Page 67 - Chemical equilibria Volume 4

Properties of States of Physico-Chemical Equilibrium 43
The fact that the system is closed yields N relations of the form [2.40] –
one for each component. This set of relations constitutes the closure
conditions for the system.
(α)
These equations may take a different form if n represents the total
(α)
quantity of matter in a phase α, and x the molar fraction of the component
k
k in that phase. We would then have:
n = x n (α) [2.42]
(α)
k
k
and relation [2.40], which is representative of the closure conditions,
becomes:

k ∑
(α)
0
xn (α) = n + R νξ ρ [2.43]
k
k ρ
ρ= 1
One of the two relations [2.40] or [2.42] can be used to impose closure of
a system.
2.3.2.2. Duhem’s theorem

Thus, the state of a closed system will be fully determined if we know:
– the physico-chemical state of the phases by their p external intensive
values and their compositions, giving us p + N variables;
– the total quantities of matter in each phase, which include φ variables.

Between all these variables, there are a certain number of relations which
are:
– the φ relations which express the fact that, in each phase, the sum of the
molar fractions is equal to 1;
– the conditions of equilibrium between the R independent
transformations;
– the N closure relations of the form of equation [2.40]. Beware, though:
if we know the initial quantities of matter, these last relations give us the R
variables of the reactional extents of the transformations at equilibrium,
which need to be added to values remaining to be determined.