Page 52 - Thermodynamics of Biochemical Reactions

46 Chapter 3 Chemical Equilibrium in Aqueous Solutions

This differentiation yields
N,SO N,,"
A,S"(iso) = riA,SP - R rilnri (3.5- 16)
i= 1 i= 1
The same form of equation can be used to calculate the standard molar entropy
S:(iso) of the isomer group. The entropy of formation of the isomer group is equal
to the mole-fraction-weighted entropy of formation plus the entropy of mixing the
isomers.
The equation for the standard molar heat capacity of formation of an isomer
group can be obtained by using

i3Af H"(iso)
A,C;(iso) = - (3.5- 17)

This differentiation yields

N,\O
C;,(iso) = 1 riC;,(i) + RT~ (T rL(AfIQ2 - (AfHo(iso))') (3.5-18)
i=l
~
i= 1
Equation (3.5-18) has been written in terms of molar heat capacities C&i), rather
than heat capacities of formation, because the heat capacities of the elements are
on both sides and cancel. The second term of this equation is always posi-
tive because the weighted average of the squares is always greater than the
square of the average. Equation 3.5-18 is in accord with LeChatelier's principle:
As the temperature is raised, the equilibrium shifts in the direction that
causes the absorption of heat. Equation 3.5-18 can also be derived using
C, = - T(d2G/dT2), (equation 2.5-25).
Equations 3.5-14 and 3.5-16 can be substituted in A,G"(iso) =
A,H"(iso) - TA,S"(iso) to obtain another form for the standard Gibbs energy of
formation of an isomer group.

NiSO N,,,
A,G"(iso) = c riAfGP + RT rilnri (3.5-1 9)
i= 1 i= 1
In other words, the standard Gibbs energy of formation of an isomer group at
equilibrium is equal to the mole fraction-weighted average of standard Gibbs
energies of formation of the isomers plus the Gibbs energy of mixing.
The fundamental equation for G of a system made up of isomer groups is
Niso
dG = -SdT + VdP + c pi(iso)dni(iso) (3.5-20)
i= 1
where Niso is the number of isomer groups. In this equation an isomer group may
consist of a single species. This equation can be used to derive the equilibrium
expressions for reactions written in terms of isomer groups. Since isomer groups
can be treated like species in chemical thermodynamics, they can be referred to a
pseudospecies. Equation 3.5-20 is based on the assumption that the species in an
isomer group are in equilibrium with each other. The number of natural variables
for a one-phase system consisting of Niso isomer groups is D = Niso + 2 prior to
application of the constraints due to reactions between the isomer groups. If the
reactions between the isomer groups are at equilibrium, the number of compo-
nents replaces the number of isomer groups and D = C + 2.

3.6 EFFECT OF IONIC STRENGTH ON EQUILIBRIA
IN SOLUTION REACTIONS

The activity coefficient yi of an ion depends on the ionic strength (I = ($z:ci,
where zi is the charge number) according to the Debye-Huckel theory in the limit
of low ionic strengths. As discussed in Section 1.2, this equation can be extended