Phase Coexistence and the Clausius-Clapeyron Relation

The Condition for Coexistence

Consider a single-component substance that can exist in two distinct phases, labelled 1 and 2. For these phases to coexist in equilibrium, there must be no net driving force for matter to transfer from one phase to the other. In thermodynamics, the quantity that governs the transfer of matter is the chemical potential, \(\mu\). For a pure substance, the chemical potential is identical to the molar Gibbs free energy, \(g\). Therefore, the fundamental condition for phase equilibrium is that the molar Gibbs free energy of phase 1 must equal that of phase 2 at the same temperature \(T\) and pressure \(P\). If \(g_1\) were greater than \(g_2\), molecules would spontaneously migrate to phase 2 to lower the total free energy of the system. Conversely, if \(g_2\) were greater, migration would occur to phase 1. Only when \(g_1(T,P) = g_2(T,P)\) is the system stable against such transfers. This equality defines a specific curve in the \(P\)-\(T\) plane. As we move along this curve, both \(T\) and \(P\) change, but the equality of the Gibbs free energies must be maintained at every point. The slope of this curve, \(dP/dT\), is the quantity we seek to determine.

Deriving the Slope from Differential Changes

To find the slope of the coexistence curve, we consider an infinitesimal change in temperature and pressure along the boundary. Since \(g_1 = g_2\) at all points on the curve, the differential change in \(g_1\) must equal the differential change in \(g_2\). The fundamental thermodynamic relation for the molar Gibbs free energy is \(dg = -s \, dT + v \, dP\), where \(s\) is the molar entropy and \(v\) is the molar volume. Applying this to both phases, we have \(dg_1 = -s_1 \, dT + v_1 \, dP\) and \(dg_2 = -s_2 \, dT + v_2 \, dP\). Equating these differentials gives \(-s_1 \, dT + v_1 \, dP = -s_2 \, dT + v_2 \, dP\). Rearranging terms to isolate the slope \(dP/dT\), we obtain \((v_2 - v_1) \, dP = (s_2 - s_1) \, dT\). Dividing by \(dT\), we find that the slope is given by the ratio of the change in molar entropy to the change in molar volume.

This result is exact and relies only on the fundamental definitions of thermodynamic potentials. It states that the slope of the phase boundary is determined by the entropy difference and volume difference between the two phases. While this form is general, it involves entropy, which is often difficult to measure directly. We can express this in terms of more accessible quantities by using the definition of latent heat.

Introducing Latent Heat

The change in entropy during a phase transition at constant temperature is directly related to the latent heat. The latent heat \(L\) is the energy required to transform one mole of the substance from phase 1 to phase 2 at constant \(T\) and \(P\). By the second law, for a reversible phase transition, the change in entropy is \(\Delta s = s_2 - s_1 = L/T\). Substituting this into our previous expression for the slope, we obtain the Clausius-Clapeyron relation in its most common form. This relation connects the slope of the phase boundary to the latent heat of the transition and the change in molar volume.

Here, \(L\) is the molar latent heat, \(T\) is the absolute temperature, and \(\Delta v = v_2 - v_1\) is the change in molar volume. This equation is powerful because it relates a geometric property of the phase diagram (the slope) to physical properties of the substance (latent heat and volume change). It applies to any first-order phase transition, including solid-liquid, liquid-gas, and solid-gas transitions.

Approximations for Gas-Liquid Equilibrium

In many practical applications, such as the vaporisation of a liquid, we can make simplifying assumptions to obtain a more useful form of the relation. For a liquid-gas transition, the molar volume of the gas phase \(v_g\) is typically much larger than the molar volume of the liquid phase \(v_l\). Therefore, we can approximate \(\Delta v \approx v_g\). If we further assume that the gas behaves ideally, then \(v_g = RT/P\), where \(R\) is the molar gas constant. Substituting this into the Clausius-Clapeyron relation gives \(dP/dT = L / (T \cdot RT/P) = LP / (RT^2)\). Rearranging this differential equation, we get \(dP/P = (L/RT^2) \, dT\). If we assume that the latent heat \(L\) is approximately constant over the temperature range of interest, we can integrate this equation. The integral of \(dP/P\) is \(\ln P\), and the integral of \(1/T^2 \, dT\) is \(-1/T\). This leads to the integrated form of the Clausius-Clapeyron equation.

where \(C\) is an integration constant. This equation shows that the natural logarithm of the vapour pressure is linearly proportional to the inverse of the absolute temperature. This linear relationship is a hallmark of ideal vaporisation and is widely used in experimental physics and chemistry to determine latent heats from vapour pressure data. The slope of a plot of \(\ln P\) versus \(1/T\) is \(-L/R\), allowing for the direct extraction of \(L\).