1.14.56: Properties: Equilibrium and Frozen
A given closed system having Gibbs energy \(\mathrm{G}\) at temperature \(\mathrm{T}\), pressure \(\mathrm{p}\), molecular composition (organization \(\xi\)) and affinity for spontaneous change \(\mathrm{A}\) is described by equation (a).
\[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi] \nonumber \]
In the state defined by equation (a), there is an affinity for spontaneous chemical reaction \(\mathrm{A}\). Starting with the system in the state defined by equation (a) it is possible to change the pressure and perturb the system to a series of neighboring states for which affinity remains constant. The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant \(\mathrm{A}\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\). Returning to the original state characterized by \(\mathrm{T}\), \(\mathrm{p}\) and \(\xi\), we imagine that it is possible to perturb the system by a change in pressure in such a way that the system remains at fixed extent of reaction, \(\xi\). The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant \(\xi\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\). We explore these dependences of \(\mathrm{G}\) on pressure at fixed temperature and at
- fixed composition, \(\xi\) and
- fixed affinity for spontaneous change, \(\mathrm{A}\).
The procedure for relating \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\), and \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\) is a standard calculus operation. At fixed temperature,
\[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{p}} \nonumber \]
This interesting equation shows that the differential dependence of Gibbs energy (at constant temperature) on pressure at constant affinity for spontaneous change does NOT equal the corresponding dependence at constant extent of chemical reaction. This inequality is not surprising. But our interest is drawn to the case where the system under discussion is, at fixed temperature and pressure, at thermodynamic equilibrium where \(\mathrm{A}\) is zero, \(\mathrm{d} \xi / \mathrm{dt}\) is zero, Gibbs energy is a minimum AND, significantly, \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Hence
\[\mathrm{V}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \mathrm{A}=0}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi^{\mathrm{eq}}} \nonumber \]
The dependence of \(\mathrm{G}\) on pressure for differential displacements at constant ‘\(\mathrm{A} = 0\)’ and \(\xi^{\mathrm{eq}}\) are identical. We confirm that the volume \(\mathrm{V}\) of a system is a ‘strong’ state variable. These comments seem trivial but the point is made if we go on to consider the volume of a system as a function of temperature at constant pressure. We use a calculus operation to derive equation (d).
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{A}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\xi}-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\xi} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}} \nonumber \]
Again we are not surprised to discover that in general terms the differential dependence of \(\mathrm{V}\) on temperature at constant affinity does not equal the differential dependence of \(\mathrm{V}\) on temperature at constant composition/organization. Indeed, unlike the simplification we could use in connection with equation (b), {namely that at equilibrium \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero} we cannot assume that the volume of reaction, \((\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero at equilibrium. In other words for a closed system at thermodynamic equilibrium at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\) {when \(\mathrm{A}=0\), \(\xi = \xi^{\mathrm{eq}}\) and \(\mathrm{d} \xi^{\mathrm{eq}} / \mathrm{dt}=0\)}, there are two thermal expansions, at constant \(\mathrm{A}\) and at constant \(\xi\)ξ.
We consider a closed system in equilibrium state I defined by the set of variables,\(\left\{\mathrm{T}[\mathrm{I}], \mathrm{p}, \mathrm{A}=0, \xi^{\mathrm{eq}}[\mathrm{I}]\right\}\). The equilibrium composition is \(\xi^{\mathrm{eq}}[\mathrm{I}]\) at zero affinity for spontaneous change. This system is perturbed to two nearby states at constant pressure.
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State I is displaced to a nearby equilibrium state II defined by the set of variables, \(\left\{\mathrm{T}[\mathrm{I}]+\delta \mathrm{T}, \mathrm{p}, \mathrm{A}=0, \xi^{\mathrm{eq}}[\mathrm{II}]\right\}\). This equilibrium displacement is characterized by a volume change;
\[\Delta \mathrm{V}(\mathrm{A}=0)=\mathrm{V}[\mathrm{II}]-\mathrm{V}[\mathrm{I}] \nonumber \]
\[\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=\left[\frac{\mathrm{V}[\mathrm{II}]-\mathrm{V}[\mathrm{I}]}{\Delta \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}=0} \nonumber \]
The equilibrium isobaric expansibility,
\[\alpha_{\mathrm{p}}(\mathrm{A}=0)=\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0) / \mathrm{V} \nonumber \]
In order for the system to move from one equilibrium state, I with composition \(\xi^{\mathrm{eq}}[\mathrm{I}]\) to another equilibrium state, II with composition \(\xi^{\mathrm{eq}}[\mathrm{II}]\), the system changes by a change in chemical composition and/or molecular organization.
