## The Clapyron Equation

The phase rule (Eq (1.21)) allows one degree of freedom for a two-phase, one-component system. As the temperature is changed, the pressure must also adjust in order to maintain both phases in equilibrium. The criterion of Eq (5.2) determines the p - T relationship for co-existence of two-phases.

If Eq (5.2) is satisfied at a particular combination of T and p, and the temperature is incremented by dT, the pressure must change by dp to maintain both phases in equilibrium. As a result of these changes, the molar Gibbs free energies of the two phases change by dgI and dgII. For equilibrium at the new conditions, the molar Gibbs free energies must also be equal, or gI + dgI = gII + dgII. Because gI = gII at the initial state, then dgI = dgII. These free energy increments can be expressed in terms of other properties using the fundamental differential of Eq (1.18a):

Since the left hand sides of the above equations are equal, equating the right-hand sides yields:

Making use of Eq (5.4) gives the final form of the Clapyron equation dp _ Ahtr tr (5.5)

dT TAvtr

The subscripts on p and T in the above equation have been omitted because the designations vary with the application. In vaporization or sublimation, for example, T is the independent variable and the dependent variable is the vapor pressure, psat. For melting of a solid, on the other hand, pressure is the independent property and the dependent property is the melting temperature. These two applications are developed below.