The Latent Heats and Clapeyrons Equation

The Latent Heats and Clapeyron s Equation.—There is a very important thermodynamic relation concerning the equilibrium between phases, called Clapeyron s equation, or sometimes the Clapeyron-Clausius equation. By way of illustration, let us consider the vaporization of water at constant temperature and pressure. On our P-V-T surface, the process we consider is that in which the system is carried along an isothermal on the ruled part of the surface, from the state whore it is all liquid, with volume Fz, to the state where it is all gas, with volume F . As we go along this path, we wash to find ihe amount of heat absorbed. We can find this from one of Maxwell s relations, Eq. (4.12), Chap. II  

The path is one of constant temperature, so that if wc multiply by T this relation gives the amount of heat absorbed per unit increase of volume. But on account of the nature of the surface, (0P/dT)v is the same for any point corresponding to the same temperature, no matter what the volume is it is simply the slope of the equilibrium curve on the P-T diagram, wdiich is often denoted simply by dP/dT (since in the P-T diagram there is only one independent variable, and we do not need partial derivatives). Then wc can integrate and have the latent heat L 

Clapeyron s equation holds, as we can see from its method of derivation, for any equilibrium between phases. In the general ease, the difference of volumes on the right side of the equation is the volume after absorbing the latent heat L, minus the volume before absorbing it. 

There is another derivation of Clapeyron s equation which is very instructive. This is based on the use of the Gibbs free energy G. In the last section we have seen that this quantity must be equal for two phases in equilibrium at the same pressure and temperature, and that if one phase has a lower value of G than another at given pressure and temperature, it is the stable phase and the other one is unstable. Wo can verify these results in an elementary way. We know that in going from liquid to vapor, the latent heat L is the difference in enthalpy between gas and liquid, orL = // — Hi. Bui if the change is carried out in equilibrium, the heat absorbed will also equal T (IS, so that the latent heat will be T(SU — Si). Equating these values of the latent heat, we have  

Clapeyron s equation, as an exact result of thermodynamics, is useful in several ways. In the first place, we may have measurements of the equation of state but not of the latent heat. Then we can compute the latent heat. This is particularly useful for instance at high pressures. 

---