Lecture 7 (9/27/2006)
Crystal Chemistry
Part 6:
Phase Diagrams
Gibbs Free Energy

G – the energy of a system in excess of its internal
energy. (This is the energy necessary for a reaction to
proceed)
G = E + PV - TS
dG = VdP – SdT
at constant T (δG/δP)T = V
at constant P (δG/δT)P = -S
Stable phases strive to have the lowest G
Therefore, the phase with the highest density at a given
pressure and the highest entropy at a given temperature
will be preferred
Relationship of Gibbs Free Energy to Phase
Equilibrium
Clapeyron Equation

Defines the state of equilibrium between
reactants and product in terms of S and V
dGr = VrdP – SrdT
dGp = VpdP – SpdT
at equilibrium: VrdP – SrdT = VpdP – SpdT
or: (Vp –Vr) dP = (Sp –Sr) dT
or: dP/dT = ΔS / ΔV
The slope of the equilibrium curve will be positive
if S and V both decrease or increase with
increased T and P
Reactants -Products
Vice > Vlw -ΔV
Sice < Slw +ΔS
Slope of Phase
Reaction Boundaries
dP/dT = ΔS / ΔV
Reactants -Products
Vlw < Vwv +ΔV
Slw < Swv +ΔS
Variables

Extensive Variables – dependent on the amount
of material present




mass
volume
moles of atoms
Intensive Variables – independent on the
amount of material present




pressure
temperature
density
compositional proportions
Gibbs Phase Rule
F=C–Φ+2
F – number of degrees of freedom
of intensive variables (p, t, x) that
will still preserve chemical
equilibrium
C – number of components
Φ – number of phases
One Component Phase Diagrams
C
Al2SiO5
Illustrate
Polymorphism
Isochemical
P & T are intensive
variables
SiO2
CaCO3
Phase Rules:
divariant fields F=2
univariant lines F=1
invariant points F=0
Two Component Phase Diagrams
Solid Solution
Crystallization
Eutectic Crystallization
• Usually portrayed as isobaric T-X diagrams
• For igneous systems, magma/melt is a phase of a simplified
composition defined by the mineral phases of interest
•Liquidus – denotes the temperature at which the liquid of a particular
compositions will begin to crystallize
•Solidus denotes the temperature at which the liquid of a particular
composition will be completely crystallized
Eutectic Crystallization of
Anorthite (plagioclase) and Diopside (pyroxene)
Eutectic
Point
Lever Rule
Proportions
Anorthite
Diopside
Solid Solution Crystallization
Limited Solid Solution
and Subsolidus
Exsolution:
e.g. Alkali Feldspar
Increasing Pressure
Exsolution Textures
Subsolidus Unmixing
Pyroxene
Hypersthene (Opx) exsolution
lamellae
in Augite (Cpx) host
Alkali Feldspar
Albite exsolution
(perthite)
in Microcline host
Multi-component Phase Diagrams
Igneous Systems – Liquidus Diagrams
Liquidus
Surface
Eutectic
Point
Cotectic
Lines
Multi-component
Phase Diagrams
Metamorphic Systems
Chemographic Diagrams
e.g. ACF
Equilibrium assemblages in metabasalts
A = Al2O3 +Fe2O3-Na2O-K2O
C = CaO – 3.3P2O5
F = FeO + MgO + MnO
Shows equilibrium assemblages
at specified P & T
Next Lecture
50-minute Test on Crystal Chemistry
Lectures 1-7 (see Powerpoints on Website)
Klein Chapters 1 (p. 1-16), 3 (p. 38-103) and 4
(p. 107-131)
See CD module 1 for help with ionic
coordination
Q & A in Lab on tomorrow (Tuesday)