Statistical Physics

8.3 Coexistence of Phases 1218.3 Coexistence of PhasesAs shown in the previous section, at given T and P a system is in the phasewhich has the lowest value of E + PV − S(E,V,N)T , and this lowest value isequal to G. The phase may be a solid, liquid, or gas depending on the temperatureand pressure. Experimentally, these phase transitions occur as thetemperature or pressure is changed, and so the minimum should be transferredfrom one phase to another at the transition point. That is, the coexistencecondition is the condition that the Gibbs free energy has the same value for thetwo phases. Since the Gibbs free energy divided by the number of moleculesis the chemical potential µ(T,P)=G(T,P,N)/N , we can also say that thechemical potentials of the two phases are equal when they coexist.Let us see how the transfer of the minimum occurs. First, we considerthe solid-to-liquid phase transition. In general, the liquid phase has a slightlylarger molar volume and a slightly higher internal energy. The molecules in theliquid require a larger nearest-neighbor distance to move around, and so thevolume expands, and the average potential energy of interaction is higher. Themolecules or atoms in the liquid phase can be anywhere in the volume, but inthe solid phase they can move only around their lattice points. Therefore, thenumber of possible microscopic states is much larger in the liquid phase than inthe solid phase, and the entropy of the liquid phase should be larger. Near thephase transition point, there should be two local minima of E+PV−S(E,V )Tin the E–V plane. One of these minima corresponds to the solid phase and theother to the liquid phase. The quantity E+PV is lower at the solid-phase localminimum, as explained above. Therefore, at lower temperatures, where thecontribution ST is small, the solid phase has a lower value of E +PV −ST.Asthe temperature becomes higher, the liquid-phase minimum decreases fasterthan the solid-phase minimum owing to the larger value of S, and so a transferof the minimum from one phase to the other occurs at some temperature.Next we consider the liquid-to-gas phase transition. In this case the energyis higher in the gas phase because there is practically no energy gain from theattractive interaction of the molecules. The volume and entropy are muchlarger in the gas phase. Therefore, both temperature and pressure can beeffective in causing a transition between the phases. The liquid phase is favoredat high pressure because of the PV term. The gas phase is favored at hightemperature because of the −ST term. These tendencies are in accordancewith our everyday experience.The coexistence condition defines lines in the P –T plane. On the coexistenceline, the volume and the internal energy of the system are somewhatarbitrary. For example, let us consider the case of a gas–liquid transition,where the system consists of one mole of molecules of some kind. When thesystem becomes a liquid it has a smaller molar volume V l , and when it becomesa gas it has a larger molar volume V g . At the coexistence point, the volumecan take values between V l and V g . Namely, if a mass fraction x is in the liquidphase and the remaining fraction 1 − x is in the gas phase, the total volume is