9.3: Pressure Dependence of ΔG
\[ \left(\dfrac{\partial G}{\partial P} \right)_{T,\{n_i\}}=V \nonumber \]
We can now turn the attention to the second coefficient that gives how the Gibbs free energy changes when the pressure change. To do this, we put the system at constant \(T\) and \(\{n_i\}\), and then we consider infinitesimal variations of \(G\). From Equation 8.2.6 :
\[ dG = VdP -SdT +\sum_i\mu_i dn_i \quad \xrightarrow{\text{constant}\; T,\{n_i\}} \quad dG = VdP, \label{9.3.1} \]
which is the differential equation that we were looking for. To study changes of \(G\) for macroscopic changes in \(P\), we can integrate Equation \(\ref{9.3.1}\) between initial and final pressures, and considering an ideal gas, we obtain:
\[\begin{equation} \begin{aligned} \int_i^f dG &= \int_i^f VdP \\ \Delta G &= nRT \int_i^f \dfrac{dP}{P} = nRT \ln \dfrac{P_f}{P_i}. \end{aligned} \end{equation} \label{9.3.2} \ \]
If we take \(P_i = P^{-\kern-6pt{\ominus}\kern-6pt-}= 1 \, \text{bar}\), we can rewrite Equation \ref{9.3.2} as:
\[ G = G^{-\kern-6pt{\ominus}\kern-6pt-}+ nRT \ln \dfrac{P_f}{P^{-\kern-6pt{\ominus}\kern-6pt-}}, \label{9.3.3} \]
which is useful to convert standard Gibbs free energies of formation at pressures different than standard pressure, using:
\[ \Delta_{\text{f}} G = \Delta_{\text{f}} G^{-\kern-6pt{\ominus}\kern-6pt-}+ nRT \ln \dfrac{P_f}{\underbrace{P^{-\kern-6pt{\ominus}\kern-6pt-}}_{=1 \; \text{bar}}} = \Delta_{\text{f}} G^{-\kern-6pt{\ominus}\kern-6pt-}+ nRT \ln P_f \label{9.3.4} \]
For liquids and solids, \(V\) is essentially independent of \(P\) (liquids and solids are incompressible), and Equation \ref{9.3.1} can be integrated as:
\[ \Delta G = \int_i^f VdP = V \int_i^f dP = V \Delta P.\label{9.3.5} \]
The plots in Figure \(\PageIndex{1}\) show the remarkable difference in the behaviors of \(\Delta_{\text{f}} G\) for a gas and for a liquid, as obtained from eqs. \ref{9.3.2} and \ref{9.3.5}.