PHASE EQUILIBRIUM

REVIEW PROBLEMS

4.7 PHASE EQUILIBRIUM

The chemical potential of substance i in the phase is a state function that depends on the temperature, pressure, and composition of the phase. Since m_iis the ratio of infini-tesimal changes in two extensive properties, it is an intensive property. From m_i⬅ the chemical potential of substance i gives the rate of change of the Gibbs energy G of the phase with respect to the moles of i added at constant T, P, and other mole numbers. The state function m_i was introduced into thermodynamics by Gibbs.

Because chemical potentials are intensive properties, we can use mole fractions instead of moles to express the composition dependence of m. For a several-phase system, the chemical potential of substance i in phase a is

(4.85) Note that, even if substance i is absent from phase a (n^a_i  0), its chemical poten-tial m_i^ain phase a is still defined. There is always the possibility of introducing sub-stance i into the phase. When dn^a_i moles of i is introduced at constant T, P, and n_ji, the Gibbs energy of the phase changes by dG^aand m^a_i is given by dG^a/dn^a_i.

The simplest possible system is a single phase of pure substance i, for example, solid copper or liquid water. Let G_m,i(T, P) be the molar Gibbs energy of pure i at the temperature and pressure of the system. By definition, G_m,i⬅ G/ni, so the Gibbs en-ergy of the pure, one-phase system is G niG_m,i(T, P). Partial differentiation of this equation gives

(4.86)*

For a pure substance, m_iis the molar Gibbs free energy. However, m_iin a one-phase mixture need not equal G_mof pure i.

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of substance j flow spontaneously from phase b to phase d. For this irreversible process, the inequality (4.15) gives dG  S dT  V dP. But dG for this process is given by (4.81) as dG S dT  V dP  aim^a_i dn^a_i. Therefore the inequality dG S dT  V dP becomes

For the spontaneous flow of dn_jmoles of substance j from phase b to phase d, we have aim^a_i dn^a_i  mjbdn^b_j  mdj dn_j^d mjbdn_j mdj dn_j 0, and

(4.89) Since dn_jis positive, (4.89) requires that m_j^d mjbbe negative: m^d_j  mjb. The sponta-neous flow was assumed to be from phase b to phase d. We have thus shown that for a system in thermal and mechanical equilibrium:

Substance j flows spontaneously from a phase with higher chemical potential M_j to a phase with lower chemical potential M_j.

This flow will continue until the chemical potential of substance j has been equalized in all the phases of the system. Similarly for the other substances. (As a substance flows from one phase to another, the compositions of the phases are changed and hence the chemical potentials in the phases are changed.) Just as a difference in temperature is the driving force for the flow of heat from one phase to another, a difference in chemical potential m_iis the driving force for the flow of chemical species i from one phase to another.

If T^b T^d, heat flows spontaneously from phase b to phase d until T^b T^d. If P^b P^d, work “flows” from phase b to phase d until P^b P^d. If m_j^b mjd, substance j flows spontaneously from phase b to phase d until m^b_j  mjd. The state function T de-termines whether there is thermal equilibrium between phases. The state function P determines whether there is mechanical equilibrium between phases. The state func-tions m_idetermine whether there is material equilibrium between phases.

One can prove from the laws of thermodynamics that the chemical potential m_j^d of substance j in phase d must increase when the mole fraction x_j^dof j in phase d is increased by the addition of j at constant T and P (see Kirkwood and Oppenheim, sec. 6-4):

(4.90)

EXAMPLE 4.5

Change in M_iwhen a solid dissolves

A crystal of ICN is added to pure liquid water and the system is held at 25°C and 1 atm. Eventually a saturated solution is formed, and some solid ICN remains undissolved. At the start of the process, is m_ICNgreater in the solid phase or in the pure water? What happens to m_ICNin each phase as the crystal dissolves? (See if you can answer these questions before reading further.)

At the start of the process, some ICN “flows” from the pure solid phase into the water. Since substance j flows from a phase with higher m_jto one with lower m_j, the chemical potential m_ICNin the solid must be greater than m_ICNin the pure water. (Recall from Sec. 4.6 that m_ICNis defined for the pure-water phase even though there is no ICN in the water.) Since m is an intensive quantity and the

10mjd>0xjd2T,P, n^dij 7 0 1mj

d mj

b2 dnj 6 0 a

a

a

i

m_i^a dn_i^a 6 0

S dT  V dP  a

a

a

i

m_i^a dn_i^a 6 S dT  V dP

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Section 4.7 Phase Equilibrium 131 temperature, pressure, and mole fraction in the pure-solid phase do not change

as the solid dissolves, m_ICN(s)remains constant during the process. As the crystal dissolves, x_ICN in the aqueous phase increases and (4.90) shows that m_ICN(aq) increases. This increase continues until m_ICN(aq) becomes equal to m_ICN(s). The system is then in phase equilibrium, no more ICN dissolves, and the solution is saturated.

Exercise

The equilibrium vapor pressure of water at 25°C is 24 torr. Is the chemical po-tential of H₂O(l) at 25°C and 20 torr less than, equal to, or greater than m of H₂O(g) at this T and P? (Hint: The vapor pressure of water at temperature T is the pressure of water vapor that is in equilibrium with liquid water at T.) (Answer:

greater than.)

Just as temperature is an intensive property that governs the flow of heat, chemi-cal potentials are intensive properties that govern the flow of matter from one phase to another. Temperature is less abstract than chemical potential because we have experi-ence using a thermometer to measure temperature and can visualize temperature as a measure of average molecular energy. One can get some feeling for chemical potential by viewing it as a measure of escaping tendency. The greater the value of m_j^d, the greater the tendency of substance j to leave phase d and flow into an adjoining phase where its chemical potential is lower.

There is one exception to the phase-equilibrium condition m_j^b mjd, which we now examine. We found that a substance flows from a phase where its chemical potential is higher to a phase where its chemical potential is lower. Suppose that substance j is ini-tially absent from phase d. Although there is no j in phase d, the chemical potential m^d_j is a defined quantity, since we could, in principle, introduce dn_jmoles of j into d and measure (or use statistical mechanics to calculate m_j^d). If initially m_j^b m_j^d, then j flows from phase b to phase d until phase equilibrium is reached.

However, if initially m_j^d m^b_j, then j cannot flow out of d (since it is absent from d). The system will therefore remain unchanged with time and hence is in equilibrium. Therefore when a substance is absent from a phase, the equilibrium condition becomes

(4.91) for all phases b in equilibrium with d. In the preceding example of ICN(s) in equilib-rium with a saturated aqueous solution of ICN, the species H₂O is absent from the pure solid phase, so all we can say is that m_H

2Oin the solid phase is greater than or equal to m_H

2Oin the solution.

The principal conclusion of this section is:

In a closed system in thermodynamic equilibrium, the chemical potential of any given substance is the same in every phase in which that substance is present.

EXAMPLE 4.6

Conditions for phase equilibrium

Write the phase-equilibrium conditions for a liquid solution of acetone and water in equilibrium with its vapor.

Acetone (ac) and water (w) are each present in both phases, so the equilib-rium conditions are m_ac^l  m^yacand m_w^l  m^yw, where m_ac^l and m^y_acare the chemical potentials of acetone in the liquid phase and in the vapor phase, respectively.

m_j^d mjb phase equilib., j absent from d 7

7

10G^d>0n^jd2^T,P,ndij  mjd

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Exercise

Write the phase-equilibrium conditions for a crystal of NaCl in equilibrium with an aqueous solution of NaCl. (Answer: m^s_NaCl m^aNq

aCl.)