Phase diagram
• Need to represent how mineral reactions
at equilibrium vary with P and T
 S R   VR 
  
 


V

T
P
 R 
S R
 P 

 
 T  G 0 VR
CP
 S R 

 
 T  P T
P-X stability and mixing
Gibbs Phase Rule
• The number of variables which are
required to describe the state of a system:
• p+f=c+2
f=c-p+2
– Where p=# of phases, c= # of components,
f= degrees of freedom
– The degrees of freedom correspond to the
number of intensive variables that can be
changed without changing the number of
phases in the system
Variance and f
• f=c-p+2
• Consider a one
component (unary)
diagram
• If considering
presence of 1 phase
(the liquid, solid,
OR gas) it is
divariant
• 2 phases =
univariant
• 3 phases = invariant
Melts
• Liquid composed of predominantly silica and
oxygen. Like water, other ions impart greater
conductivity to the solution
• Si and O is polymerized in the liquid to differing
degrees – how ‘rigid’ this network may be is
uncertain…
• Viscosity of the liquid  increases with increased
silica content, i.e. it has less resistance to flow with
more SiO2… related to polymerization??
• There is H2O is magma  2-6% typically – H2O
decreases the overall melting T of a magma, what
does that mean for mineral crystallization?
Thermodynamic definitions
• Gi(solid) = Gi(melt)
• Ultimately the relationships between these is related to the
entropy of fusion (S0fus), which is the entropy change
associated with the change in state from liquid to crystal
 dT  RT fus

 
0
dX

S
fus
 i
• These entropies are the basis for the order associated with
Bowen’s reaction series  greater bonding changes in
networks, greater entropy change  lower T equilibrium
Melt-crystal equilibrium
• Precipitated crystals
react with cooling
liquid, eventually will
re-equilibrate back,
totally cooled magma
xstals show same
composition
• UNLESS it cools so
quickly the xstal
becomes zoned or the
early precipitates are
segregated and
removed from contact
with the bulk of the
melt
Why aren’t all feldspars
zoned?
• Kinetics, segregation
• IF there is sufficient time, the crystals will
re-equilibrate with the magma they are in
– and reflect the total Na-Ca content of
the magma
• IF not, then different minerals of different
composition will be present in zoned
plagioclase or segregated from each other
physically
Exsolution
P-X stability and mixing
• More than 1 crystal can precipitate from a melt –
different crystals, different stabilities…
– 2+ minerals that do not share equilibrium in a melt are
immiscible (opposite of a solid solution)
– Liquidus  Line describing equilibrium between melt and
one mineral at equilibrium
– Solidus  Line describing equilibrium with melt and solid
– Eutectic  point of composition where melt and solid can
coexist at equilibrium
Diopside is a pyroxene
Anorthite is a feldspar
Eutectic
Solidus
Liquidus
• Melt at composition X cools to point Y where anorthite
(NOT diopside at all) crystallizes, the melt becomes more
diopside rich to point C, precipitating more anorthite with
the melt becoming more diopside-rich
• This continues and the melt continues to cool and shift
composition until it reaches the eutectic when diopside
can start forming
At eutectic, diopside
AND anorhtite crystals
precipitate
Lever Rule 
diopside/anorthite
(42%/58%) crystallize
until last of melt
precipitates and the rock
composition is Z
A
B
C
S1
S2
Z
• Melting  when heated to eutectic, the
rock would melt such that all the heat goes
towards heat of fusion of diopside and
anorthite, melts so that 42% diopside /
58% anorthite…
• When diopside gone, temperature can
increase and rest of anorthite can melt
(along liquidus)
• How does free energy change with T and P?
• From G=H-TS:
T2
T2
T1
T1
GP2 ,T2  GP1 ,T1  S P1,T 1 (T2  T1 )   CP( P1) dT  T 2 
CP( P1)
T
P2
dT   VT2 dP
P1
• T and P changes affect free energy and can drive
reactions!!
Volume Changes (Equation of State)
For Minerals:
Volume is related to energy changes:
 dG 

 V
 dP T
Mineral volume changes as a function of T: , coefficient of thermal expansion
1  V 
  
V  T  P
Mineral volume changes as a function of P: , coefficient of isothermal expansion
1  V 
   
V  P T
Volume Changes (Equation of
State)
• Gases and liquids undergo significant volume
changes with T and P changes
• Number of empirically based EOS solns..
• For metamorphic environments:
– Redlich and Kwong equation:
aRw
RT
P
 1/ 2
V  bRK T V (V  bRK )
• V-bar denotes a molar quatity, aRw and bRK are
constants
Phase Relations
• Rule: At equilibrium, reactants and products have
the same Gibbs Energy
– For 2+ things at equilibrium, can investigate the P-T
relationships  different minerals change with T-P
differently…
• For GR = SRdT + VRdP, at equilibrium,
G0, rearranging:
S R
 P 

 
 T  G 0 VR
Clausius-Clapeyron equation
Remember that a line on a phase diagram describes equilibrium, GR=0!!
S R
 P 

 
 T  G 0 VR
SR change with T or P?
CP
 S R 




T

 P T
 S   V 
  R    R 
 VR T  T  P
V = Vº(1-P)
 S 
S P  S 0     dP  S 0    VdP
P T
P2 
P2
P1
P1



 S 0  V 0 P  ( P22  P12 
2


V for solids stays nearly constant as P, T change,
V for liquids and gases DOES NOT
• Solid-solid reactions linear  S and V nearly
constant, S/V constant  + slope in diagram
• For metamorphic reactions involving liquids or
gases, volume changes are significant, V terms
large and a function of T and P (and often
complex functions) – slope is not linear and can
change sign (change slope + to –)
S R
 P 

 
 T  G 0 VR
Example – Diamond-graphite
• To get C from
graphite to
diamond at 25ºC
requires 1600 MPa
of pressure, let’s
calculate what P it
requires at 1000ºC:
graphite
diamond

(K-1)
1.05E-05
7.50E-06

(MPa-1)
3.08E-05
2.27E-06
Sº
(J/mol K)
5.74
2.38
Vº
(cm3/mol)
5.2982
3.417
Clausius-Clapyron Example