1.5.13: Chemical Potentials- Solutes- Mole Fraction Scale

For the most part we use either the molality scale or the concentration scale to express the composition of aqueous solutions. Nevertheless, the mole fraction scale is often used. Hence we express the chemical potential of solute \(j\), \(\mu_{j}\) as a function of mole fraction of solute \(j\), \(\mathrm{x}_{j} \left[=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)\right]\). Note that we are relating the property \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) to the composition of the solution using a different method from that used where the composition is expressed in terms of the molality or concentration of a solutes. By definition at fixed temperature and fixed pressure,

\[\begin{aligned}
&\mu_{j}(a q ; T ; p)= \\
&\quad \mu_{j}^{0}(a q ; T ; x-\text { scale })+R \, T \, \ln \left(x_{j} \, f_{j}^{*}\right)+\int_{p^{0}}^{p} V_{j}^{\infty}(a q ; T) \, d p
\end{aligned} \nonumber \]

\[\text { By definition, } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{f}_{\mathrm{j}}^{*}=1 \text { at all } \mathrm{T} \text { and } \mathrm{p} \text {. } \nonumber \]

\(f_{j}^{*}\) is the asymmetric solute activity coefficient on the mole fraction scale. The word 'asymmetric', although rarely used, emphasises the difference between \(f_{j}^{*}\) and the rational activity coefficients.

For solutions at ambient pressure, the integral term in equation (a) is negligibly small. At pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\),

\[\mu_{j}(a q)=\mu_{j}^{0}(a q ; x-\text { scale })+R \, T \, \ln \left(x_{j} \, f_{j}^{*}\right) \nonumber \]

For an ideal solution, \(f_{j}^{*}\) is unity. The reference state for the solute is the solution where the mole fraction of solute-\(j\) is unity. This is clearly a hypothetical solution but we assume that the properties of the solute j in this solution can be obtained by extrapolating from the properties of solute-\(j\) at low mole fractions [1]. For an ideal solution at ambient pressure and temperature,

\[\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}\right) \nonumber \]

\[\text { or, } \mu_{j}(\mathrm{aq} ; \mathrm{id})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}\right) \nonumber \]

Because, \(\mathrm{x}_{\mathrm{j}}<1.0, \quad \ln \left(\mathrm{x}_{\mathrm{j}}\right)<0\) Hence, \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})<\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })\). The solute is at a lower chemical potential than in the solution reference state [1].

Footnote

[1] The extrapolation defining the reference state as a solution wherein the mole fraction of solute is unity might seem strange. In fact such long extrapolations are common in chemical thermodynamics. For example at \(0.1 \mathrm{~MPa}\) and \(298.15 \mathrm{~K}\), liquid water is the stable phase. At \(0.1 \mathrm{~MPa}\) and \(273.15 \mathrm{~K}\) both water(\(\ell\)) and ice-1h are the stable phases. Nevertheless we know that, assuming water(\(g\)) is an ideal gas, the volume occupied by \(0.018 \mathrm{~kg}\) of water(\(g\)) at \(273 \mathrm{~K}\) and \(0.1 \mathrm{~MPa}\) equals \(22.4 \mathrm{~dm}^{3}\). The fact that this involves a long extrapolation into a state where water vapour is not the stable phase does not detract from the usefulness of the concept.