Quantitative approach for changes of T

3.3 Quantitative approach for changes of T

From Eq. (3.8) we find the van’t Hoff equation

\begin{equation*} \frac{d \ln K}{dT} = - \frac{1}{R} \frac{d(\Delta_r G^0 / T)}{d T} = \frac{\Delta_r H^0}{R T^2} \quad . \label{vantHoff_1} \end{equation*}

(3.11)

Here we used the Gibbs-Helmholtz equation

\begin{equation*} \frac{\partial G}{\partial T} = - S = \frac{G-H}{T} \quad \mbox{resp.} \quad \frac{\partial (G/T)}{\partial T} = \frac{1}{T} \frac{\partial G}{\partial T} + G \frac{d (1/T)}{d T} = \frac{1}{T} \left( -S - \frac{G}{T} \right)= - \frac{H}{T^2} \end{equation*}

(3.12)

A second representation of the van’t Hoff equation directly extracted from Eq. (3.11) is

\begin{equation*} \frac{d \ln K}{d(1/T)} = - \frac{\Delta_r H^0}{R} \quad . \label{vantHoff_2} \end{equation*}

(3.13)

As illustrated in Fig. 3.1 b) the Arrhenius-like plot allows to extract reaction enthalpies from the slope of the curve \(\ln K\) vs. \(1 / T\).

Assuming \(\Delta_r H^0\) to be independent of \(T\) the integration of Eq. (3.13) leads to

\begin{equation*} - \int_{K(T_1)}^{K(T_2)}d \ln K= \frac{\Delta_r H^0}{R} \int_{1/T_1}^{1/T_2} d(1/T) \quad \mbox{thus} \quad \frac{\ln K(T_2)}{\ln K(T_1)} = \frac{\Delta_r H^0}{R} \left(\frac{1}{T_1} - \frac{1}{T_2} \right) \quad . \label{vantHoff_int_1} \end{equation*}

(3.14)

Assuming Kirchhoff’s law

\begin{equation*} \Delta_r H^0(T) = \Delta_r H^0(T=0) + \int_0^T \Delta C_p^0 dT \end{equation*}

(3.15)

and taking the Taylor expansion

\begin{equation*} \Delta C_p^0 = \sum_i \nu_i C_{p,i} = A + B T + C T^2 + ... \end{equation*}

(3.16)

Eq. (3.11) translates into

\begin{equation*} \begin{split} \frac{d \ln K}{d T} & = \frac{\Delta_r H^0}{R T^2} = \frac{\Delta_r H^0(T=0)}{RT^2} + \frac{1}{RT^2} \int_0^T \left(A + B T + C T^2 + ...\right) dT\\ & = \frac{\Delta_r H^0(T=0)}{RT^2} + \frac{1}{R T^2} \left( A T + \frac{B T^2}{2} + \frac{C T^3}{3} + ...\right) \end{split} \end{equation*}

(3.17)

and finally the integration gives

\begin{equation*} \ln K = - \frac{\Delta_r H^0(T=0)}{RT} + \frac{A}{R} \ln T + \frac{B}{2 R} T + \frac{C}{6R} T^2 + ... + I \end{equation*}

(3.18)

where \(I\) is an additional integration constant determined by a measurement of \(K\) at one temperature.