EQUILIBRIUM IN GEOLOGICAL SYSTEMS - Chemical Fundamentals of Geology and Environmental Geoscience - page 20

A variance of zero means that the three-phase equilib-
rium assemblage completely constrains the state of the
system to a particular combination of P and T . Such a
situation is called invariant . There is no latitude (no
degree of freedom) for P or T to change at all, if the
assemblage is to remain stable. Any such variation
would lead to the disappearance of one or more of the
three phases, thus altering the character of the equilib-
rium. So the discovery of this mineral assemblage in a
metamorphic rock (a rare event, since only about a
dozen natural occurrences are known) ties down very
precisely the conditions under which the rock must
have been formed, assuming that:

exact conditions of origin only in conjunction with other
information about P or T .
At point A, where kyanite occurs alone, the variance
is equal to 2:

Point A (1 phase, ky)

ϕ = 1

C = 1 (1 component, Al 2 SiO 5 )

1 + F = 1 + 2

Therefore F = 2 a divariant condition

Within the bounds of the divariant kyanite field, there-
fore, P and T can vary independently (two degrees of
freedom) without upsetting the equilibrium phase
assemblage (just kyanite). The one-phase assemblage
is therefore little help in establishing the precise state
of the system, because it leaves two variables ( P and T )
still to be specified.
The variance cannot be greater than 2 in a one-com-
ponent system like Figure 2.1. In the more complex
systems we shall meet in the following section, the
phases present may consist of different proportions of
several components. A complete definition of the state
of such a system must then include the compositions of
one or more phases, in addition to values of P and T .
Such compositional terms ( X a , X b , etc., representing the
mole fractions of a, b, etc. in a phase) contribute to the
total variance, which can therefore, in multicomponent
systems, adopt values greater than 2.
Variance can be summarized in the following way.
The 'state' of a system - whether we consider a simple
experimental system or a real metamorphic rock in the
making - is defined by the values of certain key inten-
sive variables, including pressure ( P ), temperature ( T )
and, in multi-component systems, the compositions ( X
values) of one or more phases. For a given equilibrium
between specific phases, some of these values are auto-
matically constrained in the phase diagram by the
equilibrium phase assemblage. The variance of this
equilibrium is the number of the variables that remain
free to adopt arbitrary values, which must be deter-
mined by some other means if the state of the system is
to be defined completely.

(a) the kyanite-andalusite-sillimanite assemblage rep-
resents an equilibrium state obtained as the rock
formed, and not simply an uncompleted reaction
from one assemblage to another; and
(b) the P-T coordinates of the invariant point in
Figure 2.1 are accurately known from experimental
studies. (Whether this requirement is satisfied in
the case of the Al 2 SiO 5 polymorphs is arguable -
see Box 2.1 - but in the present discussion we shall
ignore this difficulty.)

The two-phase equilibrium between, say, kyanite
and sillimanite is less informative. The coexistence of
these two minerals indicates that the state of the system
in which they crystallized must lie somewhere on the
kyanite-sillimanite phase boundary, but exactly where
along this line remains uncertain unless we can specify
one of the coordinates of point B. We only need to spec-
ify one coordinate (for example, temperature) because
the other is then fixed by intersection of the specified
coordinate with the phase boundary. According to the
Phase Rule, the variance at B is equal to 1:

Point B (2 phases, ky + sill)

ϕ = 2

C = 1 (1 component, Al 2 SiO 5 )

2 + F = 1 + 2

Therefore F = 1 a univariant equilibrium

The one degree of freedom indicates that the state of the
system is only unconstrained in one direction in
Figure 2.1, along the phase boundary. One additional
piece of information is required (either T or P ) to tie
down the state of the system completely. The coexistence
of kyanite and sillimanite in a rock will pinpoint the

Phase diagrams in P-T space

The need to represent phase equilibrium data in visual
form on a two-dimensional page leads to the use of
various forms of phase diagram, each having its own