The Phase Rule 


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

ミ・ミセムムひクミスミウ ミセム uCoz