phase rule from rutstein

8/3/2019 Phase Rule From Rutstein

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And now, THERMODYNAMICS!And now, THERMODYNAMICS!



Thermodynamics need not be so hard if you think of it asThermodynamics need not be so hard if you think of it as

heat and chemical ³flow´ between ³phases´.heat and chemical ³flow´ between ³phases´.



Derivation of Phase RuleDerivation of Phase Rule

Let¶s do a ³bookLet¶s do a ³book--keeping´ exercisekeeping´ exercise

and evaluate the number of and evaluate the number of

minerals (phases) that can cominerals (phases) that can co--existexist

in a chemical system under certainin a chemical system under certain

P,T conditions.P,T conditions.

(Derivation adapted from Prince, 1967,(Derivation adapted from Prince, 1967,

Alloy Phase Equilibria)Alloy Phase Equilibria)



F F = C= C -- P + 2P + 2F = # degrees of freedom, or..F = # degrees of freedom, or..

The number of The number of intensiveintensive

parameters that must beparameters that must be

specified in order to completelyspecified in order to completely

determine the systemdetermine the system

What does this MEAN?What does this MEAN?

The Gibbs Phase RuleThe Gibbs Phase Rule



F=

F= CC -- PP + 2+ 2

P = # of phasesP = # of phases

phases arephases are mechanicallymechanically

separableseparable constituentsconstituents

The Phase RuleThe Phase Rule--P & CP & C

C =C = minimum #minimum # of componentsof components

(the # of chemical constituents(the # of chemical constituentsthat must be specified in order tothat must be specified in order to

define all phases)define all phases)



The Phase RuleThe Phase Rule--´2´´2´

F = CF = C -- P +P + 22

2 = the number of 2 = the number of intensiveintensive

parametersparameters

Usually = 2 for Usually = 2 for TemperatureTemperature andand

PressurePressure and this is especiallyand this is especially

useful for geologistsuseful for geologists



Derivation of Phase RuleDerivation of Phase Rulea balancing of a balancing of

FIXED PARAMETERSFIXED PARAMETERSandand

SYSTEM VARIABLESSYSTEM VARIABLES

??which means???????which means?????



HOW MANY VARIABLES AREHOW MANY VARIABLES ARE

THERE IN A CHEMICALTHERE IN A CHEMICALSYSTEM?SYSTEM?

Simplistically, ³3´,Simplistically, ³3´,

Pressure, Temperature, Composition,Pressure, Temperature, Composition,

BUT, for more than one phase, what isBUT, for more than one phase, what isthe TOTAL number of variables?the TOTAL number of variables?



AssignAssign

CC components betweencomponents between PP phasesphases

For each Phase, composition is definedFor each Phase, composition is defined

byby (C(C--1)1) concentration terms.concentration terms.

For ALL Phases in the system,For ALL Phases in the system, P(CP(C--1)1) ==

the number of concentration terms.the number of concentration terms.

Can also vary Pressure & Temperature,Can also vary Pressure & Temperature,

or P + T, which =or P + T, which = 22 more variables.more variables.



Therefore,Therefore,

the TOTAL NUMBER OF the TOTAL NUMBER OF

VARIABLES =VARIABLES =

P(CP(C--1) +21) +2



NOW, LET¶S EXAMINENOW, LET¶S EXAMINE

HOW MANY FACTORSHOW MANY FACTORS

EXIST THAT FULLYEXIST THAT FULLY

DESCRIBE THEDESCRIBE THE

SYSTEM and ARESYSTEM and ARE

³FIXED´ BY THE SET³FIXED´ BY THE SETFACTORS.FACTORS.



Since the system is inSince the system is in

equilibrium,equilibrium, BY DEFINITIONBY DEFINITION, we, we

have already implicitly definedhave already implicitly defined

some of the variables.some of the variables.

µµ = chemical potential or = chemical potential or

chemical flux or energychemical flux or energybetween two minerals.between two minerals.



So, if system is ³in equilibrium´,So, if system is ³in equilibrium´,

and if there isand if there is NO NET CHANGENO NET CHANGE ininthe net ³amounts´ of chemicalsthe net ³amounts´ of chemicalsmoving between phases that are inmoving between phases that are indynamic equilibrium,dynamic equilibrium,

(e.g., NO NET MOVEMENT or (e.g., NO NET MOVEMENT or

CHEMICAL CHANGE, PLUS ORCHEMICAL CHANGE, PLUS ORMINUS BETWEEN PHASES),MINUS BETWEEN PHASES), thenthen



AAµµ == AAµµ == AAµµ«.. =«.. = AAµµ

BBµµ == BBµµ == BBµµ«.. =«.. = BBµµ

CCµµ == CCµµ == CCµµ«.. =«.. = CCµµ

The chemical potential or the chemical fluxThe chemical potential or the chemical flux

of a given chemical must be the same in allof a given chemical must be the same in allphases coexisting at equilibriumphases coexisting at equilibrium--No NETNo NETChange!Change!



