1.9.5: Entropies- Solutions- Limiting Partial Molar Entropies

A key equation relates the chemical potential and partial molar entropy of solute-\(j\). For a given solute \(j\) in an aqueous solution,

\[\mathrm{S}_{\mathrm{j}}(\mathrm{aq})=-\left[\partial \mu_{\mathrm{j}}(\mathrm{aq}) / \partial T\right]_{\mathrm{p}}\]

In order to appreciate the importance of equation (a) we initially confine our attention to the properties of a solution whose thermodynamic properties are ideal. A given aqueous solution contains solute \(j\) at temperature \(\mathrm{T}\) and ambient pressure (which is close to the standard pressure).

\[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]

Using equation (a),

\[\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]

\(\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the partial molar entropy of solute \(j\) in an ideal aqueous solution having unit molality [1,2]. Therefore for both real and ideal solutions [3],

\[\operatorname{limit}\left(m_{j} \rightarrow 0\right) S_{j}(a q ; T ; p)=+\infty\]

In other words the limiting partial molar entropy for solute \(j\) is infinite. Interestingly if the aqueous solution contains two solutes \(j\) and \(\mathrm{k}\), then the following condition holds for solutions at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\).

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0 ; \mathrm{m}_{\mathrm{k}} \rightarrow 0\right)\left[\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{S}_{\mathrm{k}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\right]=\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{S}_{\mathrm{k}}^{0}(\mathrm{aq})\]

Similarly

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0 ; \mathrm{m}_{\mathrm{k}} \rightarrow 0\right)\left[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{k}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\right]=\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{k}}^{0}(\mathrm{aq})\]

The partial molar entropy of solute \(j\) in a real solution is given by equation (g).

\[\begin{aligned}
&S_{j}(a q ; T ; p)= \\
&S_{j}^{0}(a q ; T ; p)-R \, \ln \left(m_{j} / m^{0}\right)-R \, \ln \left(\gamma_{j}\right)-R \, T \,\left[\frac{\partial \ln \left(\gamma_{j}\right)}{\partial T}\right]_{p}
\end{aligned}\]

A given aqueous solution having thermodynamic properties which are ideal contains a solute \(j\), molality \(\mathrm{m}_{j}\). The partial molar entropy of the solvent is given by equation (h).

\[\mathrm{S}_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]

Hence,

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{S}_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]

When an ideal solution is diluted the partial molar entropy of the solvent approaches that of the pure solvent.

A given aqueous solution is prepared using \(1 \mathrm{~kg}\) of solvent and \(\mathrm{m}_{j}\) moles of solute \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the latter being close to the standard pressure. The entropy of the solution is given by equation (j).

\[\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{S}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}(\mathrm{aq})\]

In the event that the thermodynamic properties of the solution are ideal the entropy of the solution is given by equation (k).

\[\begin{aligned}
&\mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)= \\
&\quad \mathrm{M}_{1}^{-1} \,\left[\mathrm{S}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]
\end{aligned}\]

Interestingly,

\[\begin{aligned}
&\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)= \\
&\mathrm{M}_{1}^{-1} \, \mathrm{S}_{1}^{*}(\ell)+(0) \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \ln \left(0 / \mathrm{m}^{0}\right)\right]
\end{aligned}\]

But

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{m}_{\mathrm{j}} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)=0\]

In other words the entropy for an ideal solution in the limit of infinite dilution is given by the entropy of the pure solvent. For a real solution,

\[\mathrm{S}_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\phi \, \mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]

Hence,

\[\begin{aligned}
&\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right) \\
&=\mathrm{M}_{1}^{-1} \,\left[\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\phi \, \mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right] \\
&\quad+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right]
\end{aligned}\]

The difference \(\left[\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)-\mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)\right]\) yields the excess entropy, \(\mathrm{S}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)\).

Footnotes

[1] For a salt solution, the standard partial molar entropy of the salt is given by the sum of standard partial molar entropies of the ions. For a 1:1 salt, \(S_{j}^{0}(a q)=S_{+}^{0}(a q)+S_{-}^{0}(a q)\)

[2] Y. Marcus and A. Loewenschuss, Annu. Rep. Prog. Chem., Ser. C, Phys. Chem., 1984, 81, chapter 4.

[3] For comments on the entropy of dilution of salt solutions see (a classic paper), H. S. Frank and A. L. Robinson, J. Chem. Phys.,1940,8,933.

[4] For comments on partial molar entropies of apolar solutes in aqueous solutions see, H. S. Frank and F. Franks, J. Chem. Phys.,1968,48,4746.