Page 51 - Thermodynamics of Biochemical Reactions

3.5 Isomer Group Thermodynamics 45

Substituting these expressions in niso = Eni yields
N,,"
-
p:so = -RTln 1 exp [ ~ (3.5-6)
i= 1
since piso = pi. The corresponding expressions for the standard enthalpy, entropy,
and heat capacity can be obtained by using the derivatives of equation 2.5-6
indicated by the fundamental equation for G (Alberty, 1983).
The corresponding derivation for ideal solutions is a little simpler. The
chemical potential for the isomer group and for an individual isomer at chemical
equilibrium are given by
Piso = &o + RTlnCBisoI (3.5-7)
where [BiS0] is the concentration of the isomer group. At equilibrium the chemical
potential of isomer i is given by

pi = p: + RTln[Bi] (3.5- 8)
These two equations can be written as

(3.5-9)

(3.5 - 10)

Substituting these expressions in [Bis0] = Z[Bi] yields equation 3.5-6.
In making actual calculations, standard formation properties are used rather
than chemical potentials, and so the standard Gibbs energy of formation of an
isomer group is given by
Af GP
A,G"(iso) = - RTln c exp - (3.5-11)
[z [ RT]]
~
Note that A,G"(iso) is more negative than AfGP of the most stable isomer, as it
must be because the isomer group has a higher mole fraction in the reaction
system at equilibrium than the most stable isomer. The mole fraction yi of the ith
isomer in the isomer group at equilibrium is given by
ri = exp rf G'(is;k- Af Gq
(3.5- 12)

The summation in equation 3.5-11 has the form of a partition function, and the
distribution in equation 3.5-12 has the form of a Boltzmann distribution.
The equation for the standard enthalpy of formation of an isomer group can
be obtained by using the Gibbs-Helmholtz equation 2.5-23 in the form

A,H"(iso) = - T2 (3.5- 13)
dT P
This differentiation yields
N,,,
A,Ho(iso) = 1 ri$HP (3.5- 14)
i=l
Thus the standard enthalpy of formation or an isomer group is the mole fraction
weighted average. Equations 3.5-1 1 to 3.5-14 will be especially useful in the next
chapter.
The equation for the standard entropy of formation of an isomer group can be
obtained by using

aAf G"(iso)
A,S'(iso) = - (3.5- 15)