together with \(n_{i}\) and \(n_{j}\) moles of neutral solutes \(i\) and \(j\) respectively at temperature \(\mathrm{T}\) and pressure \(p\) (which is close to the standard pressure \(p^{0}\)). The mass of water is \(\mathrm{n}_{1} \, \mathrm{M}_{1}\) where \(\mathrm{M}_{1}\) is the molar mass of water. Hence the molalities of solutes \(i\) and \(j\) are \(\mathrm{m}_{\mathrm{i}} \left(=\mathrm{n}_{\mathrm{i}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)\) and \( \mathrm{m}_{\mathrm{j}}\left(=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)\) respectively. The chemical potential of water in the aqueous solution \(\mu_{1}(\mathrm{aq})\) is related to the molality of solutes using equation (a) where \(\phi\) is the practical osmotic coefficient and \(\mu_{1}^{*}(\lambda)\) is the chemical potential water(\(\lambda\)) at the same \(\mathrm{T}\) and \(p\).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right) \nonumber \]
The chemical potentials of the two solutes, \(\mu_{\mathrm{i}}(\mathrm{aq})\) and \(\mu_{\mathrm{j}}(\mathrm{aq})\), are related to \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\) together with corresponding activity coefficients, \(\gamma_{i}\) and \(\gamma_{j}\) using equations (b) and (c).
\[\mu_{\mathrm{i}}(\mathrm{aq})=\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right) \nonumber \]
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Here \(\mu_{i}^{0}(\mathrm{aq})\) is the reference chemical potential for solute \(i\) in a solution where \(\mathrm{m}_{\mathrm{j}}=0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\), \(\mathrm{m}_{\mathrm{i}}=1 \mathrm{~mol} \mathrm{~kg}\) and \(\gamma_{i} = 1\). A similar definition operates for solute \(j\). For the mixed solution at all \(\mathrm{T}\) and \(p\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{i}}=1 \nonumber \]
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1 \nonumber \]
In these terms, the thermodynamic properties of solute \(i\) are not ideal as a consequence of \(i-i\), \(j-j\) and \(i-j\) solute-solute interactions. A similar comment concerns the thermodynamic properties of solute \(j\). With increase in molalities \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\), so the extent to which the thermodynamic properties deviate from ideal increases; i.e. for solute \(j\), \(\gamma_{j} \neq 1\) and for solute \(i\), \(\gamma{i} \neq 1\). Such deviations can be understood in terms of \(i-i\), \(j-j\), and \(i-j\) solute-solute interactions.
In some applications of this analysis, solute \(i\) is present in trace amounts and so \(\gamma_{i}\) in the absence of solute \(j\) would be close to unity. However as the molality of solute \(j\) is increased, the thermodynamic properties of solute \(i\) deviate increasingly from ideal as a result of solute \(i\) \(\rightleftarrows\) solute \(j\) interactions. This feature can be described [1,2] quantitatively using equation \((\mathrm{f})\) where \(\beta_{1}, \beta_{2} \ldots\) describe the role of pairwise i-j , triplet \(i-j-j \ldots \ldots\) solute-solute interactions.
\[\ln \left(\gamma_{i}\right)=\beta_{1} \,\left(m_{j} / m^{0}\right)+\beta_{2} \,\left(m_{j} / m^{0}\right)^{2}+\ldots \ldots \nonumber \]
Depending on the signs of the \(\beta\) - coefficients , added solute \(j\) can either stabilise or destabilise solute-\(i\); i.e. either lower or raise \(\mu_{i}(a q)\) relative to that in a solution having ideal thermodynamic properties.
Footnotes
[1] Based on an analysis suggested by C. Wagner, Thermodynamics of Alloys, Addison-Wesley, Reading, Mass., 1952, pages 19-22.
[2] As quoted by G. N. Lewis and M. Randall, Thermodynamics, 2nd edn., revised by K. S. Pitzer and L Brewer, McGraw-Hill, New York, 1961, page 562.