1.22.4: Volume: Partial Molar: Frozen and Equilibrium
Consider the volume of a closed system defined by equation (a).
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
This system is displaced to a neighboring state by addition of a small amount of substance \(j\), \(\delta\mathrm{n}_{j}\). The change in volume at fixed affinity \(\mathrm{A}\) is related to the change in volume at fixed composition or organization. At fixed temperature, fixed pressure, and fixed \(\mathrm{n}_{1}\),
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{A}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\xi}-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\xi} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{n}_{\mathrm{j}}} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{n}_{\mathrm{j}}} \nonumber \]
For a system at equilibrium (\(\mathrm{A} = 0 \text { and } \xi = \xi^{\mathrm{eq}}\)), the triple product term on the R.H.S. of equation (b) is not zero. Hence we distinguish between two properties; \(\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{~A}=0} \text { and } \left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{eq}^{\mathrm{eq}}}\). By convention the first of these two terms is called the partial molar volume of substance \(j\) in the system.
Example 1
A given aqueous solution is prepared by dissolving \(\mathrm{n}_{j}\) moles of urea in \(\mathrm{n}_{1}\) moles of water at \(298.2 \mathrm{~K}\) and ambient pressure. This system has volume \(\mathrm{V}(\mathrm{aq})\) which is determined in part by water-water, water-urea and urea-urea interactions. We add \(\delta \mathrm{n}_{j}\) moles of urea to this system but stipulate that the water-water, water-urea and urea-urea interactions remain unchanged; i.e. frozen. The property \(\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T} ; \mathrm{p} ; \mathrm{n}(1), \xi}\) is a frozen partial molar volume of urea in the aqueous solution. On the other hand, if we stipulate that the water-water, water-urea and urea-urea interactions re-adjust in order that the system is at a minimum in Gibbs energy, the property \(\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p} ; \mathrm{n}(1), \mathrm{A}=0}\) is the equilibrium partial molar volume for urea in this aqueous solution.
Example 2
We consider an aqueous solution containing \(\mathrm{n}_{j}\) moles of ethanoic acid in \(\mathrm{n}_{1}\) moles of water at defined temperature and defined pressure. Conventionally, the chemical equilibrium operating in the system is expressed in the following form.
\[\mathrm{HA}(\mathrm{aq}) \leftrightarrow \mathrm{H}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq}) \nonumber \]
The volume of this system \(\mathrm{V}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is a state variable. We add \(\delta \mathrm{δ}(\mathrm{HA})\) moles of substance \(\mathrm{HA}\) to the system. In the frozen limit, the amounts of \(\mathrm{H}^{+}(\mathrm{aq})\) and \(\mathrm{A}^{-}(\mathrm{aq})\) in the solution do not change. In terms of composition all that happens is the amount of \(\mathrm{HA}(\mathrm{aq})\) increases. Hence, \(\left(\frac{\delta \mathrm{V}}{\delta \mathrm{n}(\mathrm{HA})}\right)\) is a measure of the ‘frozen partial molar volume’ of \(\mathrm{HA}\) in the system. If we remove the frozen restriction and allow chemical equilibrium to be re-established, the derived quantity is the equilibrium partial molar volume for \(\mathrm{HA}\) in this aqueous solution, part of added \(\delta\mathrm{P}mathrm{n}(\mathrm{HA})\) having dissociated in order that the resulting solution has zero affinity for spontaneous change. We use quotation marks ‘….’ Around the phrase ‘frozen partial molar volume’ to make the point that this property is not a proper equilibrium thermodynamic property.