12.480 Handout #6: Thermometry-Barometry Using Pyrox­
enes
Reading
Lindsley et al. (1981) Adv. in Physical Geochem., 149-175.
Davidson and Lindsley (1985) Contrib. Min. Pet., 91:383-404.
Supplementary Reading
Lindsley and Andersen (1983) J. Geophys. Res. 88 (1983): A877-A906.
Prewitt, Charles T. (ed.) Pyroxenes (Reviews in Mineralogy, Vol. 7). Washington, D. C.: Miner­
alogical Society of America, 1980. ISBN: 0939950146
This endeavor got its start thanks to an elegantly presented paper by Boyd (1973) GCA37 ­
2533-2546.
Boyd’s suggestion was that two equilibria be used to estimate the T&P of equilibration of garnet
lherzolites.
Boyd proposed that coexisting cpx-opx be used to infer temperature, and that exchange equilibria
involving garnet-opx be used to infer pressure.
We’ve come a long way from Boyd’s initial proposal to the thermometry calculations of today.
Today we’ll look at progress and pitfalls encountered in this endeavor.
opx-cpx — 2 phases with 2 associated G curves. Correct formulation treats each phase as a
distinct solution.
µ oDiopx
o cpx
µ En
G
o cpx
µDi
o opx
µ En
En ss
En
Di ss
En + Di
Di
X
At equilibrium:
opx
rxn
µcpx
En − µEn = ΔGEn = 0
opx
rxn
µcpx
Di − µDi = ΔGDi = 0
1
We could also write an exchange reaction:
Encpx + Diopx → Dicpx + Enopx
ΔGexch = ΔGDi − ΔGEn
opx
opx
cpx
= µcpx
Di + µEn − µDi − µEn
Now we need to develop a realistic and workable solution model. We have at equilibrium:
o,cpx
µo,opx
= RT ln
En − µEn
ΔGoEn = RT ln
cpx
cpx
XEn
γEn
opx + RT ln opx
XEn
γEn
cpx
cpx
XEn
γEn
opx + RT ln opx
XEn
γEn
Now we may plug in what we know and solve for T .
cpx
XEn
opx
XEn
ΔGoEn
ΔGoDi
cpx
γDi
cpx
γEn
← asymmetric
opx
γDi
opx
γEn
← symmetric
This is what Lindsley et al. do, using experimental data in Di − En.
Note — This is the right way of doing things — other ways
Assumption of Wood & Banno (1973) CMP 42 109-124 was that:
activity coefficient ratio = 1
cpx
γM
g2 Si2 O6
opx
γM
g2 Si2 O6
=1
didn’t consider constraints from CaM gSi2 O6 .
Warner & Luth (1973) Am. Min. 58, 998- assume opx-cpx obey single equation of state and used
WGs as fitting parameters.
Binary system - two phases
equilibrium conditions:
cpx
M g2 Si2 O6opx � M g2 Si2 O6
CaM gSi2 O6opx � CaM gSi2 O6cpx
cpx
µopx
En − µEn = 0;
opx
cpx
µDi
− µDi
=0
XDi dµDi + XEn dµEn = 0
Gibbs-Duhem relates Di & En components at equilibrium.
Δµo
cpx
cpx
�
��
�
XEn
γEn
cpx
o,cpx
o,opx
µEn
− µopx
=
µ
−
µ
+RT
ln
+
RT
ln
= 0
opx
opx
En
En
En
XEn
γEn
2
cpx
o,cpx
o,opx
µDi
− µopx
− µDi
+ RT ln
Di = µDi
cpx
cpx
XDi
γDi
opx + RT ln opx
XDi
γDi
Use symmetric solution model for opx and asymmetric for cpx.
o,cpx
µo,opx
= RT ln
En − µEn
cpx
XEn
opx
opx 2
cpx
cpx 2
cpx
cpx 2
opx − WG (XDi ) + 2WG1 XEn (XDi ) + WG2 (1 − 2XEn ) (XDi )
XEn
o,opx
o,cpx
µDi
− µDi
= RT ln
cpx
XDi
opx
opx 2
cpx
cpx 2
cpx
cpx 2
opx − WG (XEn ) + WG1 (1 − 2XDi ) (XEn ) + 2WG2 XDi (XEn )
XDi
Lindsley, Grover, Davidson model:
opx opx
opx opx
GXS, opx = WGopx XEn
XDi = 25XEn
XDi
2
2
cpx
cpx
cpx
cpx
GXS, cpx = WG1 XEn
(XDi
) + WG2 XDi
(XEn
)
↑
↑
(31.216 − .0061P )
(25.484 + .0812P )
Calculation of temperature from opx-cpx pairs (binary system CaM gSi2 O6 − M g2 Si2 O6 ). Note:
these two expressions should give you the same T .
�
cpx
cpx
cpx 2
T ◦ K (M g2 Si2 O6opx � M g2 Si2 O6
) = 3.561 + .0355P + 2WG1 XEn
(XDi
)
2
�
� �
X cpx
/ .0091 − R ln En
opx
XEn
2
cpx
cpx
opx
+WG2 (XDi
) (1 − 2XEn
) − WGopx (XDi
)
�
cpx
cpx 2
cpx
T ◦ K (CaM gSi2 O6cpx � CaM gSi2 O6
) = −21.178 − .0908P + WG1 (XEn
) (1 − 2XDi
)
2
�
� �
X cpx
/ −.00816 − R ln Di
opx
XDi
2
cpx
cpx
opx
+2WG2 XDi
(XEn
) − WGopx (XEn
)
So, we’ve done the pure system Di − En, but the pyroxenes we want to study are almost always
F e − Ca − M g solid solutions (at least!) — also Al, T i, Cr, N a, F e3+ etc.
How do we generalize to these complex phases?
1. We must generalize activity approx. to multi-stite phase where the species can be ordered on
sites.
2. We must generalize such an expression for two component systems to complex systems. This
has been done by Davidson and Lindsley (1985). There are also several graphical/empirical
thermometers for pyx solns. that can be used.
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