Page 70 - Chemical equilibria Volume 4

46 Chemical Equilibria
(β)
(φ)
(α)
,
The unknowns in this system of equations are Δ n , Δ n , …, Δ n
r
r
r
ξ , ξ , …, ξ , which is to say that there are φ + R unknowns. The original
1 2 R
state will therefore be indifferent if the system of equations can have a
(α)
solution such that the terms Δ n are not all null. However, the Gibbs
r
variance of that system can be written as:
v = N − (ϕ + ) R + p [2.50]
G
The indifference conditions [2.46] will therefore depend on the difference
between the number of equations N and the number of unknowns φ + R; and
depending on the values of the variance, two cases need to be considered:
1) If v < p , then N < (ϕ + ) R and the system of equations [2.49] will
G
always have fewer equations than unknowns. Thus it will have solutions
different to zero and we can arbitrarily choose a certain number of nonzero
(α)
values Δ n – that number is ( pv− G ). Hence, systems whose Gibbs
r
variance is less than the number of external intensive parameters p are
always indifferent.
2) If v ≥ p , then N ≥ (ϕ + ) R . In order for the system of
G
equations [2.49] to have an acceptable solution (at least one nonzero value of
Δ n ) it is necessary and sufficient for the determinants of order ϕ + R ,
(α)
r
formed on the basis of the unknowns, to have the value of 0. The
corresponding matrix is written as:
x 1,1 x 1,2 ... x 1,ϕ ν 1,1 ν 1,2 ... ν 1,R
x 2,1 x 2,2 ... x 2,ϕ ν 2,1 ν 2,21 ... ν 2,R [2.51]
... ... ... ... ... ... ... ...
x N ,1 x N ,1 ... x N ,ϕ ν N ,1 ν N ,2 ... ν N ,R

As there are several determinants which must be null, the number of
indifference conditions will depend on the dimensions of matrix [2.51].
If N = (ϕ + ) R , there will be only one determinant, and therefore only

one indifference condition.