9.1: Gibbs Equation

Recalling from chapter 8, the definition of \(G\) is:

\[ G = U -TS +PV = H-TS, \nonumber \]

which, taking the differential at constant \(T\) and \(P\), becomes:

\[ dG = dH \; \overbrace{-SdT}^{=0} -TdS = dH -TdS. \nonumber \]

Integrating Equation \ref{9.1.2} between the initial and final states of a process results in:

\[\begin{equation} \begin{aligned} \int_i^f dG &= \int_i^f dH -T \int_i^f dS \\ \\ \Delta G &= \Delta H -T \Delta S \end{aligned} \end{equation} \label{9.1.2} \]

which is the famous Gibbs equation for \(\Delta G\). Using Definition: Spontaneous Process, we can use \(\Delta G\) to infer the spontaneity of a chemical process that happens at constant \(T\) and \(P\) using \(\Delta G \leq 0\). If we set ourselves at standard conditions, we can calculate the standard Gibbs free energy of formation, \(\Delta_{\text{rxn}} G^{-\kern-6pt{\ominus}\kern-6pt-}\), for any reaction as:

\[\begin{equation} \begin{aligned} \Delta_{\text{rxn}} G^{-\kern-6pt{\ominus}\kern-6pt-}&= \Delta_{\text{rxn}} H^{-\kern-6pt{\ominus}\kern-6pt-}-T \Delta_{\text{rxn}} S^{-\kern-6pt{\ominus}\kern-6pt-}\\ \\ &= \sum_i \nu_i \Delta_{\mathrm{f}} H_i^{-\kern-6pt{\ominus}\kern-6pt-}+ T \sum_i \nu_i S_i^{-\kern-6pt{\ominus}\kern-6pt-}, \end{aligned} \end{equation} \nonumber \]

where \(\Delta_{\mathrm{f}} H_i^{-\kern-6pt{\ominus}\kern-6pt-}\) are the standard enthalpies of formation, \(S_i^{-\kern-6pt{\ominus}\kern-6pt-}\) are the standard entropies, and \(\nu_i\) are the stoichiometric coefficients for every species \(i\) involved in the reaction. All these quantities are commonly available, and we have already discussed their usage in chapters 4 and 7, respectively.\(^1\)

The following four options are possible for \(\Delta G^{-\kern-6pt{\ominus}\kern-6pt-}\) of a chemical reaction:

\(\Delta G^{-\kern-6pt{\ominus}\kern-6pt-}\)		\(\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}\)	\(\Delta S^{-\kern-6pt{\ominus}\kern-6pt-}\)	Spontaneous?
–	if	–	+	Always
+	if	+	–	Never
–/+	if	–	–	Depends on \(T\): \(\scriptstyle{\text{spontaneous at low } T}\)
+/–	if	+	+	Depends on \(T\): \(\scriptstyle{\text{spontaneous at high } T}\)

Or, in other words:

• Exothermic reactions that increase the entropy are always spontaneous.
• Endothermic reactions that reduce the entropy are always non-spontaneous.
• For the other two cases, the spontaneity of the reaction depends on the temperature:
  • Exothermic reactions that reduce the entropy are spontaneous at low \(T\).
  • Endothermic reactions that increase the entropy are spontaneous at high \(T\).

A simple criterion to evaluate the entropic contribution of a reaction is to look at the total number of moles of the reactants and the products (as the sum of the stoichiometric coefficients). If the reaction is producing more molecules than it destroys \(\left( \left| \sum_\text{products} \nu_i \right| > \left| \sum_\text{reactants} \nu_i \right| \right)\), it will increase the entropy. Vice versa, if the total number of moles in a reaction is reducing \(\left( \left| \sum_\text{products} \nu_i \right| < \left| \sum_\text{reactants} \nu_i \right| \right)\), the entropy will also reduce.

As we saw in section 8.2, the natural variables of the Gibbs free energy are the temperature, \(T\), the pressure, \(P\), and chemical composition, as the number of moles \(\{n_i\}\). The Gibbs free energy can therefore be expressed using the total differential as (see also, last formula in Equation 8.4.2):

\[ dG(T,P,\{n_i\}) = \mkern-18mu \underbrace{\left(\dfrac{\partial G}{\partial T} \right)_{P,\{n_i\}}}_{\text{temperature dependence}} \mkern-36mu dT + \underbrace{\left(\dfrac{\partial G}{\partial P} \right)_{T,\{n_i\}}}_{\text{pressure dependence}} \mkern-36mu dP + \sum_i \underbrace{\left(\dfrac{\partial G}{\partial n_i} \right)_{T,P,\{n_{j \neq i}\}}}_{\text{composition dependence}} \mkern-36mu dn_i. \label{9.1.5} \]

If we know the behavior of \(G\) as we vary each of the three natural variables independently of the other two, we can reconstruct the total differential \(dG\). Each of these terms represents a coefficient in Equation \ref{9.1.5}, which are given in Equation 8.4.3.

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1. ︎It is not uncommon to see values of \(\Delta_{\text{f}} G^{-\kern-6pt{\ominus}\kern-6pt-}\) tabulated alongside \(\Delta_{\mathrm{f}} H^{-\kern-6pt{\ominus}\kern-6pt-}\) and \(S_i^{-\kern-6pt{\ominus}\kern-6pt-}\), which simplifies even further the calculation. In fact, a comprehensive list of standard Gibbs free energy of formation of inorganic and organic compounds is reported in the appendix of this book 16. For cases where \(\Delta_{\text{f}} G^{-\kern-6pt{\ominus}\kern-6pt-}\) are not reported, they can always be calculated by their constituents.