Page 144 - Thermodynamics of Biochemical Reactions

142 Chapter 8 Phase Equilibrium in Aqueous Systems

extensive degrees of freedom D have been discussed in Chapter 3, and F' and D'
at specified pH have been discussed in Chapter 4. That discussion is continued
here. The distribution of carbon dioxide between the gas phase and aqueous
solution is discussed as a function of pH and ionic strength.

8.1 TWO-PHASE SYSTEMS WITHOUT CHEMICAL
REACTIONS

One Species, Two Phases
This system is not useful for representing a biochemical system but is needed as
a foundation. The fundamental equation for G for a system containing alpha and
beta phases

dG = - SdT+ VdP + pAzd/TA1 + pAgdMA,j (8.1-1)
This shows that the natural variables for G for this system before phase
equilibrium is established are 7; P, nAa, and t7!,. When A is transferred from one
phase to the other. di?,, = - dnA8. Substituting this conservation relation into
equation 8.1 -1 yields
dG = - SdT+ VdP + (~1.4~ - /L,%p)dlZA1 (8.1 -2)

which shows that /iAS = pAO = p, at phase equilibrium. Substituting this equilib-
rium condition in equation 8.1-1 yields

dG = - SdT+ VdP + p,(dn,, + drtA0) = - SdT+ VdP + /[,d/l,q (8.1-3)
where IZ,, is the amount of component A in the two-phase system. This
fundamental equation shows that the system at phase equilibrium has D = 3
natural variables, which seems to suggest 7; P, and ncA. However, we will see in
the next paragraph that this is not a suitable choice of natural variables. The
Gibbs-Duhem equations for the two phases at phase equilibrium are

0 = S,dT+ ljdP - nA,d/lA (8.1-4)
~
0 = - S,dT+ VOdP - ltAadL1, (8.1-5)

Eliminating dp, between these two equations yields the Clapeyron equation
dP AH,,
- (8.1 -6)
d T TA V,,

where AH = HmAa - H,,, = T(S,,, - S,,& is the change in molar enthalpy and
AVm is the change in molar volume in the phase change. Thus the pressure can
be taken to be a function of the temperature, or the temperature can be taken to
a function of the pressure. This indicates that a two-phase system with a single
species has a single independent intensive variable, in agreement with the phase
rule F = C -p + 2 = 1 - 2 + 2= 1.
Using the Clapeyron equation to eliminate dP from equation 8.1-1 and
substituting /iAa = 1-1,~~ yields

(8.1-7)

This form of the fundamental equation, which applies at equilibrium, indicates
that the natural variables for this system are 7: nAar and nA8. Alternatively, P, nAZ,
and nAp could be chosen. Specification of the natural variables gives a complete
description of the extensive state of the system at equilibrium, and so the criterion
of spontaneous change and equilibrium is dG < 0 at constant 7; it4z, and H,,,~ or