van't Hoff equation
The van't hoff equation links the equilibrium constant \(K\) to temperature through the standard reaction enthalpy \(\Delta H^\circ\), quantifying how equilibrium shifts when \(T\) changes.
Thermodynamic foundation
Equilibrium constants and standard Gibbs energy are connected by \[ \Delta G^\circ = -RT\ln K, \] while standard thermochemistry connects \(\Delta G^\circ\), \(\Delta H^\circ\), and \(\Delta S^\circ\) through \[ \Delta G^\circ = \Delta H^\circ - T\Delta S^\circ. \] Combining these relations yields a linear dependence of \(\ln K\) on \(1/T\) when \(\Delta H^\circ\) is approximately constant over the temperature range.
Linear form (constant \(\Delta H^\circ\), \(\Delta S^\circ\) over the interval): \[ \ln K = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R}. \]
Two-temperature (integrated) form: \[ \ln\!\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right). \]
Differential form: \[ \frac{d(\ln K)}{dT}=\frac{\Delta H^\circ}{RT^2}. \]
Meaning of the sign of \(\Delta H^\circ\)
The sign of \(\Delta H^\circ\) controls the temperature trend. Endothermic reactions satisfy \(\Delta H^\circ > 0\), giving larger \(K\) at higher \(T\). Exothermic reactions satisfy \(\Delta H^\circ < 0\), giving smaller \(K\) at higher \(T\). This trend is a quantitative expression of Le Châtelier’s principle.
Standard forms and typical use
| Form | What it provides | Common context in general chemistry | Key assumptions |
|---|---|---|---|
| \(\ln K = -\dfrac{\Delta H^\circ}{R}\left(\dfrac{1}{T}\right) + \dfrac{\Delta S^\circ}{R}\) | Straight-line relation of \(\ln K\) vs \(1/T\); slope \(= -\Delta H^\circ/R\) | Extracting \(\Delta H^\circ\) from multiple \(K\) values measured at different \(T\) | \(\Delta H^\circ\) and \(\Delta S^\circ\) treated as constant over the range |
| \(\ln\!\left(\dfrac{K_2}{K_1}\right) = -\dfrac{\Delta H^\circ}{R}\left(\dfrac{1}{T_2}-\dfrac{1}{T_1}\right)\) | Estimating \(K_2\) at a new temperature from \(K_1\) and \(\Delta H^\circ\) | Relating \(K_c\) or \(K_p\) to temperature changes for an equilibrium system | \(\Delta H^\circ\) approximately constant; temperatures in kelvin |
| \(\dfrac{d(\ln K)}{dT}=\dfrac{\Delta H^\circ}{RT^2}\) | Instantaneous sensitivity of \(K\) to \(T\) | Conceptual interpretation of how strongly equilibrium responds near a given \(T\) | Thermodynamic standard state; smooth \(K(T)\) behavior |
Numerical example (two-temperature form)
A representative endothermic reaction has \(\Delta H^\circ = +50.0\ \text{kJ}\cdot\text{mol}^{-1}\). At \(T_1 = 298\ \text{K}\), the equilibrium constant is \(K_1 = 1.50\). The equilibrium constant at \(T_2 = 350\ \text{K}\) follows from the integrated van’t Hoff equation:
\[ \ln\!\left(\frac{K_2}{1.50}\right) = -\frac{50\,000\ \text{J}\cdot\text{mol}^{-1}}{8.314\ \text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}} \left(\frac{1}{350}-\frac{1}{298}\right). \] \[ \left(\frac{1}{350}-\frac{1}{298}\right)\ \text{K}^{-1} \approx -4.9856\times 10^{-4}\ \text{K}^{-1}, \quad \ln\!\left(\frac{K_2}{1.50}\right) \approx 2.998. \] \[ \frac{K_2}{1.50} \approx e^{2.998} \approx 20.05, \quad K_2 \approx 1.50 \times 20.05 \approx 30.1. \]The increase in \(K\) with temperature is consistent with \(\Delta H^\circ > 0\).
Visualization: \(\ln K\) versus \(1/T\)
Common pitfalls
Temperature scale: kelvin is required in every form of the van’t Hoff equation; Celsius introduces large systematic error.
Logarithm base: natural logarithm is implied by \(\ln\); converting to \(\log_{10}\) changes constants by a factor of \(\ln 10\).
Units consistency: \(\Delta H^\circ\) in joules per mole matches \(R\) in \(\text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\).
Meaning of \(K\): equilibrium constants are defined in terms of activities; concentration-based constants (\(K_c\)) and pressure-based constants (\(K_p\)) use standard conventions and approximations.
Temperature range: strong curvature in \(\ln K\) vs \(1/T\) signals non-constant \(\Delta H^\circ\) (heat-capacity effects), requiring a more detailed treatment.