1.12.7: Expansions- Isobaric and Isentropic
The volume of a closed system at thermodynamic equilibrium containing two chemical substances is defined by equation (a).
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]
At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the chemical composition / organisation \(\xi^{\mathrm{eq}}\) corresponds to the state where the affinity for spontaneous change is zero.
Isobaric Expansions
The system is displaced by a change in temperature to a neighbouring state where the affinity for spontaneous change is also zero; the organisation/composition changes to \(\xi^{\mathrm{eq}}(\mathrm{new})\).
\[\mathrm{V}(\text { new })=\mathrm{V}\left[\mathrm{T}(\text { new }), \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}(\text { new })\right] \nonumber \]
The differential dependence on temperature of the volume defined in equation. (a) is the equilibrium isobaric thermal expansion, \(\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)\) [1].
\[\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{V}}{\partial T}\right)_{\mathrm{p}, \mathrm{A}=0} \nonumber \]
The chemical composition/organisation changes to hold the affinity for spontaneous change at zero. Indeed the perturbation in the form of a change in temperature might have to be extremely slow so that the change in organisation/chemical composition keeps in step with the change in temperature.
The isobaric expansion \(\mathrm{E}_{\mathrm{p}}(\mathrm{aq})\) for an aqueous solution containing solute \(j\) is related to the partial molar isobaric expansions of solute and solvent; equation (d).
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{pl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}(\mathrm{aq}) \nonumber \]
Alternatively using the concept of an apparent molar property, we define an (equilibrium) apparent molar isobaric expansion for solute \(j\), \(\phi\left(E_{p j}\right)\).
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) \nonumber \]
\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
Isentropic Expansions
Generally little interest has been shown in either partial molar or apparent molar isentropic expansions of solutes. Complications are encountered in understanding isentropic expansions without the redeeming feature of practical accessibility via an analogue of the Newton -Laplace equation. The isentropic expansions \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is defined by equation (g).
\[\mathrm{E}_{\mathrm{S}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})} \nonumber \]
The constraint on the partial derivative refers to the entropy of the solution \(\mathrm{S}(\mathrm{aq})\). As we change the amount of solute nj for fixed temperature and fixed pressure and amount of solvent \(\mathrm{n}_{1}\), so both \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{S}(\mathrm{aq})\) change yielding a new isentropic thermal expansion, \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) at a new entropy \(\mathrm{S}(\mathrm{aq})\). For a series of solutions having different molalities of solute, comparison of \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is not straightforward because \({\mathrm{S}(\mathrm{aq})\) is itself a function of solution composition. Further comparison cannot be readily drawn with the isentropic thermal expansion of the solvent, \(\mathrm{E}_{\mathrm{SI}}^{*}(\ell)\); equation (h).
\[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell)=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right) \quad \text { at constant } \quad \mathrm{S}_{1}^{*}(\ell) \nonumber \]
\(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is a non-Gibbsian property. Consequently familiar thermodynamic relationships involving partial molar properties are not valid in the case of partial molar isentropic thermal expansions which are non-Lewisian properties. \(\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})}\) is a semi-partial property.
Footnote
[1] For a system at equilibrium where \(\mathrm{A} = 0\), \(\frac{\partial^{2} G}{\partial T \, \partial p}=\frac{\partial^{2} G}{\partial p \, \partial T}\)
Therefore, \(\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}\)