Page 145 - Thermodynamics of Biochemical Reactions

8.1 Two-Phase Systems without Chemical Reactions 143

dG < 0 at constant P, itAa, and itAB. The natural variables for a multiphase system
must include extensive variables that are related to the sizes of all the phases, since
the amounts of the various phases are independent variables. The number D of
natural variables is in agreement with D = F + p because this yields
D = 1 + 2 = 3. In summary, a suitable choice of natural variables includes F
intensive variables and p extensive variables, which may be taken as the amounts
in the p phases.

Two Species, Two Phases

The fundamental equation for G is
dG = - SdT+ VdP + pAadnAz + pBrdYlg, + pAlidnAli + pBsdnBO (8.1-8)

This can be used to derive two equilibrium expressions that convert this
fundamental equation to its form at phase equilibrium, which is
dG = - SdT+ VdP + ,uAdn,,\ + pUdnCB (8.1-9)

where piL4 is the amount of the A in the system and ncB is the amount of B in the
system. This shows that the system at phase equilibrium has D = 4 natural
variables, which can be taken to be 7; P, nLA and ncR. The Gibbs-Duhem
equations for the two phases at equilibrium are
0 = - S,dT+ I/,dP - nA,dpA - nBzdpB (8.1 - 10)
(8.1-11)

dpB can be eliminated between these two equations to obtain pA as a function of
T and P. Thus F = 2. The relation D = F + p is satisfied, and the natural
variables can be taken to be 7; P, ltcA. and ncB, although it might be more
convenient to use 7; P, nv, and no, where n, = nAa + nu,.

N, Species, Two Phases
When there are N, species the fundamental equation for G can be written
N. N jj
dG = - SdT+ VdP + C piadnix + 1 pigdnig (8.1-12)
i=l i= 1
where there is a term for each species in each phase. This equation can be used
to derive the N, equilibrium conditions pj, = pis. Substituting these equilibrium
conditions in the fundamental equation yields

dG = - SdT+ VdP + C ,Liidncj (8.1 - 1 3)
i= 1
The number C of components is equal to the number of terms in the summations
in equation 8.1-12 minus the number N, of independent equilibria between phases,
that is, C = 2N, - N, = N,. Equation 8.1-13 shows that there are
D = C + 2 = N, + 2 natural variables. The Gibbs-Duhem equations for the two
phases are

N,
0 = - S,dT+ V,dP - ni,dpi (8.1 - 14)
i=l
NS
0 = - SsdT+ I/;,dP - 2 ttisdpi (8.1 - 1 5)
i= 1
Since there are N, + 2 intensive variables and two relations between them,
F = N,. This is in agreement with D = F + p. The criterion for spontaneous
change and equilibrium for this system is dG < 0 at constant ?; P, and (nCij.