Page 87 - Chemical equilibria Volume 4

Molecular Chemical Equilibria 63
3.1.4. Systems composed of a set of equilibria
We now consider a system comprising a set of equilibria. As the set of
balance equations of these reactions constitutes a vector space E (see section
2.2.1), we can deduce that any linear combination of a certain number of
balance equations E 1, E 2, … E m, such that:

E = λ 1 .E ⊕ λ 2 .E ⊕ ...⊕ λ m .E m [3.22]
1
2
r
is also a balance equation for the system.
For the expressions of the different affinities of the corresponding
reactions – e.g. for a reaction ρ in which the algebraic stoichiometric number
of the component A i is ν – we have:
i ρ
∑
A ρ =− νμ i [3.23]
k i ρ

Thus, the affinity of the reaction corresponding to the balance equation E r
will be:

A = ∑ λ A ρ = − ∑∑ λ ν μ i [3.24a]
ρ
ρ
ρ ρ i i ρ
A similar relation can be established for the reference affinity of reaction
E r:

A 0 = ∑ λ A ρ 0 = − ∑∑ λ ν μ i 0 [3.24b]
ρ
ρ
i ρ
ρ ρ i
The equilibrium constants are linked to the reference affinities by
relations of the form:

T
Rln K = A ρ 0 [3.25]
ρ
Relation [3.24] enables us to write the following for the equilibrium
constant of the reaction E r:

Rln K = RT ∑ λ ρ ln K ρ [3.26]
T
r
ρ