Clapeyron equation

The Clapeyron equation is derived from the condition of equal chemical potentials of two phases at equilibrium. For a pure substance, the Gibbs free energy of each phase must be equal along the coexistence curve, leading to the relation dP/dT = L/(T ΔV), where L is the latent heat (enthalpy of transition) and ΔV is the molar volume change between phases.¹ The equation applies to any first-order phase transition, including solid–liquid, liquid–vapor, and solid–vapor boundaries.

For vaporization, when the vapor behaves as an ideal gas and the molar volume of the liquid is negligible, the equation simplifies to the Clausius–Clapeyron relation, which is often used to estimate vapor pressure as a function of temperature.²

Benoît Clapeyron, a French engineer and physicist, introduced the equation in 1834 in his seminal paper on the motive power of heat, which also helped popularize Sadi Carnot's ideas.³ Clapeyron's work provided a mathematical framework for analyzing phase equilibria, building on the earlier qualitative observations of latent heat.

Later, Rudolf Clausius extended the equation to include the ideal-gas approximation, giving rise to the Clausius–Clapeyron relation. The original Clapeyron equation remains a cornerstone of thermodynamics, taught in introductory courses and used in advanced applications.

The Clapeyron equation is used to calculate the pressure–temperature slope of phase boundaries, which is essential for constructing phase diagrams of materials, from water to metals and minerals.⁴ It also explains phenomena such as the lowering of the melting point of ice under pressure, which is why ice skates glide on ice.

In geology, the equation helps model metamorphic reactions and the behavior of minerals deep within the Earth's crust. In engineering, it is applied to design processes involving evaporation, condensation, and refrigeration cycles.

While the Clapeyron equation is often presented for pure substances, it can be extended to mixtures and solutions, where the chemical potentials of components must be considered. This leads to more complex expressions but retains the same underlying principle.

An often-overlooked application is in the study of solid–solid transitions, such as the transformation between graphite and diamond. The equation predicts the slope of the coexistence line, which is crucial for industrial diamond synthesis. Additionally, the equation is used in cryogenics to understand the behavior of superfluids and superconductors, where phase transitions occur at extremely low temperatures.

Latent heat

The heat absorbed or released during a phase change at constant temperature.

Phase boundary

The line on a pressure–temperature diagram separating two phases in equilibrium.

Molar volume

The volume occupied by one mole of a substance.