Page 66 - Chemical equilibria Volume 4

42 Chemical Equilibria
The number of independent components is the difference between the
number of components in the system and the number of independent
transformations – i.e. the dimension of the vector space of the balance equations.
This number must be decreased by 1 in the case of the presence of ions, due to
the electrical neutrality condition, and decreased by 1 in the case of the species
belonging to a solid phase if the condition of conservation of the
crystallographic sites is imposed.

Remember that the variance is the number of free variables in the system,
among the p external intensive physical variables and the molar fractions.
This number gives us the maximum number of degrees of freedom of the
system; that maximum number of degrees of freedoms may be reduced by
the imposition of specific constraints. Gibbs’ phase rule applies equally to
open systems and closed systems, and pertains solely to the intensive
variables.

2.3.2. Duhem’s phase rule in closed systems

Duhem’s phase rule enables us to define a Duhem variance, which
applies only to closed systems and pertains to the external intensive
variables, the composition variables and the quantities of matter. The goal is
always to determine the number of free variables.

2.3.2.1. Closure conditions
As before, we suppose that the system contains N components in φ phases
and is home to R independent transformations. The variation, over an
arbitrary length of time, of the quantity n of one of the components k in a
k
phase in the system will be given as a function of the reactional extents ξ ρ

of the different transformations by:

k ∑
dn = R ν dξ ρ [2.40]
ρ= 1 k ρ
By integrating between the initial time, when n = n , and ξ = 0, we
0
ρ
k
k
find:
k ∑
n = n + R νξ ρ [2.41]
0
k
ρ= 1 k ρ