1.5.7: Chemical Potentials- Solutions- Raoult's Law
A given closed system contains an aqueous solution; the solute is chemical substance \(j\). The system is at equilibrium at temperature \(\mathrm{T}\). The chemical potential of water in the aqueous solution is related to the mole fraction \(\mathrm{x}_{1}\) of water using equation (a) which is based on Raoult’s Law for the solvent.
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp} \nonumber \]
\[\text { By definition, at all } \mathrm{T} \text { and } \mathrm{p}, \operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{\mathrm{1}}=1 \nonumber \]
If ambient pressure is close to the standard pressure \(\mathrm{p}^{0}\), the chemical potential of solvent water in the aqueous solution is given by equation (c).
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
For an ideal solution, \(\mathrm{f}_{1} = 1\).
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
But for a solution \(\mathrm{x}_{1} < 1.0\) and so \(\ln \left(x_{1}\right)<0\). In other words, by adding a solute to water (forming an ideal solution) we stabilise the solvent. We define a quantity \(\Delta(\ell \rightarrow a q) \mu_{1}(\mathrm{~T}, \mathrm{p})\) using equation (e) which measures the change in chemical potential of water when one mole of water is transferred from water(\(\ell\)) to an ideal aqueous solution.
\[\Delta(\ell \rightarrow \mathrm{aq}) \mu_{1}(\mathrm{~T}, \mathrm{p})=\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
i.e. \(\Delta(\ell \rightarrow \mathrm{aq}) \mu_{1}(\mathrm{~T}, \mathrm{p} ; \mathrm{id})<0\)
In the case of a real solution, the extent of stabilisation depends on whether \(\mathrm{f}_{1}\) is either larger or smaller than unity. [Note that \(\mathrm{f}_{1}\) cannot be negative]. This line of argument leads to an important theme in the description of the properties of aqueous solutions. We compare the chemical potentials of water in real and in the corresponding ideal solutions. The difference is the excess chemical potential, \(\mu_{1}^{E}(a q ; T ; p)\).
\[\text { By definition, } \quad \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id}) \nonumber \]
\[\text { Hence } \quad \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{\mathrm{l}}\right) \nonumber \]
If \(\mathrm{f}_{1} > 1.0\), then \(\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})>0\); if \(\mathrm{f}_{1} < 1.0\), then \(\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})<0\). In the latter case, interactions involving solute and solvent are responsible for the fact that the properties of a given solution are not ideal and the fact that these interactions stabilise the solvent relative to that for an ideal solution.