1.5.16: Chemical Potentials- Solute- Molality and Mole Fraction Scales
The chemical potential of solute j in aqueous solution at temperature \(\mathrm{T}\) and at close to ambient pressure is related to the molality \(\mathrm{m}_{j}\) and mole fraction \(\mathrm{x}_{j}\) [1].
\[\begin{aligned}
\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \\
&=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}} \, \mathrm{f}_{\mathrm{j}}^{*}\right)
\end{aligned} \nonumber \]
Therefore,
\[\begin{aligned}
\ln \left(\mathrm{f}_{\mathrm{j}}^{*}\right)=\ln \gamma_{\mathrm{j}} &+\ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{x}_{\mathrm{j}} \, \mathrm{m}^{0}\right) \\
&+(\mathrm{l} / \mathrm{R} \, \mathrm{T})\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x} \text { - scale })\right]
\end{aligned} \nonumber \]
For dilute solutions [1], \(\left(1 / \mathrm{M}_{1}\right)>\mathrm{m}_{\mathrm{j}}\). Hence \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}\right)\) equals \(\left(m^{0} \, M_{1}\right)^{-1}\), a dimension-less quantity. Therefore,
\[\begin{aligned}
\ln \left(f_{j}^{*}\right)=\ln \gamma_{j} &-\ln \left(m^{0} \, M_{1}\right) \\
&+(1 / R \, T)\left[\mu_{j}^{0}(a q ; T ; p)-\mu_{j}^{0}(a q ; T ; p ; x-s c a l e)\right]
\end{aligned} \nonumber \]
It is unrealistic to expect that \(\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) equals \(\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})\) because the two reference states for solute-\(j\) are quite different. In general terms, \(\mathrm{f}_{\mathrm{j}}^{*}\) does not equal \(\gamma_{j}\) for the same solution. Nevertheless, both \(\mathrm{f}_{\mathrm{j}}^{*}\) and \(\gamma_{j}\) tend to the same limit, unity, as the solution approaches infinite dilution.
Hence as \(\mathrm{n}_{j}\) tends to zero,
\[\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{\mathrm{l}}\right) \nonumber \]
For example, in the case of aqueous solutions at \(298.15 \mathrm{~K}\), \(\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{1}\right)\) equals \(\left(-9.96 \mathrm{~kJ} \mathrm{~mol}^{-1}\right)\) meaning that, with respect to the reference states for the two solutions, the chemical potential of solute \(j\) is higher on the mole fraction scale than on the molality scale. Combination of equations (b) and (d) yields an equation relating the two activity coefficients with the two terms describing the composition of the solution.
\[\ln \left(f_{j}^{*}\right)=\ln \gamma_{j}+\ln \left(m_{j} \, M_{1} / x_{j}\right) \nonumber \]
The term ‘unitary’ is sometimes used to describe reference chemical potentials on the mole fraction scale, \(\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{j}}=1\right)\). The term \(\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{1}\right)\) in equation (d) is called cratic [2] because it refers to different amounts of solute and solvent which are mixed to form reference states for the solute on molality, mole fraction and concentration scales. The impression is sometimes given that standard states for solutes based on the mole fraction scale (sometimes identified as the unitary scale) are more fundamental but there is little experimental evidence to support this view.
Footnotes
[1] For a solution prepared using \(\mathrm{w}_{1}\) kg of water and \(\mathrm{n}\) moles of solute \(j\),
\(\begin{gathered}
\mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\left(\mathrm{w}_{1} / \mathrm{M}_{1}\right)+\mathrm{n}_{\mathrm{j}}\right] \text { and } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1} \, \\
\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \,\left[\left(\mathrm{w}_{\mathrm{l}} / \mathrm{M}_{\mathrm{l}}\right)+\mathrm{n}_{\mathrm{j}}\right] / \mathrm{w}_{1} \, \mathrm{m}^{0} \, \mathrm{n}_{\mathrm{j}} \\
\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}=\left[\left(1 / \mathrm{M}_{1}\right)+\mathrm{m}_{\mathrm{j}}\right] / \mathrm{m}^{0}
\end{gathered}\)
[2] The terms ' unitary ' and ' cratic ' were suggested by R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.