Equilibrium Between Phases

Consider a system that consists of two phases, which we shall denote by 1 and 2. For example, these might be solid and liquid, or liquid and gas. Suppose that the system is in equilibrium with a reservoir at the constant temperature $T$ , and the constant pressure $p$ , so that the system always has the temperature $T$ and the mean pressure $p$ . However, the system can exist in either one of its two phases, or some mixture of the two. Let us begin by finding the conditions that allow the two phases to coexist in equilibrium with one another.

In accordance with the discussion in Section 9.4, the equilibrium condition is that the Gibbs free energy, $G$ , of the system is a minimum:

$$G = E-T S+p V = {\rm minimum}.$$ (9.59)

Let $\nu_i$ be the number of moles of phase $i$ present in the system, and let $g_i(T,p)$ be the Gibbs free energy per mole of phase $i$ at the temperature $T$ and the pressure $p$ . It follows that

$$G = \nu_1 g_1 + \nu_2 g_2.$$ (9.60)

Furthermore, the conservation of matter implies that the total number of moles, $\nu$ , of the substance remains constant:

$$\nu_1+\nu_2=\nu={\rm constant}.$$ (9.61)

Thus, we can take $\nu_1$ as the one independent parameter that is free to vary. In equilibrium, Equation (9.59) requires that $G$ be stationary for small variations in $\nu_1$ . In other words,

$$dG = g_1 d\nu_1+g_2 d\nu_2 = (g_1-g_2) d\nu_1 = 0,$$ (9.62)

because $d\nu_2=-d\nu_1$ , as a consequence of Equation (9.61). Hence, a necessary condition for equilibrium between the two phases is

$$g_1=g_2.$$ (9.63)

Clearly, when this condition is satisfied then the transfer of a mole of substance from one phase to another does not change the overall Gibbs free energy, $G$ , of the system. Hence, $G$ is stationary, as required. Incidentally, the condition that $G$ is a minimum (rather than a maximum) is easily shown to reduce to the requirement that the heat capacities and isothermal compressibilities of both phases be positive, so that each phase is stable to temperature and volume fluctuations.

Figure 9.1: Pressure-temperature plot showing the domains of relative stability of two phases, and the phase-equilibrium line.

Now, for a given temperature and pressure, $g_1(T,p)$ is a well-defined function characteristic of phase 1. Likewise, $g_2(T,p)$ is a well-defined function characteristic of phase 2. If $T$ and $p$ are such that $g_1<g_2$ then the minimum value of $G$ in Equation (9.59) is achieved when all $\nu$ moles of the substance transform into phase 1, so that $G=\nu g_1$ . In this case, phase 1 is the stable one. On the other hand, if $g_1>g_2$ then the minimum value of $G$ is achieved when all $\nu$ moles of the substance transform into phase 2, so that $G=\nu g_2$ . In this case, phase 2 is the stable one. Finally, if $g_1=g_2$ then the condition (9.59) is automatically satisfied, and any amount, $\nu_1$ , of phase 1 can coexist with the remaining amount, $\nu_2=\nu-\nu_1$ , of phase 2. The locus of the points in the $T$ -$p$ plane where $g_1=g_2$ then represents a phase-equilibrium line along which the two phases can coexist in equilibrium. This line divides the $T$ -$p$ plane into two regions. The first corresponds to $g_1<g_2$ , so that phase 1 is stable. The second corresponds to $g_1>g_2$ , so that phase 2 is stable. See Figure 9.1.