Geology Reference
In-Depth Information
How will the equilibrium respond if we attempt to
increase the applied pressure (without changing the
temperature)? Le Chatelier's principle suggests that
the assemblage will adjust into a more compact form,
because by taking up less space it can relieve the addi-
tional pressure applied. The system can accomplish
this by recrystallizing andalusite (density 3.2 kg dm
−
3
)
into the denser polymorph kyanite (density 3.6 kg dm
−
3
).
By allowing the proportion of kyanite to increase at the
expense of andalusite, the system can for the time
being prevent any increase in pressure, hence - in the
words of Le Chatelier's Principle - 'nullifying the
change'. Eventually, however, the andalusite becomes
exhausted and the pressure, no longer constrained by
univariant equilibrium, is able to rise into the kyanite
field. From Le Chatelier's principle one can therefore
show that in any
P-T
phase diagram the higher-den-
sity (lower molar volume) phase assemblage will be
found on the higher-pressure side of a reaction bound-
ary. Diamond and liquid water are other examples
(Box 2.2).
The free-energy change of this reaction is:
∆
GG
=
−
G
products
reactants
(
)
(2.7)
=
GGG
−
+
albite
jadeite
quartz
The molar free energy of each phase (which could be
calculated from published tables of molar enthalpy and
entropy) varies with pressure and temperature. Δ
G
therefore varies systematically across a
P-T
diagram.
The equilibrium boundary in Figure 2.2 marks the locus
of
P-T
coordinates for which Δ
G
= 0. It can be shown
(using quite simple calculus) that the condition for
remaining on the univariant equilibrium boundary as
P
and
T
are varied by small amounts d
P
and d
T
is that:
d
d
P
T
=
∆
∆
S
V
(2.8)
where Δ
S
and Δ
V
are the entropy and volume changes
occurring during the reaction in Equation 2.3:
(
)
∆
SS
=
−
S
+
S
albite
jadeite
quartz
(
)
∆
VV
=
−
VV
+
albite
jadeite
quartz
δ
P
S
and
V
represent the molar entropy and molar vol-
ume of each phase, which can be looked up in tables of
thermodynamic data for minerals (such as Holland
and Powell, 1998).
Equation 2.8 is called the
Clapeyron equation
, after its
originator, an eminent 19th-century French railway
engineer. It provides a means of estimating the gradi-
ent of a reaction boundary in a
P-T
diagram from eas-
ily obtainable thermodynamic data. It is also very helpful
in interpreting many features of phase-equilibrium
diagrams.
To predict the slope of the phase boundary in
Figure 2.2, one proceeds as follows. The relevant molar
entropy and molar volume data are:
-
δ
V
Le Chatelier's principle
A second consequence of Le Chatelier's rule is that
the phase assemblage on the high-temperature side of
an equilibrium boundary is invariably the one having
the higher enthalpy (Box 4.2).
The Clapeyron equation
A second useful application of thermodynamic data to
phase diagrams is to estimate the gradient (slope) of
an equilibrium boundary in
P-T
space. In the symbol-
ism of calculus (see 'Differentiation' in Appendix A),
this is written
d
d
P
T
, meaning the rate at which
P
S
J K
−
1
mol
−
1
V
10
−
6
m
3
mol
−
1
increases for a given increase of
T
as one follows the
univariant boundary. The gradient has a sign (positive
or negative - see Figure A1(b) in Appendix A) and a
numerical magnitude (indicating whether it is gentle
or steep).
Consider the reaction of Equation 2.3 and Figure 2.2:
jadeite (NaAlSi
2
O
6
)
133.5
60.4
quartz (SiO
2
)
41.5
22.7
albite (NaAlSi
3
O
8
)
207.4
100.1
Adding the reactants together:
−
1
−
1
−
63 1
−
jadeitequartz lbite
+
jadeitequartz
+
S
=
175 0
.
J Kmol
V
=
83110
.
mmol
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