This document last updated on 18-Mar-2010
EENS 212
Petrology
Prof. Stephen A. Nelson
Tulane University
Thermodynamics and Metamorphism
Equilibrium and Thermodynamics
Although the stability relationships between various phases can be worked out using the
experimental method, thermodynamics gives us a qualitative means of calculating the stabilities
of various compounds or combinations of compounds (mineral assemblages).
We here give an
introductory lesson in thermodynamics to help us better understand the relationships depicted
on phase diagrams.
The
First Law of Thermodynamics
states that "the internal energy, E, of an isolated system is
constant". In a closed system, there cannot be a loss or gain of mass, but there can be a change
in energy, dE.
This change in energy will be the difference between the heat, Q, gained or lost,
and the work , W done by the system.
So,
dE = dQ - dW
(1)
Work, W, is defined as force x distance.
Since Pressure, P, is defined as Force/surface area,
Force = P x surface area, and thus W = P x surface area x distance = P x V, where V is volume.
If the work is done at constant pressure, then W = PdV.
Substitution of this relationship into
(1) yields:
dE = dQ - PdV
(2)
This is a restatement of the first law of thermodynamics.
The
Second Law of Thermodynamics
states that the change in heat energy of the system is
related to the amount of disorder in the system.
Entropy is a measure of disorder, and so at
constant Temperature and Pressure:
dQ = TdS
Thus, substituting into (2) we get:
dE = TdS - PdV
(3)
The
Gibbs Free Energy, G
, is defined as the energy in excess of the internal energy as follows:
G = E + PV - TS
(4)
Differentiating this we get:
dG = dE +VdP + PdV - TdS - SdT
Thermodynamics and Metamorphism
3/18/2010
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*Sign up*Substituting (3) into this equation then gives:
dG = TdS - PdV + VdP + PdV - SdT - TdS
or
dG = VdP - SdT
(5)
For a system
in equilibrium at constant P and T
,
dG = 0.
If we differentiate equation (5) with respect to P at constant T, the result is:
(6)
and if we differentiate equation (5) with respect to T at constant P we get:
(7)
Equation (6) tells us that phases with small volume are favored at higher pressure, and equation
(7) tells us that phases with high entropy (high disorder) are favored at higher temperature.
Equation (5) tells us that the Gibbs Free Energy is a function of P and T.
We can see this with
reference to the diagram below, which shows diagrammatically how G, T, and P are related in a
system that contains two possible phases, A and B.
In the diagram, phase A has a steeply sloping free energy
surface.
Phase B has a more gently sloping surface.
Where the two surfaces intersect, Phase A is in
equilibrium with phase B, and G
A
= G
B
.
Next we look at 3 cross-sections through this figure.

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