材料科学基础课后习题答案7.docx
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材料科学基础课后习题答案7.docx
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材料科学基础课后习题答案7
SOLUTIONFORCHAPTER7
1.FIND:
LabelthephasefieldsinFigureHP7-1.
SOLUTION:
Therearetworegionsofsinglephaseequilibriumseparatedbyaregionoftwo-phaseequilibrium.Thelineseparatingthesinglephaseliquidfromthetwophaseliquidandsolidregioniscalledtheliquidus,andthelineseparatingthetwophaseliquidandsolidfromthesinglephasesolidiscalledthesolidus.
SKETCH:
2.FIND:
ThenamegiventothetypeofequilibriumdiagramshowninFig.HP7-1.
SOLUTION:
Thediagramistermedisomorphous.Abovetheliquidusthereisonlyonephase,namelytheliquid,andithasthesamestructureregardlessofcomposition.Belowthesolidusisthesolidphase,andaswellithasthesamestructureregardlessofcomposition.
3.FIND:
WhatisthecrystalstructureofcomponentB,why?
SOLUTION:
Therearenophaseboundariesbelowthesolidusconsequentlythephaseandthereforethestructuremustbethesame.IfAisFCC,thenBandallcompositionsbetweenAandBmustbethesamephaseandhencesamestructure.IfAisFCC,thenBmustalsobeFCC.
4.FIND:
SketchequilibriumcoolingcurvesforalloyXoandpurecomponentB.Explainwhytheyhavedifferentshapes.
SOLUTION:
Theslopeofthetemperatureversustimebehaviorforthealloyinthesinglephaseregioniscontrolledbythecoolingrateandtheheattransferredfromtheliquid.Attheliquidustemperatureasmallamountofsolidisformed,releasinganamountoflatentheatoffusionrelatedtothevolumeofsolidtransformed.Hence,attheliquidustemperaturetherewillbeachangeinslope.Sinceheatisbeinggeneratedtheslopewillbelessthanthatabovetheliquidus.Asthetemperatureisreducedthroughthetwo-phaseregionasmallamountofsolidisformedandacorrespondingheatreleased;oncethesolidustemperatureisreachednoadditionaltransformationtakesplaceandthereisagainachangeinslope.Theslopeisgreaterasthesolidcoolsthanastheliquidandsolidcooled.Forthepurecomponent,solidandliquidareinequilibriumatTB,andhence,ahorizontallinewhenliquidtransformstosolidatTB.
SKETCH:
5.FIND:
TheliquidustemperatureandthesolidustemperatureofalloyXo
SOLUTION:
FromfigureHP7-1theliquidustemperatureisapproximately1110oCandthesolidustemperatureisapproximately1070oC.
6.FIND:
Determinecompositionsandphasefractionsofeachphaseinequilibriumat1100oCforalloyXo.
SOLUTION:
Atequilibrium,thetemperatureoftwophasesmustbethesame,andthecompositionofthesolidisfoundwherethetielineintersectsthesolidusandtheliquidisattheintersectionofthetielinewiththeliquidus.Fromthephasediagramthecompositionofsolidis0.35Bandthatoftheliquidis0.55B.Usingtheleverrulethefractionofliquid,fL,andfractionofsolid,fsisdetermined.
7.FIND:
SketchfLandfSforalloyXoasthealloyiscooledunderequilibriumconditionsfrom1200oCtoroomtemperature.
SOLUTION:
Attheliquidustemperaturetheamountofliquidisalmost100%withonlyaverysmallamountofsolid.Converselyatthesolidustemperature,theamountofsolidisalmost100%withonlyaverysmallamountofliquid.Inatwo-phasefield:
fL+fs=1.
8.FIND:
ChangesinthecompositionsoftheliquidandsolidphasesduringquilibriumcoolingofalloyXothroughthetwo-phasefield.
