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.
Applications of Clapeyron and Clausius-Clapeyron equations
A number of common phenomena depend on the temperature and pressure dependence of phase boundaries. Ice skating is possible because ice melts under the pressure of the skates and thereby reduces the friction. Weathering occurs when ice in cracks in rocks expands due to cooling in mid-winter. The phase diagram for water-ice shows that ice expands as it cools. This expansion can produce enormous pressures. Both of these phenomena can be calculated using the Clapeyron equation.
The liquid-vapor phase boundary is useful for understanding phenomena such as cooking in a pressure cooker. The Clausius-Clapeyron equation can be used to calculate the temperature and pressure relationships. For example, given a pressure limitation from a safety valve one can calcualte the temperature inside the pressure cooker. Alternative, if we have a target temperature we can design the pressure cooker to reach a specific pressure.
