Phase Equilibrium

2. Phase Equilibrium

Water exists as a liquid at 20ºC at pressures in excess of 2.34 kPa. This is the satura- tion pressure or bubble point of water at 20ºC. During an isothermal process, a vapor bubble forms as the pressure equals the saturation pressure. As the pressure is further lowered, all of the water will exist as superheated vapor. Therefore, at a specified temperature, water can exist as a compressed liquid (P>P sat ), as a two–phase mixture (P = P sat ), or as superheated vapor (P<P sat ). In the compressed liquid or superheated states, the water exists in a single phase and two independent properties, say (T, P), are required to specify the state, i.e., there are two de- grees of freedom present. In the saturated liquid or vapor region, one independent property,

e.g., temperature, specifies P sat (conversely, the pressure specifies T ). In a mixture consisting of two components (say, methanol and water) an additional parameter related to the component

sat

concentration, e.g., the mole fraction is required to describe the state of the system. Methanol is highly volatile and has a higher P sat as compared to water at the same temperature. There- fore, a mixture may have different compositions in the corresponding liquid and vapor phases. The phase change phenomenon is schematically illustrated in Figure 1 .

a. Two Phase System Consider a two component mixture consisting of water (normal boiling point 100ºC) and methanol (normal boiling point 65ºC) in a piston–cylinder–weight assembly immersed in an isothermal bath. Suppose the molal concentration of water is 60 % and, consequently, 40 % for methanol and the mixture temperature is 20ºC. At a pressure of 200 kPa there is no phase change for this mixture (cf. Figure 2 ). The same holds true for any methanol–water mixture of arbitrary composition at the same temperature and pressure. If the pressure is decreased to 6.2 kPa (while the mixture temperature is still 20ºC), a vapor bubble appears, but inside the vapor bubble the mole fraction of water vapor at steady state is 0.23 (so that the mole fraction of methanol is 0.77) even though the liquid water content is 60 % . This is an example of phase equilibrium between a liquid mixture and a vapor bubble. Upon decreasing the pressure to 5.0 kPa additional vapor is formed, the composition of which can be determined by developing phase equilibrium criteria, as will be discussed below. During evaporation, it is possible to

P =200 kpa

T=20 C, H2O

P=6.2 kPa

T=20 C, CH3OH = 40 % H2O : 60 %

Figure 1: Illustration of phase change in a single– and two–component mix- ture.

maintain the liquid at constant composition by introducing appropriate amounts of the compo- nent(s) into the piston–cylinder–weight assembly.

Recall from Chapter 3 that ˆg α = ˆg β k k at phase equilibrium. This can also be inferred by using the results from Chapter 8 for a fixed mass system, i.e.,

d ˆg k =– ˆs k dT + ˆv k dP. (1) At constant temperature and pressure, d ˆg k = 0 so that during phase transition ˆg k remains un-

changed. (This result is true also for pure components.) The partial molal Gibbs function is related to the fugacity (since, d ˆg k = d ln ˆf k ) so that at equilibrium ˆg k α = ˆg k β or ˆf k β = ˆf k α .

b. Multiphase Systems Consider a k–component mixture consisting of π phases. At a specified temperature

and pressure (cf. Appendix A) ˆg 1(1) = ˆg 1(2) = ˆg 1(3) = ... = ˆg 1( π) , ˆg 2(1) = ˆg 2(2) = ˆg 2(3) = ... = ˆg 1( π) ,

…, and ˆg k(1) = ˆg k(2) = ˆg k(3) = ... = ˆg 1( π) , (2) where the subscripts 1(2) refer to the component 1 in phase 2 . ˆg k( α) = g k( α) (T,P) + R T ln( ˆf k( α) /f k( α) )=( g k( α),0 (T,P)+ R T ln f k( α) )+ R T ln ˆf k( α) /f k( α)

= g k( α α),0 (T,P) + R T ln ˆf k . (3) Where g k( α),0 (T,P) refers to ideal gas Gibbs free energy.

Similarly,

(4) ˆf k( α) = ˆf k( β)

ˆg k( β) = g k( β) , 0 (T,P) + R T ln ˆf k( β) , and

(5) Expanding the last equation

f ˆ 11 () = f ˆ 12 () ...ˆ f 1 () π ,ˆ f 21 () = ˆ f 22 () ...ˆ f 2 () π ˆ f k () 1 = ˆ f k () 2 ...ˆ f k () π .

c. Gibbs Phase Rule In a single phase consisting of a pure component, the temperature and pressure can be independently varied, i.e., it possesses two degrees of freedom (F = 2). In order for two phases of a single component to coexist, a single degree of freedom (F = 1) must exist, i.e., either the

temperature or pressure. For example, one can maintain vapor and liquid phases for H 2 O at say T= 20 C (P sat = 2.34kPa), T = 100 C (P sat = 100 kPa) etc. In this case “T” is an independ- ent variable and P sat is dependent on T within the wet region. If a single component simultane- ously exists in three phases of (e.g., at the triple point of the substance), there are no degrees of

P=6.2 kPa

T=13 C

Figure 2: Illustration of Raoult’s Law. The component states are described in terms of a P–v diagram for the pure compo- nents.

freedom, and F = 0 (for instance, at the triple point of ice, liquid water, and water vapor at 273.15 K and 0.0061 bar). If all three phases are desired, then neither the temperature nor the pressure can be altered.

For a single phase consisting of two components, there are three degrees of freedom (the temperature, pressure, and the mole fraction of either species). Two phases of two compo- nents must possess two degrees of freedom in order to exist. For three phases of two compo- nents to exist, there must be a single degree of freedom. In general, the number of degrees of freedom of an N–component mixture containing π phases is (cf. Appendix A)

F=K+2– π. (6)