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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