So,So,

AAµµ == AAµµ && AAµµ == AAµµ

and, they yieldand, they yield

AAµµ == AAµµ

and then,and then,

TWO independent equations determine theTWO independent equations determine the

equilibrium between 3 phases for EACHequilibrium between 3 phases for EACH

Component.Component.



For EACH Component,For EACH Component,

there are:there are:

(P(P--1)1) independent equationsindependent equations

relating the chemicalrelating the chemical

potential,potential, µµ, of that, of that

component incomponent in ALLALL of theof thePhases.Phases.



For theFor the GENERALGENERAL case of Pcase of P

phases and C components,phases and C components,

There areThere are

C(PC(P--1)1) independent equations.independent equations.

Thus, weThus, we ³FIX´ C(P³FIX´ C(P--1)1)

variablesvariables when we stipulate thatwhen we stipulate thatthe system is in equilibriumthe system is in equilibrium..



Now, the number of Now, the number of

independent variables or independent variables or the total number of the total number of

variations which can bevariations which can be

made independently =made independently =

the total number of the total number of

variables, less those thatvariables, less those thatare automatically fixed.are automatically fixed.



F= number of ³Freedom´ factorsF= number of ³Freedom´ factors

F= [P(C

F= [P(C--1) +2]1) +2] ± ± [C(P[C(P--1)]1)]

TOTALTOTAL AUTOMATICALLYAUTOMATICALLY

FIXEDFIXED



The variance of a system or theThe variance of a system or the

Degrees of Freedom =Degrees of Freedom =

F= CF= C--P +2P +2

Which is calledWhich is called

the Gibb¶s Phase Rule.the Gibb¶s Phase Rule.

For a ´dry´ system w/ no vapor,For a ´dry´ system w/ no vapor,

F =CF =C--P +1P +1



The Goldschmidt MineralogicalThe Goldschmidt Mineralogical

Phase RulePhase Rule

What is the likelihood of What is the likelihood of

being on a specificbeing on a specificreaction curve in Preaction curve in P--TT

space or being inspace or being in

³general´ P³general´ P--T space,T space,

where P & T are variables?where P & T are variables?



The Phase Rule in MetamorphicThe Phase Rule in Metamorphic

SystemsSystems

If F If F uu 2 (at least P & T are variables)2 (at least P & T are variables)which is the most common situation,which is the most common situation,then the phase rule may be adjustedthen the phase rule may be adjusted

accordingly:accordingly:F = CF = C -- P + 2, and P = CP + 2, and P = C

PP ee CC which iswhich is Goldschmidt¶sGoldschmidt¶s³Mineralogical Phase Rule´³Mineralogical Phase Rule´when solid solutions andwhen solid solutions andsystem is ´open´ andsystem is ´open´ andcomponents are ³mobile´.components are ³mobile´.



Consider each of the following three scenarios for Consider each of the following three scenarios for PP--T space for the aluminoT space for the alumino--silicate polymorphs:silicate polymorphs:

C = 1C = 1

P= 1P= 1 commoncommon

P= 2P= 2 rarerare

P= 3P= 3 only at theonly at the

specific Pspecific P--TT

conditions of theconditions of the

invariant pointinvariant point

(~ 0.37 GPa and(~ 0.37 GPa and500500ooC)C)

Calculated P-T phase diagram for

the system Al2SiO5. Winter, 2001



³Problems´ in real Rock Systems³Problems´ in real Rock Systems

Equilibrium has not been attainedEquilibrium has not been attained

The phase rule applies only toThe phase rule applies only to

systems at equilibrium, and theresystems at equilibrium, and therecould be any number of mineralscould be any number of minerals

coexisting if equilibrium iscoexisting if equilibrium is notnot

attainedattained

We didn¶t chooseWe didn¶t choose the number the number of of

components correctlycomponents correctly



Choosing the number of Choosing the number of

components correctlycomponents correctly

Components that substitute for each other Components that substitute for each other

Adding a component such as NaAlSiAdding a component such as NaAlSi33OO88

(albite) to the 1(albite) to the 1--C anorthite system wouldC anorthite system would

dissolve in the anorthite structure, resultingdissolve in the anorthite structure, resultingin ain a single solidsingle solid--solution mineralsolution mineral (plagioclase)(plagioclase)

below the solidusbelow the solidus

Fe and Mn commonly substitute for MgFe and Mn commonly substitute for Mg

Al may substitute for SiAl may substitute for Si

Na may substitute for KNa may substitute for K



Correct number of components:Correct number of components:

³Perfectly mobile´ components³Perfectly mobile´ components

Mobile components are either a freely mobileMobile components are either a freely mobile

fluid component or a component thatfluid component or a component that

dissolves readily in a fluid phase and can bedissolves readily in a fluid phase and can be

transported easily.transported easily.

The chemical activity of such components isThe chemical activity of such components is

commonly controlled by factorscommonly controlled by factors externalexternal toto

the local rock systemthe local rock system

They are commonly ignored in deriving C for They are commonly ignored in deriving C for

most rock systemsmost rock systems

phase rule from rutstein

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