Concepts from Chemical Reaction Kinetics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 191

We can define a term called the extent of the reaction , ξ , as defined by

n i + ν i ξ n i = , (5.5)

where n i is the number of moles of i when

ξ =

0 (i.e., the initial condition). Note that

when ξ = 1, all reactants have converted to products. From Equation 5.5 we have

d n i = ν i d ξ . (5.6)

Therefore, the change in free energy for the reaction is given by

i μ i ν i d

d G = ξ . (5.7)

As noted in Chapter 2, the quantity Σ i ν i μ i is called the free energy change of the

reaction , Δ G :

i ν i μ i .

Δ G = (5.8)

Thus, we have the relation

d G
d

Δ G = . (5.9)

ξ

The free energy change of a reaction is the rate of change of Gibbs free energy with
the extent of the reaction. Figure 5.1 shows a typical change in free energy with the
extent of reaction. Note that at equilibrium, d G /d

ξ = 0, that is, Δ G = 0, as required

by the laws of thermodynamics. The entropy production is

=− Δ

G
T d

ξ

d S . (5.10)

0

Since μ i = μ

i +

RT ln a i ,

i ν i μ

RT

0

Δ G =

i +

i ν i ln a i .

(5.11)

This can also be written as

RT ln
i

G 0 (a i ) ν i . Δ G = Δ + (5.12)

For the general reaction (Equation 5.1) we have

a x X a y Y

(a i ) ν i =

a A a B =

K . (5.13)

i