The Clausius-Clapeyron Equation — Derivation & Applications

The Clausius-Clapeyron equation relates the slope of a coexistence curve to measurable thermodynamic quantities. It is one of the most powerful exact results in classical thermodynamics.

Exact Clausius-Clapeyron Equation

Along any phase boundary the chemical potentials are equal: $\mu^\alpha = \mu^\beta$. Differentiating while staying on the boundary:

$$dG_m^\alpha = dG_m^\beta \implies -S_m^\alpha dT + V_m^\alpha dp = -S_m^\beta dT + V_m^\beta dp$$ $$\boxed{\frac{dp}{dT} = \frac{\Delta_{\alpha\to\beta} S_m}{\Delta_{\alpha\to\beta} V_m} = \frac{\Delta_{\text{trs}} H}{T\,\Delta_{\text{trs}} V_m}}$$

Liquid-Vapour Boundary — Approximate Form

For the liquid-vapour transition, $V_m(\text{gas}) \gg V_m(\text{liq})$ and treating vapour as ideal ($V_m^g = RT/p$):

$$\frac{dp}{dT} = \frac{p\,\Delta_{\text{vap}}H}{RT^2}$$

Rearranging and integrating from reference point $(T_1, p_1)$ to $(T_2, p_2)$:

$$\ln\frac{p_2}{p_1} = -\frac{\Delta_{\text{vap}}H}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$$

Solid-Liquid Boundary

Both phases are condensed; $\Delta V_m$ is small and roughly constant. Integrating the exact equation:

$$p_2 - p_1 = \frac{\Delta_{\text{fus}}H}{T_{\text{fus}}\,\Delta_{\text{fus}}V_m}(T_2 - T_1)$$

For water: $\Delta_{\text{fus}}H = 6.01$ kJ/mol, $\Delta_{\text{fus}}V_m = -1.63\times10^{-6}$ m³/mol (note: negative! ice is less dense), $T_{\text{fus}} = 273.15$ K, giving $dp/dT \approx -13.5$ MPa/K. Increasing pressure by 1 atm lowers the melting point by only 0.0075 K — skating on ice does not melt the ice by pressure alone.

Applications

• Vapour pressure elevation: predict $p(T)$ from one measured data point and $\Delta_{\text{vap}}H$.
• Boiling point at altitude: lower $p$ means lower boiling point ($\sim$95°C at 3000 m elevation).
• Freeze-drying: operate below the triple point pressure to sublime water directly.
• Pressure-cooking: higher $p$ raises boiling point, accelerating cooking.