1.10.4: Gibbs Energies- Equilibrium and Spontaneous Change

The Gibbs energy of a closed system at temperature \(\mathrm{T}\) is related to the enthalpy \(\mathrm{H}\) using equation (a) [1].

\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\]

The differential change in Gibbs energy at constant temperature is related to the change in enthalpy \(\mathrm{dH}\) using equation (b).

\[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}\]

For a process taking place in a closed system involving a change from state I to state II, the change in Gibbs energy is given by equation (c).

\[\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \, \Delta \mathrm{S}\]

The latter equation signals how changes in enthalpy and entropy determine the change in Gibbs energy. A given closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{\mathrm{j}}\) moles of solute \(j\). The system is at equilibrium such that the composition/organisation is represented by \(\xi^{\mathrm{eq}\) and the affinity for spontaneous change is zero. We summarise this state of affairs as follows.

\[\mathrm{G}^{e q}=\mathrm{G}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]

In a plot of \(\mathrm{G}\) against \(\xi\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) the Gibbs energy is a minimum at \(\xi^{\mathrm{eq}}\) [2]. The enthalpy \(\mathrm{H}^{\mathrm{eq}}\) of the equilibrium state can be represented in a similar fashion.

\[\mathrm{H}^{e q}=\mathrm{H}^{e q}\left[\mathrm{~T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]

However it is unlikely that \(\mathrm{H}^{\mathrm{eq}}\) at \(\xi^{\mathrm{eq}}\) corresponds to a minimum in enthalpy \(\mathrm{H}\) when \(\mathrm{H}\) is plotted as a function of \(\xi\). A similar comment applies to the entropy \(\mathrm{S}^{\mathrm{eq}}\);

\[\mathrm{S}^{\mathrm{eq}}=\mathrm{S}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]

However taken together \(\mathrm{H}^{\mathrm{eq}}\) and \(\mathrm{S}^{\mathrm{eq}}\) produce the minimum in \(\mathrm{G}\) at \(\mathrm{G}^{\mathrm{eq}}\).

\[\mathrm{G}^{\mathrm{eq}}=\mathrm{H}^{\mathrm{eq}}-\mathrm{T} \, \mathrm{S}^{\mathrm{eq}}\]

In summary; at thermodynamic equilibrium

1. \(\mathrm{A}\) is zero,
2. \(\mathrm{G}\) is a minimum and
3. the rate of change of composition/organisation \(\mathrm{d}\xi/\mathrm{dt}\) is zero.

The latter condition emerges from the conclusion that this rate is zero if there is no affinity for change. For a given system at defined \(\mathrm{T}\) and \(\mathrm{p}\), the state for which \(\mathrm{G}\) is a minimum is unique [3]. Indeed if this was not the case, chemistry would be a very difficult subject. In a given spontaneous chemical reaction proceeds until the composition/organisation reaches \(\xi^{\mathrm{eq}}\). In other words the Gibbs energy is the important thermodynamic potential, certainly forming the basis of treatments of chemical reactions in closed systems at fixed \(\mathrm{T}\) and \(\mathrm{p}\) [4]. However a word of caution is in order. The Gibbs energy of a system differs from the thermodynamic energy \(\mathrm{U}\). In fact the Gibbs energy is a somewhat contrived property but aimed at a description of closed systems at fixed \(\mathrm{T}\) and \(\mathrm{p}\). Nevertheless the Gibbs energy can be given practical significance. We consider a system at equilibrium (i.e. \(\mathrm{A} = 0\)) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) where in this state (state I) the Gibbs energy is \(\mathrm{G}[\mathrm{I}]\). The system is displaced by a change in pressure to a neighbouring equilibrium state (at constant \(\mathrm{T}\)). The equilibrium isothermal dependence of Gibbs energy \(\mathrm{G}[\mathrm{I}]\) on pressure equals the volume of the system, \(\mathrm{V}[\mathrm{I}]\) [5].

\[V=\left[\frac{\partial G}{\partial p}\right]_{T, A=0}\]

In other words we may not know the Gibbs energy of a system (in fact never know) at least we know that the pressure dependence is the volume which we can readily measure. The isobaric dependence of \(\mathrm{G}[\mathrm{I}]\) on temperature for an equilibrium displacement yields the entropy.

\[\mathrm{S}=-\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}=0}\]

Four key points can be made.

1. The equilibrium state for a system at constant \(\mathrm{T}\) and \(\mathrm{p}\) corresponds to a minimum in Gibbs energy.
2. The minimum in Gibbs energy of a given system is unique.
3. In the non-equilibrium, there is no direct relationship between the gradient \((\partial \mathrm{G} / \partial \xi)\) and the rate of spontaneous change, \((\partial \xi / \partial t)\).
4. The equilibrium state is stable; \((\partial \mathrm{A} / \partial \xi)<0\) at \(\xi^{\mathrm{eq}}\) [6].

Footnotes

[1]

\[\begin{aligned}
&\mathrm{G}=[\mathrm{J}] ; \mathrm{T} \, \mathrm{S}=[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}^{-1}\right]=[\mathrm{J}] ; \mathrm{p} \, \mathrm{V}=\left[\mathrm{N} \mathrm{m}^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}] \\
&\mathrm{A} \, \xi=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \,[\mathrm{mol}]=[\mathrm{J}]
\end{aligned}\]

[2] See for example

1. G. Willis and D. Ball, J. Chem. Educ.,1984, 61, 173, and
2. P. L. Corio, J. Phys. Chem.,1983, 87, 2416.

[3] This point is discussed in the monograph, F. Van Zeggeren and S. H. Story, The Computation of Chemical Equilibria, Cambridge University Press, 1970.

[4] The dependence of rate of reaction on composition is described using the law of mass action and rate constants. The law of mass action is in these terms, extrathermodynamic , meaning that the law does not follow from the first and second laws.

[5] For completeness we consider the case where in equilibrium state [I] at \(\xi^{\mathrm{eq}}[\mathrm{I}]\), the system is displaced by a change to a neighbouring state having the same composition/organisation, \(\xi[\mathrm{I}]\); i.e. the system is ‘frozen’. The isothermal dependence of \(\mathrm{G}[\mathrm{I}]\) on pressure at constant composition equals the volume. Thus, \(\mathrm{V}[\mathrm{I}]=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi[\mathrm{I}]}\) Similarly, \(\mathrm{S}[\mathrm{I}]=-\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi[\mathrm{I}]}\)

The identities of \(\mathrm{S}\) and \(\mathrm{V}\) at constant ‘\(\mathrm{A}=0\)’ and at \(\xi^{\mathrm{eq}}[\mathrm{I}]\) arise from the fact that \(\mathrm{V}\) and \(\mathrm{S}\) are strong state variables.

[6] Consider a simple cup in which we have placed (delicately) a steel ball near the top edge.

1. We release the ball, the ball rolls down and the ball gathers speed.
2. At the bottom of the cup the speed of the ball is a maximum (i.e. maximum in kinetic energy) where the potential energy is a minimum.
3. The ball continues through the bottom of the cup and rises to the opposite edge.
4. The ball slows down and then changes direction, falling back to the bottom of the cup.
5. If the surface of the cup is perfectly smooth ( i.e. no energy lost to friction), the ball oscillates about the bottom of the well of the cup. This mechanical model is not correct for chemical reactions in closed systems. In fact the rate of the chemical reaction decreases as a system approaches the minimum in Gibbs energy.