1.9.2: Entropy- Dependence on Temperature and Pressure
The volume of a given closed system at equilibrium prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute-\(j\) is defined by the set of independent variables shown in equation (a).
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]
The same set of independent variables defines the entropy \(\mathrm{S}\).
\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right] \nonumber \]
We envisage that the system is displaced by a change in pressure along a path where the system remains at equilibrium (i.e. \(\mathrm{A} = 0\)) and the volume remains the same as defined by equation (a). In a plot of entropy against \(\mathrm{p}\), the gradient of the plot at the point defined by the independent variables, \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \boldsymbol{\xi}^{\mathrm{eq}}\right]\) is given by equation (c).
Isochoric
\[\left(\frac{\partial S}{\partial p}\right)_{V, A=0} \nonumber \]
The set of derivatives is completed by the following partial derivatives.
Isothermal
\[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \nonumber \]
Isobaric
\[\left(\frac{\partial S}{\partial T}\right)_{p, A=0} \nonumber \]