Clausius–Clapeyron relation

Clausius–Clapeyron Relation

A phase diagram illustrating the Clausius–Clapeyron relation.

The Clausius–Clapeyron relation is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent. Named after Rudolf Clausius and Benoît Paul Émile Clapeyron, it provides a quantitative means of understanding the relationship between the pressure and temperature at which two phases coexist in equilibrium.

Mathematical Formulation

The Clausius–Clapeyron relation is expressed as:

\[ \frac{dP}{dT} = \frac{L}{T \Delta V} \]

where:

• \( \frac{dP}{dT} \) is the slope of the coexistence curve in a phase diagram.
• \( L \) is the latent heat of the phase transition.
• \( T \) is the absolute temperature.
• \( \Delta V \) is the change in volume per mole of the substance during the phase transition.

Applications

The Clausius–Clapeyron relation is used to predict the conditions under which a phase transition will occur. It is particularly useful in the study of boiling, melting, and sublimation processes. For example, it can be used to determine the boiling point of a liquid at a given pressure or the melting point of a solid under different pressures.

Derivation

The derivation of the Clausius–Clapeyron relation begins with the Gibbs free energy \( G \), which is equal for two phases in equilibrium. By considering the differential form of the Gibbs free energy and applying the conditions for equilibrium, the relation can be derived.

Limitations

The Clausius–Clapeyron relation assumes that the latent heat \( L \) and the change in volume \( \Delta V \) are constant over the temperature range of interest. This assumption is valid for small temperature ranges but may not hold for larger ranges or for substances with significant changes in \( L \) or \( \Delta V \).

Related Pages

• Phase transition
• Latent heat
• Gibbs free energy
• Phase diagram