Phase Equilibrium
A particular class of equilibrium states concerns equilibria between different phases of the same substance, e.g., liquid-vapor equilibria.
Figure 17.2 shows liquid and vapor in equilibrium in a system where pressure p and temperature T are fixed at the boundaries by the given mass of the piston, and exposure to a large reservoir at T . According to (16.1) the equilibrium state of this system is determined by a minimum of the Gibbs free energy G, which is just the sum of the Gibbs free energies of the two phases. The mass m = mL + mV within the system is constant, and thus we have
The specific free energies of the individual phases, gL and gV , depend only on the intensive variables p and T . When thermal and mechanical equilibrium are established, T and p are homogeneous throughout both phases, and the vapor mass mV is the only variable. The chemical equilibrium is assumed when G becomes a minimum, that is for dG/dmV = 0, which gives
Hence, in a two phase system in equilibrium, pressure, temperature and Gibbs free energies are homogeneous. It follows that both phases can coexist only at values for pressure and temperature (T, p) that fulfill the above condition. Solving for p gives the saturation pressure psat (T ), with the well known value of psat (100 ◦C) = 1 atm for water. Solving for T gives the saturation temperature, Tsat (p).
In case that temperature and pressure are chosen such that the Gibbs free energies of liquid and vapor are different, the Gibbs free energy (17.38) assumes a boundary minimum with either mL = m, mV = 0 (compressed liquid) or mV = m, mL = 0 (superheated vapor). In detail we have for a specified pressure p:
The phase change can be understood as a competition between energy and entropy. Recall that Gibbs free energy is g = h − T s. For small temperatures, the entropic term (−T s) is relatively small, and energetic effects dominate.
Then the Gibbs free energy is small for the liquid, where the potential energy between particles due to the molecular interaction is at a minimum, the particles are close to each other, and the volume is small. For larger temperatures, the entropic contribution becomes more important, and the Gibbs free energy becomes small for large entropies. Since vapor entropy grows with volume,1 the vapor state prevails and the volume is large. At saturation, energetic and entropic contributions are of comparable size, and both phases coexist.
Alternatively, we have for a specified temperature T :
Since vapor entropy grows with lower pressure2, the entropic term will dominate even at low temperatures, if only the pressure is sufficiently small. Thus, exposing a substance to low pressure might induce phase change.
While we used liquid and vapor as example, the above derivation is not restricted to any particular phases. For any two phases to be in equilibrium, their Gibbs free energies must agree. For an example, revisit Fig. 6.4 in Chapter 6 which shows the saturation lines for water as ice, liquid, and vapor.
At the triple point, all three phases coexist in equilibrium, and their free energies must agree (S stands for solid),
These are two conditions for T, p and thus there is only one pair of values Ttr , ptr at which three phases can coexist, the triple point (e.g., for water: Ttr = 0.01 ◦C, ptr = 611 Pa).
The conditions derived above describe the thermodynamic equilibrium of two phases, which is not always attained. Some substances can exist for very long periods in metastable states, outside of equilibrium. A typical example is tin, which below 13.2 ◦C is stable as a semiconductor phase, and is metallic above. However, the phase transition does only occur at much lower tem- peratures. Another example is carbon, for which the stable phase at room temperature is graphite, while diamond is metastable, which obviously does not diminish its value, both as a gem, and for toolmaking.