Hence we define the ‘frozen’ isobaric expansion, \(\mathrm{E}_{\mathrm{p}} (\xi = \text { fixed})\). An alternative name is the instantaneous expansion because, practically, we would have to change the temperature at a such a high rate that there is no change in molecular composition or organisation in the system.
\[\mathrm{E}_{\mathrm{p}}(\xi=\text { fixed })=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \nonumber \]
Further
\[\alpha_{p}(\xi=\text { fixed })=\frac{1}{V} \,\left(\frac{\partial V}{\partial T}\right)_{p, \xi} \nonumber \]
Similar comments apply to isothermal compressibilities, \(\mathrm{K}_{\mathrm{T}}\); there are two limiting quantities \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\) and \(\kappa_{\mathrm{T}}(\xi)\). In order to measure \(\kappa_{\mathrm{T}}(\xi)\) we have to change the pressure also in an infinitely short time.
The entropy \(\mathrm{S}\) is given by the partial differential, \(-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \xi}\). At equilibrium where \(\mathrm{A}=0, \mathrm{~S}=-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}=0}\). We carry over the argument described in the previous section but now concerned with a change in temperature. We consider the two pathways, constant \(\mathrm{A}\) and constant \(\xi\).
\[\begin{aligned}
&(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}}= \\
&\quad(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \xi}-(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}} \,(\partial \mathrm{A} / \partial \mathrm{T})_{\mathrm{p}, \bar{\xi}} \,(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}
\end{aligned} \nonumber \]
But at equilibrium, \(\mathrm{A}\) which equals \(-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero, and so \(\mathrm{S}(\mathrm{A}=0)\) equals \(\mathrm{S}\left(\xi^{\mathrm{eq}}\right)\). Then just as for volumes, the entropy of a system is not a property concerned with pathways between states; entropy is a strong function of state. Another important link involving Gibbs energy and temperature is provided by the Gibbs-Helmholtz equation. We explore the relationship between changes in \((\mathrm{G} / \mathrm{T})\) at constant affinity \(\mathrm{A}\) and at fixed \(\xi\), following perturbation by a change in temperature.
\[\begin{aligned}
&{[\partial(\mathrm{G} / \mathrm{T}) / \partial \mathrm{T}]_{\mathrm{p}, \mathrm{A}}=} \\
&{[\partial(\mathrm{G} / \mathrm{T}) / \partial \mathrm{T}]_{\mathrm{p}, \xi}-(1 / \mathrm{T}) \,(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}} \,(\partial \mathrm{A} / \partial \mathrm{T})_{\mathrm{p}, \xi} \,(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}
\end{aligned} \nonumber \]
But at equilibrium, \(\mathrm{A}\) which equals \(-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Then \(\mathrm{H}(\mathrm{A}=0)=\mathrm{H}\left(\xi^{\mathrm{eq}}\right)\). In other words, the variable enthalpy is another strong function of state. This is not the case for isobaric heat capacities.
\[\begin{aligned}
&(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}}= \\
&(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}-(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}} \,(\partial \mathrm{A} / \partial \mathrm{T})_{\mathrm{p}, \xi} \,(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}
\end{aligned} \nonumber \]
We cannot assume that the triple product term in the latter equation is zero. Hence, there are two limiting isobaric heat capacities; the equilibrium isobaric heat capacity \(C_{p}(A=0)\) and the frozen isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}(\xi \mathrm{eq})\). In other words, an isobaric heat capacity is not a strong function of state because it is concerned with a pathway between states. Unless otherwise stated, we use the symbol \(\mathrm{C}_{\mathrm{p}}\) to indicate an equilibrium transformation, \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\).