The phase rule tells that there is only one degree of freedom along a phase bounadry (e.g. solid-liquid or liquid-vapor). This means that it is not possible to change the temperature without changing the pressure in a defined way. Temperature and pressure are functionally related along the coexistence curve. There are a number of ways to derive this relationship. Fortunately, no matter how we derive the relationship we obtain the same answer. One simple way to see the relationship along the phase boundary is to use the temperature ddependence of the free energy. For example, along the liquid-vapor phase boundary the equilirium constant is just the vapor pressure. If we consider the equilibrium:
then we can see that the equilibrium constant is nothing more than the vapor pressure of the liquid. Recall that we do not consider solid substnacces or pure liquids contributing to the equilibrium constant. In this case we can derive the temperature dependence of the equilibrium constant.
