1.22.1: Volume: Partial Molar: General Analysis
At temperature \(\mathrm{T}\), pressure \(\mathrm{p}\) and equilibrium, the volume of a closed system containing i-chemical substances where the amounts can be independently varied, is defined by the following equation.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \ldots \ldots \mathrm{n}_{\mathrm{i}}\right] \nonumber \]
Or, in general terms according to Euler’s theorem,
\[\mathrm{V}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}} \nonumber \]
where
\[\mathrm{V}_{\mathrm{j}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i} \neq \mathrm{j}}} \nonumber \]
The general differential of equation (b) has the following form.
\[\mathrm{dV}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{dV} \mathrm{V}_{\mathrm{j}}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{V}_{\mathrm{j}} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}} \nonumber \]
The general differential of equation (a) has the following form
\[\mathrm{dV}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i} \neq \mathrm{j}}} \, \mathrm{dn}_{\mathrm{j}} \nonumber \]
Comparison of equations (d) and (e) shows that
\[0=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dT}-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{dV} \mathrm{j}_{\mathrm{j}} \nonumber \]
Equation (f) is the Gibbs-Duhem Equation with respect to the volumetric properties of a closed system at equilibrium.
Application
A given closed system contains \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The system is at equilibrium where \(\mathrm{G}\) is a minimum, the affinity for spontaneous change \(\mathrm{A}\) is zero and the composition-organisation \(\xi^{\mathrm{eq}}\). The dependent variable volume \(\mathrm{V}\) is defined using a set of independent variables; equation (g).
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
Equation (k) has an interesting property. If we multiply the extensive variables \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\) by a factor \(\mathrm{k}\), the volume of the system equals (\(\mathrm{V}. \mathrm{~k}\)). In terms of Euler’s Theorem [1], the variable \(\mathrm{V}\) linked to the variables \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\) is a homogeneous function of the first degree. The important consequence is the following key relation.
\[\mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}} \nonumber \]
where
\[\mathrm{V}_{1}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})} \nonumber \]
and
\[\mathrm{V}_{\mathrm{j}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
We do not have to specify the conditions ‘at constant \(\mathrm{T}\) and \(\mathrm{p}\)’ in conjunction with equation (h) which is a mathematical identity.
Footnote
[1] Degree of Homogeneity
At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the volume of a closed system containing \(\mathrm{n}_{j}\) moles of each chemical substance \(j\) is given by
\[\mathrm{V}=\mathrm{V}\left[\mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots \ldots . \ldots \mathrm{n}_{\mathrm{k}}\right] \nonumber \]
The property volume has unit degree of homogeneity. That is to say – if the amount of each substance is increased by a factor \(\lambda\) then the volume increases by the same factor. Thus
\[\mathrm{V}\left[\lambda \mathrm{n}_{1}, \lambda \mathrm{n}_{2} \ldots \ldots \ldots \ldots . . \lambda \mathrm{n}_{\mathrm{k}}\right]=\lambda \, \mathrm{V}\left[\mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots \ldots . \ldots \mathrm{n}_{\mathrm{k}}\right] \nonumber \]