Equilibrium Between Phases

A change in temperature or pressure may cause a substance to transform from one phase into another. This transformation is called a phase transition.

The principal phase transitions are:

• Melting: Transformation from solid to liquid.
• Freezing: Transformation from liquid to solid.
• Vaporization: Transformation from liquid to vapour.
• Condensation: Transformation from vapour to liquid.
• Sublimation: Direct transformation from solid to vapour.
• Deposition: Direct transformation from vapour to solid.

During phase equilibrium, two phases coexist without any net transfer of matter from one phase to the other, although molecular exchange may continue in both directions.

Thermodynamic Condition for Phase Equilibrium

Consider a closed system containing two phases, denoted by phase 1 and phase 2, of the same pure substance. The phases are in thermal and mechanical contact and are allowed to exchange molecules.

The system is maintained at constant temperature \(T'\) by a heat reservoir and at constant external pressure \(P'\) by a movable piston.

Fig. : Schematic representation of two phases of a pure substance in thermal and mechanical equilibrium at constant temperature and pressure.

For a closed system maintained at constant temperature and pressure, the condition of stable thermodynamic equilibrium is that the Gibbs free energy must be minimum.

The Gibbs free energy is defined as

\[ \boxed{ G=U-TS+PV } \tag{1} \]

where

\[ \begin{aligned} G &= \text{Gibbs free energy},\\ U &= \text{Internal energy},\\ T &= \text{Absolute temperature},\\ S &= \text{Entropy},\\ p &= \text{Pressure},\\ V &= \text{Volume}. \end{aligned} \]

At equilibrium with the surroundings, \(T=T'\) and \(P=P'\). Therefore, the equilibrium condition is

\[ \boxed{ G=\text{minimum} } \tag{2} \]

Let the internal energies, entropies, volumes, and Gibbs free energies of the two phases be represented by the subscripts 1 and 2.

Since the system consists of two phases of the same substance, its total thermodynamic properties are additive.

\[ U=U_1+U_2 \]

\[ S=S_1+S_2 \]

\[ V=V_1+V_2 \]

Consequently, the total Gibbs free energy is

\[ \boxed{ G=G_1+G_2 } \tag{3} \]

Let \(N_1\) and \(N_2\) be the numbers of molecules present in phases 1 and 2, respectively.

If \(g_1\) and \(g_2\) denote the Gibbs free energies per molecule of the two phases, then

\[ G_1=N_1g_1 \]

\[ G_2=N_2g_2 \]

Substituting these expressions into Eq. (4), we obtain

\[ \boxed{ G=N_1g_1+N_2g_2 } \tag{4} \]

Differentiating Eq. (4), we obtain

\[ dG=d(N_1g_1+N_2g_2) \]

Applying the product rule,

\[ \begin{aligned} dG={}&N_1\,dg_1+g_1\,dN_1\\ &+N_2\,dg_2+g_2\,dN_2 \end{aligned} \]

Therefore,

\[ \boxed{ dG=g_1\,dN_1+g_2\,dN_2 }  \because g=constant. \tag{5} \]

The system is closed, so the total number of molecules remains constant:

\[ N_1+N_2=N=\text{constant} \]

Differentiating,

\[ dN_1+dN_2=0 \tag{6} \]

Hence,

\[ \boxed{ dN_2=-dN_1 } \tag{7} \]

Substituting Eq. (7) into Eq. (5),

\[ dG=g_1\,dN_1-g_2\,dN_1 \]

Taking \(dN_1\) as a common factor,

\[ \boxed{ dG=(g_1-g_2)dN_1 } \tag{8} \]

At thermodynamic equilibrium, the Gibbs free energy is stationary with respect to an infinitesimal transfer of molecules between the two coexisting phases. Therefore,

\[ dG=0 \]

Using Eq. (8),

\[ (g_1-g_2)dN_1=0 \]

Since an arbitrary infinitesimal transfer of molecules is possible between the two coexisting phases, \(dN_1\) need not be zero. Consequently,

\[ g_1-g_2=0 \]

Therefore,

\[ \boxed{ g_1=g_2 } \tag{9} \]

When these equilibrium values are plotted on a pressure–temperature (\(P-T\)) diagram, the resulting curve is called the phase equilibrium curve or phase coexistence curve.

Fig. : Schematic phase equilibrium curve separating the stability regions of two phases of a pure substance. The curve represents \(g_1=g_2\).

The phase equilibrium curve divides the pressure–temperature diagram into regions in which different phases are thermodynamically stable.

At a given temperature and pressure, the stable phase is the one having the lower Gibbs free energy per molecule.

Case I: When \(g_1<g_2\)

Phase 1 is the thermodynamically stable phase, and the substance tends to transform completely into phase 1.

Case II: When \(g_1>g_2\)

Hence, phase 2 is thermodynamically stable, and the substance tends to transform completely into phase 2.

Case III: When \(g_1=g_2\)

Consequently, both phases can coexist in equilibrium at the same temperature and pressure.