Clausius-Clapeyron Equation

a thermodynamic equation relating to the transition processes of a substance from one phase to another (such as vaporization, fusion, sublimation, polymorphic transformation). According to the Clausius-Clapeyron equation, the heat of a phase transition (for example, heat of vaporization, heat of fusion) under equilibrium conditions is defined by the equation

where T is the temperature of transition (isothermal process), dp/dT is the value of the derivative of pressure with respect to temperature at a given temperature of the transition, and (V₂ — V₁) is the change in the volume of the substance during its transition from the first phase to the second.

The equation was initally derived in 1834 by B. P. E. Clapeyron from analysis of the Carnot cycle for the condensation of steam in thermal equilibrium with the liquid. R. Clausius refined the equation in 1850 and extended it to other phase transitions. The Clausius-Clapeyron equation is applicable to any phase transition accompanied by either absorption or evolution of heat (the phase transition of the first order) and is a direct result of the conditions of phase equilibrium from which it is derived.

The Clausius-Clapeyron equation may be used for calculating any of the quantities entering into the equation if the remaining quantities are known. In particular, the equation is used for calculating the heats of evaporation, which are difficult to determine experimentally.

The Clausius-Clapeyron equation may be formulated in terms of the derivatives dp/dT or dT/dp:

For evaporation and sublimation processes, dp/dT expresses the change in the saturation vapor pressure p with the temperature T, whereas for the processes of fusion and polymorphic transformation, dT/dp determines the change in the transition temperature with the pressure. In other words, the Clausius-Clapeyron equation is a differential equation of the curve of phase equilibrium in terms of the variables p and T.

In order to solve the Clausius-Clapeyron equation it is necessary to know how the quantities L, V₁, and V₂ change with temperature and pressure, which is a difficult task. This relationship is usually determined empirically and the Clausius-Clapeyron equation is solved numerically.

The Clausius-Clapeyron equation may be applied to pure substances and to solutions and the separate components of solutions. In the latter cases, the Clausius-Clapeyron equation relates the partial pressure of unsaturated vapor of a given component to its partial heat of vaporization.