Page 148 - Thermodynamics of Biochemical Reactions

146 Chapter 8 Phase Equilibrium in Aqueous Systems

/iCz = ,uCg = p,. Substitution of this relation in equation 8.3-3 yields

dG = - SdT+ VdP + p,dnCC + (d>/j - d),)dQ (8.3-4)
where n,, is the amount of the component C. This indicates that there are four
natural variables for this system, D = 4. Integration of equation 8.3-4 at constant
values of the intensive variables yields
G = Ycncc + (d/~ - d’aJQ (8.3-5)

The Gibbs-Duhem equations for the two phases at phase equilibrium are
0 = - S,dT+ I/,dP - nCadpc - Q,dqh, (8.3-6)

(8.3-7)
This looks like there are five intensive variables, but there are not because only
the difference in electric potentials between the phases is important. We can take
qha = 0 and delete the electric work term in equation 8.3-6. Since there are four
intensive variables and two equations, F = 2, in agreement with F =
C - p + 2 = 2 - 2 + 2 = 2. Note that Qp is taken as a component. This leads to
D = F + p = 2 + 2 = 4 in agreement with equation 8.3-4.
In considering the thermodynamics of systems in which there are electric
potential differences, the activity ui of an ion can be defined in terms of its
chemical potential p; and the electric potential 4i of the phase it is in by

illi = p: + RTln ai + Fzi4, (8.3-8)
where py is the standard chemical potential in a phase where the electric potential
is zero, F is the Faraday constant, and zi is the charge number. This is the
arbitrary introduction of a property of a species, the activity. that is more
convenient in making calculations than the chemical potential of the species.
According to equation 8.3-8 the chemical potential of an ion is a function of d)i
as well as ui. The activity here has the same functional dependence on intensive
variables in the presence of electric potential differences as in the absence of
electric potential differences. When equation 8.3-8 is substituted in p,, = ,ucca, we
obtain

(8.3-9)

or

(8.3-10)

based on the convention that = 0. This is referred to as the membrane equation,
and it has been very useful in research on ion transport and nerve conduction. It
is really a form of the Nernst equation (equation 9.1-4).

W 8.4 TWO-PHASE SYSTEM WITH A CHEMICAL
REACTION AND A MEMBRANE PERMEABLE BY A
SINGLE ION

In this system the reaction A + B = C occurs in both phases, but only C can
diffuse through the membrane (Alberty, 1997d). The fundamental equation for G
is