Page 58 - Chemical equilibria Volume 4

34 Chemical Equilibria
2.2.1. Property of the set of balance equations of a system
We consider the set E in the mathematical sense of the term of the R
R
balance equations of the transformations that take place in the system:
E R = { ,E E 2 ,...,E ρ ,...,E R } [2.26]
1

We shall demonstrate a very important property relative to balance
equations for the reactions in a system when those reactions are at
equilibrium.

THEOREM 2.1.– all the balance equations of reactions at equilibrium
occurring in a system constituting a vector space.

For the sake of comfort, we often speak of the vector space of equilibria.

To demonstrate our theorem, we shall show that on the set E R, we can
define an internal composition law of addition and an external multiplication
law on the set of real numbers R.

2.2.1.1. Internal composition law of addition
Consider two balance equations E 1 and E 2 from that set E R. For the
reaction E 1 we have:

0 = ∑ ν 1 k A k [2R.2]
k
and for reaction E 2:

0 = ∑ ν 2 k A k [2R.3]
k

Let us choose the sum of E 1 and E 2, which are components of balance
equation E Σ defined by:

E Σ = E ⊕ E 2 [2.27]
1
Constructed with the sum of the terms of E 1 and E 2 and such that:

k ∑
0 = ∑ ν 1 k A + ν 2 k A k [2R.4]
k k