Page 71 - Chemical equilibria Volume 4

Properties of States of Physico-Chemical Equilibrium 47
2.4.3. Set of indifferent points of equilibrium
The indifference conditions of a system are unrelated to the states of
equilibrium of that system. This means that certain indifferent states could
possibly be equilibrium states, but there are also indifferent states which are
not equilibrium states.

We have seen that in intensive variables, the equilibrium states constitute
a set of dimension v (Gibbs variance); the elements in that set also belong
G
to the set of indifferent states, and therefore, satisfying v + p − 1 additional
G
−
conditions will constitute a set of dimension: v + p − 1 v = p − 1 .
G
G
Therefore, we can state Saurel’s theorem:
THEOREM 2.3.– The subset of the indifferent equilibrium states of a system
is represented by a set of dimension p – 1, independent of the variance of the
system.

For example, by bringing together the conditions of equilibrium and
those of indifference, we can show that in a monovariant system with two
external intensive variables (P and T), all the equilibrium states are
indifferent states.

2.4.4. Gibbs–Konovalov theorem

We shall now look at a theorem developed by Gibbs and Konovalov. For
the moment, we shall simply accept this theorem as true. We shall
1
demonstrate it later when we examine equilibria between phases.
THEOREM 2.4.– In a system with two external intensive variables – pressure
and temperature – at equilibrium, for any shift at constant temperature (or
respectively at constant pressure), the pressure (or respectively the
temperature) reaches an extremum for an indifferent state, and vice versa. If
amongst the values of the pressure (or respectively the temperature) which
keep the system at equilibrium at constant temperature (or respectively
constant pressure), there is an extremum value, then the state corresponding
to that value is indifferent.

1 See Volume 5 of this Set of books: Phase Transformations.