The Clapeyron-Clausius equation is a differential equation giving the interdependence of the pressure and temperature along the phase equilibrium curve of a pure substance. This equation was suggested by B. Clapeyron in 1834 and improved by R. Clausius in 1850.

According to the general conditions of thermodynamic equilibrium, when two phases of a pure substance are in equilibrium, the following equations are valid:

where the subscripts 1 and 2 refer to the respective phases; g is the specific Gibbs energy.

Since the specific Gibbs energy is a function of temperature and pressure, it follows that there is an additional interdependence between temperature and pressure:

This interdependence cannot be expressed in an explicit form because the specific Gibbs energy is defined in thermodynamics only in terms of arbitrary constants included in the enthalpy and entropy expressions. Therefore, the interdependence between T and p is considered in a differential form.

Equation (2) has to be obeyed in every state of phase equilibrium, which means that the following equation has to be valid:

where dg is the change of g along the phase equilibrium curve when both temperature and pressure are changed. Therefore:

From thermodynamics, it is known that:

where s and v are, respectively, the specific entropy and specific volume of the phase.

As a result, the equilibrium condition can be written as follows:

or

Equation (5) is the initial form of the Clapeyron-Clausius equation.

Proceeding from the fact that an equilibrium phase transition in a pure substance is an isothermal and isobaric one, it can be concluded:

where h_{12} is the specific enthalpy of phase transition from Phase 1 to Phase 2 and h_{2} and h_{1} are the respective specific enthalpies of the phases.

According to Eq. (6), the Clapeyron-Clausus equation for different phase equilibria can be written:

**solid-vapor equilibrium**(7)**solid-liquid equilibrium**(8)**liquid-vapor equilibrium**(9)

Here h_{SG}, h_{LS} and h_{LG} are respectively the heats of sublimation, melting and evaporation. The subscripts G, S and L—refer respectively to the vapor, solid and liquid phases.

To integrate the Clapeyron-Clausius equation, it is in general necessary to know the explicit temperature and pressure relations for the enthalpy of phase transition and specific volume. As a rule, the respective functions are complicated and unknown; but in some particular cases, the integration can be carried out.

At moderate pressures, the vapor specific volume is several orders of magnitude greater than the liquid or solid specific volume. It is therefore possible to neglect the values v_{s} and v_{L} in Eqs. (7) and (9). At the same time, the vapor's specific volume can, with reasonable accuracy, be derived from the perfect gas equation of state:

Using these assumptions, Eqs. (7) and (9) can be written as:

As a first approximation, the heats of phase transition h_{SG} and h_{LG} at moderate pressures can be regarded as constants. With this assumption, Eqs. (10) and (11) are easily integrated:

The integration constants C_{1} and C_{2} can be found when the temperature and pressure at the triple point T_{tr}, P_{tr} and the normal boiling temperature T_{n·b} are known. Equations (12) and (13) are then expressed as:

In a semilogarithmic plot, both equations are presented as segments of straight lines with slopes h_{SG}/R and h_{LG}/R having an intersection at the triple point.

Since the specific volumes v_{L} and v_{S} are of the same order, the above approach is invalid for the solid-liquid equilibrium. Moreover, there exist substances (for instance, water) having v_{S} > v_{L}. In this case, dp/dT < 0, i.e., with growing pressure the melting temperature for such substances decreases.

Heat & Mass Transfer, and Fluids Engineering