Geology Reference
InDepth Information
Box 2.3 Tielines and the Lever Rule
In
TX
,
PX
or
PTX
diagrams, tielines link together the
compositions of two different phases that can coexist in
equilibrium under specific conditions. Any composition
lying between the ends of a tieline must therefore repre
sent a physical
mixture
of the two phases. From the posi
tion of that point on the tieline, 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
cd
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
twophase 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 beambalance (Figure 2.3.1b), in which the
weight of a body
C
is inversely proportional to the dis
tance from the fulcrum (
cx
) 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
)
leading to
D
xc
dc
−
−
=
(a)
Melt field
x
c
T
1
d
Solidsolution
(crystal) field
Twophase 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 beambalance.
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