Geology Reference
In-Depth Information
Box 2.3 Tie-lines and the Lever Rule
In T-X , P-X or P-T-X diagrams, tie-lines link together the
compositions of two different phases that can coexist in
equilibrium under specific conditions. Any composition
lying between the ends of a tie-line must therefore repre-
sent a physical mixture of the two phases. From the posi-
tion of that point on the tie-line, one can work out the
relative proportions of the two phases in the mixture.
Figure 2.3.1a shows part of a phase diagram in which
complete solid solution exists between two compounds, A
and B (cf. Figure 2.5). The tie line c-d depicts equilibrium
at temperature T 1 between a melt of composition c on the
liquidus, and a solid solution of composition d on the soli-
dus; both c and d are expressed in mass % B. Composition
x lies in the two-phase field between c and d , and must
signify a physical mixture of these two distinct phases. Let
C and D represent the mass fractions ( i.e. C + D = 1.00) in
which c and d are mixed to form x . We can then express
the composition of x as a weighted average of c and d :
Substituting D = 1 - C into Equation 2.3.1, we can show in
a similar fashion that:
C xd
cd
dx
dc
(
)
=
=
thechangeinsigncancelsout
The mass ratio in which c and d are present in x is there-
fore given by:
C
D
dx
dc
xc
dc
dx
xc
(
)
=
=
aftercancelling denominators
(
) = (
)
Inother words: CxcDdx
(2.3.2)
This useful equation is known as the Lever Rule, since
it can also be applied to the 'lever effect' of the old-
fashioned beam-balance (Figure  2.3.1b), in which the
weight of a body C is inversely proportional to the dis-
tance from the fulcrum ( c-x ) at which it balances an
opposing weight D :
weight of C
weight ofD =
xd
cx
(2.3.1)
xCcDd
=+
Since C = 1 - D , this can be rewritten:
Qualitatively, the closer the composition of a mixture
plots (in composition space) to one of its constituents,
the greater the percentage of that constituent in the
mixture.
= (
) += +
x
1-
D cDdcDc Dd
-
Therefore x - c = D ( d - c )
dc
=
(a)
Melt field
x
c
T 1
d
Solid-solution
(crystal) field
Two-phase field
Composition
A
B
(b)
C
D
c
d
x
Figure 2.3.1 (a) Part of a phase diagram similar to Figure 2.5 to illustrate the Lever Rule. (b) The analogous geometry
of the beam-balance.                           Search WWH ::

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