1.12.17: Expansions- Isentropic- Solutions- Apparent and Partial Molar
A given solution is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute \(j\). The volume of the system is defined by equation (a).
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
We consider the case where the closed system is at equilibrium and hence where the affinity for spontaneous change is zero. The entropy of the system (at equilibrium) is defined by the same set of independent variables. Thus
\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
The system is perturbed at constant pressure by a change in temperature. The path followed by the system is such that the affinity for spontaneous change remains at zero (i.e. at equilibrium) and that the entropy of the system \(\mathrm{S}(\mathrm{aq})\) remains constant at that given by equation (b).
The equilibrium isentropic expansion of the system is defined by equation (c).
\[\mathrm{E}_{\mathrm{s}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq}), \mathrm{A}=0} \nonumber \]
\(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\) is an extensive property of the system. Nevertheless it is convenient to consider an intensive property. For example, \(\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{A}=0 ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)\) is the equilibrium isentropic expansion of a solution molality \(\mathrm{m}_{j}\) prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)).
For a system comprising pure solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\) we define a molar (equilibrium) isentropic expansion, \(\mathrm{E}_{\mathrm{S}}^{*}(\ell)\); equation (d).
\[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{A}=0)=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\ell), \mathrm{A}=0} \nonumber \]
The volume of a solution, molality \(\mathrm{m}_{j}\), prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) is related to the composition using either equations (e) or (f).
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
A key problem emerges. We note that the conditions on the partial differential in equation (c) relate to the entropy of the aqueous solution. The latter condition is not the same as that invoked in equation (d) which refers to the molar entropy of the pure solvent. We could of course differentiate equation (e) with respect to temperature at fixed entropy S(aq). However we would encounter a term \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})}\). This is a complicated derivative where we might have hoped for a term \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\ell)}\). The way forward is to accept the problem and define a property, by analogy with the corresponding isobaric property, a property \(\phi\left(\mathrm{E}_{\mathrm{S} j} ; \text { def }\right)\) which has the appearance of proper thermodynamic apparent property. Then,
\[\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{A}=0 ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell ; \mathrm{A}=0)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right) \nonumber \]
There is a subtle problem with respect to equation (f) which can be differentiated with respect to \(\mathrm{T}\) at constant \(\mathrm{S}(\mathrm{aq})\) as defined by equation (b). Then
\[\mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})}+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})} \nonumber \]
Partial molar isentropic expansions \(\mathrm{E}_{\mathrm{S}1}(\mathrm{aq})\) and \(\mathrm{E}_{\mathrm{S}j}(\mathrm{aq})\) are defined by the following equations.
\[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq})=\left(\frac{\partial \mathrm{E}_{\mathrm{s}}(\mathrm{aq})}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})} \nonumber \]
\[\mathrm{E}_{\mathrm{Sj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{E}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
But \(\mathrm{E}_{\mathrm{S}1}\) and \(\mathrm{E}_{\mathrm{S}j}\) are non-Lewisian partial molar properties. Hence
\[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq}) \neq\left(\frac{\partial \mathrm{V}_{1}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})} \nonumber \]
\[\mathrm{E}_{\mathrm{S} j}(\mathrm{aq})=\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})} \nonumber \]
Then,
\[\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq}) \nonumber \]
In practical terms equation (n) follows from equation (g),
\[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{A}=0 ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{A}=0)\right] \nonumber \]
Two practical equations follow from equation (n) allowing \(\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)\) to be calculated from the isentropic expansibilities of solutions and solvent, both volume intensive variables [1].
\[\phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{sl}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
\[\phi\left(E_{\mathrm{Sj}} ; \text { def }\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{S} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
where
\[\alpha_{\mathrm{s}}(\mathrm{aq})=\frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})} \nonumber \]
\[\alpha_{\mathrm{S} 1}^{*}(\ell)=\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\ell)} \nonumber \]
Footnotes
[1] From equation (n),
\[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{sl}}^{*}(\ell)\right] \nonumber \]
We use equation (m) for a solution prepared using \(1 \mathrm{~kg}\) of water.
\[\mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq}) \nonumber \]
Then
\[\phi\left(\mathrm{E}_{\mathrm{S} j} ; \mathrm{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq})-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{Sl} 1}^{*}(\ell)\right] \nonumber \]
Or,
\[\phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=\mathrm{E}_{\mathrm{S} j}(\mathrm{aq})+\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq})-\mathrm{E}_{\mathrm{s} 1}^{*}(\ell)\right] \nonumber \]
Hence using equation (m),
\[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell) \nonumber \]
Using equations (q) and (r),
\[\phi\left(\mathrm{E}_{\mathrm{s}} ; \mathrm{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq}) \, \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell) \nonumber \]
Or,
\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{s} j} ; \text { def }\right)=&\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq}) \,\left[\left(1 / \mathrm{M}_{1}\right) \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
&-\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\mathrm{l}) \, \mathrm{V}_{1}^{*}(\ell)
\end{aligned} \nonumber \]
Or
\[\phi\left(E_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{V}_{1}^{*}(\ell) / \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right) \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Or
\[\phi\left(E_{\mathrm{s} j} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Also \(\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}=\left[1 / \mathrm{c}_{\mathrm{j}}\right]-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) Then,
\[\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)=\left[\left(1 / \mathrm{c}_{\mathrm{j}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{sl}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Or,
\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{c}_{\mathrm{j}}\right) \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right] \\
&-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq})+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell)+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Hence,
\[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{c}_{\mathrm{j}}\right) \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{S} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell) \nonumber \]
For further details see----
J.C.R.Reis, G. Douheret, M.I.Davis, I.J.Fjellanger and H.Hoiland, Phys. Chem. Chem. Phys., 2008,10, 561.