10.2: Temperature Dependence of Keq
To study the temperature dependence of \(K_{\text{eq}}\) we can use Equation 10.1.14 for the general equilibrium constant and write:
\[ \ln K_{\text{eq}} = -\dfrac{\Delta G^{-\kern-6pt{\ominus}\kern-6pt-}}{RT}, \label{10.2.1} \]
which we can then differentiate with respect to temperature at constant \(P,\{n_i\}\) on both sides:
\[ \left( \dfrac{\partial \ln K_{\text{eq}}}{\partial T} \right)_{P,\{n_i\}} = -\dfrac{1}{R} \left[ \dfrac{\partial \left( \dfrac{\Delta G^{-\kern-6pt{\ominus}\kern-6pt-}}{T} \right)}{\partial T} \right]_{P,\{n_i\}}, \label{10.2.2} \]
and, using Gibbs-Helmholtz equation (Equation \ref{9.9}) to simplify the left hand side, becomes:
\[ \left( \dfrac{\partial \ln K_{\text{eq}}}{\partial T} \right)_{P,\{n_i\}} = -\dfrac{1}{R} \left( -\dfrac{\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}}{T^2} \right) = \dfrac{\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}}{RT^2}, \label{10.2.3} \]
which gives the dependence of \(\ln K_{\text{eq}}\) on \(T\) that we were looking for. Equation \ref{10.2.3} is also called van ’t Hoff equation, \(^1\) and it is the mathematical expression of Le Chatelier’s principle. The simplest interpretation is as follows:
- For an exothermic reaction (\(\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}< 0\)): \(K_{\text{eq}}\) will decrease as the temperature increases.
- For an endothermic reaction (\(\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}> 0\)): \(K_{\text{eq}}\) will increase as the temperature increases.
If we integrate the van ’t Hoff equation between two arbitrary points at constant \(P\), and assuming constant \(\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}\), we obtain the following:
\[ \int_1^2 d \ln K_{\text{eq}} = \dfrac{\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}}{R} \int_1^2 \dfrac{dT}{T^2}, \label{10.2.4} \]
which leads to the linear equation:
\[ \ln K_{\text{eq}}(2) = \ln K_{\text{eq}}(1) - \dfrac{\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}}{R} \left( \dfrac{1}{T_2}-\dfrac{1}{T_1} \right). \label{10.2.5} \]
which is the equation that produces the so-called van ’t Hoff plots , from which \(\Delta H^{-\kern-6pt{\ominus}\kern-6pt-}\) can be experimentally determined:
- named after Jacobus Henricus “Henry” van ’t Hoff Jr. (1852–1911).