Clapeyron Equation

To find out the dependence of pressure on equilibrium temperature when two phases coexist.

Along a phase transition line, the pressure and temperature are not independent of each other, since the system is univariant, that is, only one intensive parameter can be varied independently.

When the system is in a state of equilibrium, i.e., thermal, mechanical and chemical equilibrium, the temperature of the two phases has to be identical, the pressure of the two phases has to be equal and the chemical potential also should be the same in both the phases.

Representing in terms of Gibbs free energy, the criterion of equilibrium is:

at constant T and P

or,

Consider a system consisting of a liquid phase at state 1 and a vapour phase at state 1’ in a state of equilibrium. Let the temperature of the system is changed from T₁ to T₂ along the vaporization curve.

For the phase transition for 1 to 1’:

or

or

In reaching state 2 from state 1, the change in the Gibbs free energy of the liquid phase is given by:

Similarly, the change in the Gibbs free energy of the vapour phase in reaching the state 2’ from state 1’ is given by:

Therefore,

Or

Where the subscript sat implies that the derivative is along the saturation curve.

The entropy change associated with the phase transition:

Hence,

Which is known as the Clapeyron equation

Since is always positive during the phase transition, _sat will be positive or negative depending upon whether the transition is accompanied by expansion (>0) or contraction (<0).

Consider the liquid-vapour phase transition at low pressures. The vapour phase may be approximated as an ideal gas. The volume of the liquid phase is negligible compared to the volume of the vapour phase( >> )and hence = = =RT/P.

The Clapeyron equation becomes:

or

which is known as the Clausius-Clapeyron equation.

Assume that is constant over a small temperature range, the above equation can be integrated to get,

or +constant

Hence, a plot of lnP versus 1/T yields a straight line the slope of which is equal to –(h_fg/R).

Kirchoff Equation

Kirchoff relation predicts the effect of temperature on the latent heat of phase transition.

Consider the vaporization of a liquid at constant temperature and pressure as shown in figure. The latent heat of vaporization associated with the phase change 1 to 1’ is (-) at temperature T. When the saturation temperature is raised to (T+dT), the latent heat of vaporization is (-). The change in latent heat,

The variation in the enthalpy associated with the variation in the independent variables T and P is given by:

or,

Substituting for (dP/dT)_sat from the clapeyron equation,

This is known as Kirchoff relation.

For a solid-to-liquid transition, it is a reasonably good approximation to assume that the molar heat capacity and the molar volume are constant in each phase and the coefficient of volume expansion b is negligible for each phase. Then,

where is the latent heat of fusion.

For the transition from liquid phase to vapour phase, the molar volume of the liquid phase can be neglected compared to the molar volume of the gas phase, and b_g>>b_f. The vapour phase may be approximated as an ideal gas. Then b_g=1/T. It is clear that v_gb_g> v_fb_f. Hence,

Phase Equilibrium- Gibbs Phase Rule

The number of independent variables associated with a multi component, multiphase system is given by the Gibbs Phase Rule, expressed as,

F=C+2-P

Where,

F= The number of independent variables

C= The number of components

P= The number of phases present in the equilibrium

· For a single component (C=1) two phase (P=2) system, one independent intensive property needs to be specified (F=1).

· At the triple point, for C=1, P=3 and thus F=0. None of the properties of a pure substance at the triple point can be varied.

· Two independent intensive properties need to be specified to fix the equilibrium state of a pure substance in a single phase.

Phase diagram for a single component system is given in figure.