An Overview of Chemical Reaction Analysis - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 105

It can be shown that the free ene rg y change in a system of chemically interacting species is
related to the chemical potentials G j of each species through the relationship

X

N S

X

N S

D G ¼ n j G j ¼ n j G j (3.71)

j ¼1 j ¼1

where G j is the Gibbs free energy per mole of species j and D G is the Gibbs free energy change
per mole in the reaction, as illustrated in Fig. 3.2 . We call G j ¼ v G/ v n j the chemical potential of
species j which is actually the partial molar free energy. The chemical potential of species j is
related to its chemical potential in the standard state (the state in which the activity a j of
species j is unity) by the relation

G j þ

G j ¼ RT ln a j (3.72)

At chemical equilibrium, at constant temperature and pressure, the Gibbs free energy of the
system is a minimum and D G ¼ 0. Therefore, we have

X

N S

X

N S

X

N S

n j G j þ

0 ¼ DG ¼ n j G j ¼ n j RT ln a j (3.73)

j ¼1 j ¼1 j ¼1

at chemical equilibrium. In this expression a j is the activity of species j, which is a m eas ur e
of the amount of a species defined such that a j ¼

0
j

1 in the standard state where G j ¼G .

For gases, the standard state is usually defined as the ideal gas state at 1 bar (1 bar ¼

1.023 atm), while for liquids, it may be either the pure material or the material in a solution
at a concentration of 1 mol/L. The definitions of standard state and activity are somewhat
arbitrary, but they are uniquely related by the definition of unit activity in the
standard state. Once the standard state is defined, the situation is well defined. Typical
standard states are listed in Tabl e 3 . 2 . In all, one can observe that the activity is
dimensionless.
Next, dividing the preceding equation by RT and taking exponentials on both sides,
we obtain

0 1

RT X

N S

Y

N S

n j G j a n j

j

@ A ¼

exp (3.74)

j ¼1 j ¼1

Since we define P N S
j ¼

0
j ¼ DG

0

1 n j G

R , the Gibbs free energy change of the reaction in the stan-

dard state, we obtain

!

Y

N S

D G R
RT

a n j j ¼ exp ¼ K eq (3.75)

j ¼1

where K eq is the equilibrium constant as defined by this equation. [We note in passing that
this notation is misleading in that the “equilibrium constant” is constant only for fixed
temperature, and it usually varies strongly with temperature. To be consistent with our defi-
nition of the “rate coefficient”, we should use “equilibrium coefficient” for the equilibrium
constant, but the former designation has become the accepted one.]