 | The Phase Rule |
C http://ijolite.geology.uiuc.edu/08SprgClass/geo436/lectures.html
http://ijolite.geology.uiuc.edu/08SprgClass/geo436/436%20lectures/L5-Phase1.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. Free-energy Surface in G-T-P 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 cross-sections: G-T and G-P: 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 P-T section, commonly used for phase diagrams;all phase diagrams are projections from G-T-P 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 P-T 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 1-component 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 (andalusite-kyanite-sillimanite) - common in metamorphic rocks
-C = 1 = Al2SiO5
ボﾟ = 1, 2, or 3 = 3 polymorphs