# Clausius Clapeyron equation

Having established that the specific Gibbs energies for liquid and gaseous phases are the same along the phase boundary, we can now proceed to calculate the slope of the phase boundary in the T-p plane. This slope is the rate of change of the vapor pressure with respect to temperature as the system is allowed to move along the phase boundary. This slope measures the rate at which the saturation vapor pressure increases for incremental changes in temperature - an important quantity in meteorology.

First consider such a reversible change of the composite system (gas and liquid in equilibrium) along the phase boundary. We have:

Then (using infinitesimal notation instead of A)

where the small letters s and v refer to specific entropy and volumes respectively, and henceforth we denote the saturation vapor pressure as es. Rearranging:

First, notice that vg » vl (e.g., for one gram mole of vapor 22.4 x 103 cm3 versus 18 cm3 of water) so that vl can be neglected. The specific volume of an ideal gas is RT/p, where R is the gas constant for the particular species (here water vapor). The difference of specific entropies can be calculated. This difference is the change in specific entropy as we convert a unit mass of liquid into gas form at a fixed temperature: AHvap/T = L/T, where L is the enthalpy of vaporization (per kilogram). We arrive at the Clausius-Clapeyron equation:

 des Les dT = RT2

[Clausius-Clapeyron equation]. (5.13)

Before proceeding to integrate this equation to find an expression for es(T), it should be noted that the procedure just employed is very general and can be applied to many other problems. While we will not pursue it here, it is perhaps clear that the equilibrium we speak of could be that of chemical species instead of phases, or it could be a combination of both. In physical chemistry texts the technique of equilibrium boundaries utilizing the Gibbs energy can be found to lead to such diverse rules as the temperature dependence of reaction rate coefficients.