Using the state postulate, take the specific entropy, , for a homogeneous substance to be a function of specific volume, , and temperature, .
During a phase change, the temperature is constant, so
Using the appropriate Maxwell relation gives
Since temperature and pressure are constant during a phase change, the derivative of pressure with respect to temperature is not a function of the specific volume. Thus the partial derivative may be changed into a total derivative and be factored out when taking an integral from one phase to another,
For a closed system undergoing an internally reversible process, the first law is
Using the definition of specific enthalpy, , and the fact that the temperature and pressure are constant, we have
After substitution of this result into the derivative of the pressure, one finds
where the shift to capital letters indicates a shift to extensive variables. This last equation is called the Clausius-Clapeyron equation, though some thermodynamics texts just call it the Clapeyron equation, possibly to distinguish it from the approximation below.
When the transition is to a gas phase, the final specific volume can be many times the size of the initial specific volume. A natural approximation would be to replace with . Furthermore, at low pressures, the gas phase may be approximated by the ideal gas law, so that , where R is the mass specific gas constant (forcing and to be mass specific). Thus,
This leads to a version of the Clausius-Clapeyron equation that is simpler to integrate:
These last equations are useful because they relate saturation pressure and saturation temperature to the enthalpy of phase change, without requiring specific volume data. Note that in this last equation, the subscripts 1 and 2 correspond to different locations on the pressure versus temperature phase lines. In earlier equations, they corresponded to different specific volumes and entropies at the same saturation pressure and temperature.
Suppose two phases, I and II, are in contact and at equilibrium with each other. Then the chemical potentials are related by . Along the coexistence curve, we also have . We now use the Gibbs-Duhem relation , where and are, respectively, the entropy and volume per particle, to obtain
Hence, rearranging, we have
From the relation between heat and change of entropy in a reversible process δQ = T dS, we have that the quantity of heat added in the transformation is
Combining the last two equations we obtain the standard relation.
The Clausius-Clapeyron equation for the liquid-vapor boundary may be used in either of two equivalent forms.
This can be used to predict the temperature at a certain pressure, given the temperature at another pressure, or vice versa. Alternatively, if the corresponding temperature and pressure is known at two points, the enthalpy of vaporization can be determined.
The equivalent formulation, in which the values associated with one P,T point are combined into a constant (the constant of integration as above), is
For instance, if the p,T values are known for a series of data points along the phase boundary, then the enthalpy of vaporization may be determined from a plot of against .
In meteorology, a specific derivation of the Clausius-Clapeyron equation is used to describe dependence of saturated water vapor pressure on temperature. This is similar to its use in chemistry and chemical engineering.
It plays a crucial role in the current debate on climate change because its solution predicts exponential behavior of saturation water vapor pressure (and, therefore water vapor concentration) as a function of temperature. In turn, because water vapor is a greenhouse gas, it might lead to further increase in the sea surface temperature leading to runaway greenhouse effect. Debate on iris hypothesis and intensity of tropical cyclones dependence on temperature depends in part on “Clausius-Clapeyron” solution.
Clausius-Clapeyron equations is given for typical atmospheric conditions as
One can solve this equation to give
Thus, neglecting the weak variation of (T+243.5) at normal temperatures, one observes that saturation water vapor pressure changes exponentially with .
To provide a rough example of how much pressure this is, to melt ice at -7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass = 1000 kg) on a thimble (area = 1 cm²).