
The Phase Rule  
C http://ijolite.geology.uiuc.edu/08SprgClass/geo436/lectures.html
http://ijolite.geology.uiuc.edu/08SprgClass/geo436/436%20lectures/L5Phase1.html
A. Observations
B. General
Cooling melts crystallize over a range of T and P.
Several minerals crystallize; number increases as T decreases.
Minerals crystallize in a recognizable sequence.
Some minerals change composition during crystallization.
Melt composition also changes with cooling.
Specific minerals and sequence depend on magma T and composition.
Pressure affects melting T; may affect what minerals form.
Type and pressure of volatiles (H2O, CO2) also affect Tmelt and particular minerals.
II. Simple systems
A. Definitions
Recall system = part of reality
Open system exchanges heat and matter with surroundings
Closed system heat only
Isolated system  neither
Phase = physically separate part of system (solid vs liquid, or albite vs anorthite)
Component = chemical description of ingredients making up the phase(s)  defined as minimum # needed to completely describe phases
III. Freeenergy Surface in GTP Space
A. Consider a closed system with one phase a
G = f(T, P) and dG = S dT + V dP
Assuming S and V are constants, integrate from 298 K and 0.1 MPa to T and P:
G at T and P = G at 298 and 0.1  S ( T  298) + V ( P  0.1)
This is the equation of a planar surface in 3D: G, T, P
In reality, S and V = f (T, P) so slope would vary and surface would be curved.
B. Add a phase b = polymorph of a
For b. S and V are different, so surfaces are not parallel.
Surfaces intersect along a line where G for a = G for b
For the reaction aハΠハ bハホハdG = 0, i.e., equilibrium
All points along the line G of a = G of bハ must satisfy both planar equations above:3 variables (G, T, P);2 equations => one variable can be chosen freely, then others are fixed; this line is called univariant.
Consider crosssections: GT and GP: 1 variable fixed, surfaces project as lines, and line of intersection projects as a point. The point represents the only place in these diagrams where a and b coexist: invariant ; the most useful projection is PT section, commonly used for phase diagrams;all phase diagrams are projections from GTP space.
Equilibrium line separates a field from b field. In each, T and P may be changed independently: divariant
C. Clapeyron Equation
Recall dP / dT = dS / dV
For most melting reactions, dS and dV > 0 => positive slope for univariant lines in PT diagrams
Ice > water is an exception: dV < 0 => negative slope for melting line
III. The Phase Rule
A. More definitions regarding the state of a system (i.e., complete set of properties)
Extensive variable = one that depends on the size of system: # moles, volume, mass; dividing one extensive variable by another > property that doesn''t depend on extent of system.
Intensive variable = one that is intrinsic to system: density, molar volume, T, P
Variance = degrees of freedom = # of intensive variables that must be specified before all other intensive are determined
B. Variance = f = C + 2  oハwhere C = # components, 2 = T and P, oハ= # phases
Adding a component => must specify one more IV to constrain system
Adding a phase => must specify one less IV to constrain system
Phase Rule applies only to systems in equilibrium.
C. Application of phase rule to 1component system: H2O
ボﾟ = 1, 2, or 3: critical point: above 374 oC and 21.8 MPa, no distinction between liquid and vapor => material is called a supercritical fluid; there is steep negative slope of ice = water univariant line: result of dV in Clapeyron equation for melting of ice; there is other forms of ice at higher pressure  these have + slopes => expand on melting; there loss of liquid field between ice and vapor at low P => sublimation
D. Al2SiO5 system (andalusitekyanitesillimanite)  common in metamorphic rocks
C = 1 = Al2SiO5
ボﾟ = 1, 2, or 3 = 3 polymorphs
