ClausiusClapeyron

The Clausius–Clapeyron equation is a fundamental relation in thermodynamics that describes how the pressure of a substance at phase equilibrium changes with temperature. It is commonly applied to liquid–vapor transitions to relate vapor pressure to temperature.

The general form is derived from the Clapeyron equation: dP/dT = ΔS/ΔV, where ΔS is the entropy change and ΔV is the volume change during a phase transition. For a liquid–vapor equilibrium, ΔS is the latent heat divided by temperature (ΔS ≈ L/T) and the liquid’s volume is small compared with the vapor volume, so ΔV ≈ V_g. Assuming the vapor behaves as an ideal gas (V_g ≈ RT/P), the equation becomes dP/dT = (L P)/(R T^2). This can be rearranged to d(ln P)/dT = L/(R T^2). If the latent heat L is treated as constant, integration yields ln P = -L/(R T) + C, or, for two temperatures T1 and T2, ln(P2/P1) = -ΔH_vap/R (1/T2 - 1/T1). In many practical uses, the integrated form is written as P(T) ≈ P0 exp[-ΔH_vap/R (1/T - 1/T0)].

Applications include estimating vapor pressures, boiling points under varying pressure, and atmospheric science relating to water vapor. Limitations arise from non-ideal gas behavior, temperature dependence of latent heat, and deviations near the critical point or at high pressures.

Historically, the relationship was developed in the 19th century by Benoît Clapeyron and Rudolf Clausius; it is now commonly referred to as the Clausius–Clapeyron equation.