Page 149 - Thermodynamics of Biochemical Reactions

8.4 Two-Phase System with a Chemical Reaction 147

The reactions in the system are represented as
A, + B, = C, (8.4-2)

(8.4-3)
(8.4-4)
Note that the phase transfer is treated like a chemical reaction. When the
equilibrium conditions for these three reactions are introduced into equation
8.4-1, we obtain

dG = - SdT+ VdP + PA&~A, + ~ASdn,,qj + ~cdl~,c + (4, - 4JdQ (8.4-5)
The partial derivative of the Gibbs energy with respect to the amount of a
component yields the chemical potential of a species (Beattie and Oppenheim,
1979).

(8.4-6)

The criterion for equilibrium based on G is dG < 0 at constant 7; P, nLAa, IZ,~,,
nee, and Q. Integration of equation 8.4-1 at constant values of the intensive
variables yields
+
G = PAAA~ pAfinc4p + Pc’,c + (4~ 4JQ (8.4-7)
-
The Gibbs-Duhem equations for the two phases are
0 = - S,d T + I/,dP - ncA,dpAa - ncsadpc (8.4-8)
0 = - S,dT+ I/;,dP - ncABdpA, - nCcBdpc - Qdq5,J (8.4-9)
where 4, has been taken equal as zero. Since there are six variables and two
equations, F = 4, which can be taken to be 7; P, pAa, and &. The number of
independent intensive variables can also be calculated using the phase rule:
F = C - p + 3 = 3 - 2 + 3 = 4, where the 3 is for 7; P, and (PR. The number D
of natural variables is given by D = F + p = 4 + 2 = 6.
The equilibrium expressions for reactions 8.4-2 to 8.4-4, which are derived
from the equilibrium conditions using equation 8.3-8, are

%a
K, = ~ = exp[ - p: - pl- ,ui)/R T] (8.4- 10)
‘Aa‘Ba
uc,
K -- = exPC-(P: - Pi - PWTI (8.4-1 1)
’ - uApaBp

(8.4-12)

The effect of the electric potential cancels in a chemical reaction in a phase. Note
that uc, and uCa are not independent variables in the chemical reaction system.
Substituting the equilibrium concentrations of C from equations 8.4-10 and 8.4-1 1
in equation 8.4-12 yields

(8.4- 13)

or

( 8.4- 14)

This shows how a chemical reaction can establish an electric potential difference
between phases. This potential difference can then be used to transport other ions
between the phases against their concentration gradients.