Phase Transition

The curve separating the liquid and gas phases ends in the critical point V_c, τ_c, where (∂P/∂V)_{τ = τc} = 0. Using arguments based on thermodynamic stability, determine

at the critical point.

SOLUTION

For a system at equilibrium with an external reservoir, the Gibbs free energy G = F + P₀V is a minimum. Any deviation from equilibrium will raise G:

(1)

where P₀ is the pressure of the reservoir. Expanding δF in δV, we have

(2)

Since ∂F/∂V = -P, we may rewrite (2) as

(3)

At the critical point, ∂P/∂V = 0, so (3) becomes

(4)

For (4) to hold for arbitrary δV, we have

(5)