Geology Reference
In-Depth Information
changes usually lead eventually to a situation where,
for every element present, the flux of atoms across the
crystal boundary is the same in both directions, result-
ing in zero net flow, and no change of composition
with time. This is what we mean by
equilibrium
.
The rate at which equilibrium is achieved varies
widely and, as Chapter 3 will show, disequilibrium is
found to be a common condition in geological systems,
particularly at low temperatures.
A
B
6
Kyanite
4
Sillimanite
C
D
2
The Gibbs Phase Rule
Andalusite
A natural question to ask is: how many phases can be
in equilibrium with each other at any one time? In
Figure 1.3a we looked at a simple system in which only
two phases occurred. Most actual rocks, however, are
not so simple. What factors determine the mineralog-
ical complexity of a natural rock? Which aspect of a
chemical equilibrium controls the number of phases
that participate in it?
This question was addressed in the 1870s by the
American engineer J. Willard Gibbs, the pioneer of
modern thermodynamics. The outcome of his work
was a simple but profoundly important formula called
the
Phase Rule
, which expresses the number of phases
that can coexist in mutual equilibrium (
ϕ
) in terms of
the number of components (
C
) in the system and
another property of the equilibrium called the
variance
(
F
). The Phase Rule can be stated symbolically as:
0
200
400
600
800
T
/°C
Figure 2.1
A
P-T
diagram showing phase relations between
the
aluminium silicate
minerals (composition Al
2
SiO
5
). The
pressure axis is graduated in units of 10
8
pascals, equal in
magnitude to the traditional pressure units, kilobars
(1 kbar = 10
3
bars = 10
8
Pa). Kyanite is a triclinic mineral that
usually occurs as pale blue blades in hand specimen.
Sillimanite is orthorhombic and is commonly fibrous or
prismatic in habit. Andalusite is also orthorhombic and is
characteristically pink in hand specimen.
of the equilibrium assemblage. Point A lies within a
field where only one phase, kyanite, is stable. Point B
lies on the phase boundary between two stability fields,
where two minerals, kyanite and sillimanite, are stable
together. Point C, at the
triple point
where the three
stability fields (and the three phase boundaries) meet,
represents the only combination of pressure and tem-
perature in this system at which all three phases can
exist stably together.
It is clear that the three-phase assemblage
(kyanite + sillimanite + andalusite), when it occurs,
indicates very precisely the state of the system (that is,
the values of
P
and
T
) in which it is produced, because
there is only one set of conditions under which this
assemblage will crystallize in equilibrium. Using the
Phase Rule (Equation 2.2), one can calculate that the
variance
F
at point C is zero:
FC
2
(2.2)
φ+=+
The variance is alternatively known as the
number of
degrees of freedom
(hence the symbol
F
used to represent
it). The concept is most easily introduced through an
example. Figure 2.1 illustrates the equilibrium phase
relations between the minerals kyanite, sillimanite and
andalusite. These minerals are all
aluminium silicate
2
polymorphs
of identical composition. A single compo-
nent (Al
2
SiO
5
) is therefore sufficient to cover the com-
positional 'range' of the entire system.
Points A, B and C are three different points in the
'
P-T
space' covered by the diagram; they represent
three classes of equilibrium that can develop in the sys-
tem. The obvious difference between them is the nature
Point C
(3 phases, ky + sill + andal)
ϕ
= 3
C
= 1
(1 component, Al
2
SiO
5
)
3 +
F
= 1 + 2
Therefore
F
= 0
an
invariant
equilibrium
Not to be confused with
aluminosilicate
minerals discussed
in Chapter 9.
2
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