CHAP. 7: PHASE EQUILIBRIA [CONTENTS] 190
7.5 Phase equilibria of pure substances
For a pure component the intensive criterion of phase equilibrium (7.2) simplifies to
G(1)m =G(2)m , (7.11)whereG(mj)is the molar Gibbs energy in thejthphase, which in a pure component is identical
with the chemical potential. Similarly, equation (7.3) simplifies to
f(1)=f(2). (7.12)7.5.1 Clapeyron equation.
From the extensive criterion of equilibrium (7.1) and from the Gibbs equation (3.36) we have
dp
dT=
S(2)−S(1)
V(2)−V(1)
=
H(2)−H(1)
T[V(2)−V(1)]
=
∆H
T∆V
=
∆Hm
T∆Vm, [phase equilibrium], (7.13)whereS(j),H(j),V(j)are the entropy, enthalpy and volume of phase j, and ∆Hm, ∆Vmare
the molar changes in enthalpy and volume during the phase transition. This relation is called
theClapeyron equation. It expresses the relation between a change in temperature and a
change in pressure under the conditions of equilibrium between two phases.
Note: The Clapeyron equation applies exactly to all first-order phase transitions [see
7.2.1].7.5.2 Clausius-Clapeyron equation.
If one of the phases undergoing a phase transition is a gas at not very high pressures, the
Clapeyron equation (7.13) may be simplified to
d lnps
dT=
∆Hm
RT^2, (7.14)
which is called theClausius-Clapeyron equation. It can be derived from (7.13) on condition
that the gaseous phase is formed by an ideal gas, and that the volume of the liquid or solid phase