SOLUTION:
Attheliquidustemperature,thecompositionofthesolidisapproximately0.30B.Asequilibriumsolidificationprogresses,thecompositionofthesolidincreasesto0.5B,themaximumvalueitcanreach.TheliquidontheotherhandisinitiallythecompositionofthealloyXo,andasequilibriumsolidificationprogressesthecompositionincreasesinBtothemaximumcompositionintheliquidofapproximately0.7B.
9.FIND:
Fromthefollowingdataconstructaplausibleequilibriumphasediagram.ComponentAmeltsat800︒CandBmeltsat1000︒C;AandBarecompletelysolubleinoneanotheratroomtemperature;andifsolidαcontaining0.3Bisheatedunderequilibriumconditions,thesolidtransformstoliquidhavingthesamecompositionat500︒C.
GIVEN:
Meltingtemperaturesforthetwocomponents,congruentmeltingtemperatureandcomposition.
SOLUTION:
Thesketchshownbelowsatisfiesalloftherequirementsstatedintheproblem.ThemeltingtemperatureofpureAis800︒CandthatforpureBis1000︒C.Thecongruentmeltingtemperatureisat500︒Catacompositioncontaining0.3B.Belowthecongruentmeltingtemperaturethereisacontinuousαsolidsolution.
10.FIND:
Foralloyscontaining10,22,25,27,and40wt%Ni,determinethenumberofphasespresentandthecompositionofthephasesinequilibriumat1200oC.
SOLUTION:
Alloycomposition
in%Ni
Numberofphasesinequilibrium
Compositionof
Phases(%Ni)
10
1(Liquid)
10%
22
1(Liquid)
22%
25
2(Liquid+Solid)
liquid:
22%,solid:
32%
27
2(Liquid+Solid)
liquid:
22%,solid:
32%
40
1(Solid)
40%
11.FIND:
Beginningwithastatementofmassbalance,derivetheleverruleinatwo-phasesystem.
SOLUTION:
InatwophasefieldforanalloyofsomeoverallcompositionXo,thesoluteisdistributedinthetwophases:
xo=fαxα+fβxβwherethecompositionsareexpressedintermsofoneofthecomponents,fα+fβ=1.Then,fα=1-fβ.Substituting:
xo=(1-fβ)xα+fβxβ
xo=xα-xαfβ+xβfβ
xo-xα=(xβ-xα)fβ
12.FIND:
DiscusseachofthefactorsthatpermittheCu-Nisystemtobeisomorphousoverthetemperaturerange350-1000oC.
SOLUTION:
TheempiricalrulesofHume-Rotheryidentifythecharacteristicsthattwoelementsmusthaveincommonforextensivesolubility.Thisshouldrequirethat
a.thetwocomponentsmusthavethesamecrystalstructure
b.theatomicradiiofthetwoatomsmustbesimilar
c.thetwocomponentshavethecomparableelectronegativities,and
d.thetwocomponentshavethesimilarvalence.
13.FIND:
Whatdoesthetemperature322OCrepresentinFigureHP7-2?
SOLUTION:
322oCiscalledthecriticaltemperature.Atthecriticaltemperaturethereisacorrespondingcriticalcomposition.Foranalloyofthiscomposition,coolingunderequilibriumconditionsfromabovethecriticaltemperaturetobelowthistemperatureresultsintheformationoftwophasesfromonephaseofthecriticalcomposition.Ascoolingcontinuesthetwophasesthatformhavedifferentcompositions.
14.FIND:
InFigureHP7-2,areα1andα2differentcrystalstructures?
SOLUTION:
α1andα2aretwophaseshavingthesamestructurebutdifferentcompositions.Thecompositionofthetwophasesaredeterminedasinanytwophasesystembyusingthetie-line.Wherethetie-lineattheequilibriumtemperatureintersectsthephaseboundariesdeterminesthecompositionofthetwophaseinequilibrium.
15.FIND:
LocationoftheequilibriumphaseboundaryattemperatureT.
GIVEN:
Alloy1containing30%Bandalloy2containing50%B,whenequilibratedattemperatureTareinthesametwo-phase(L+S)region.Thefractionofliquidinalloy1is0.8andthefractionofliquidinalloy2is0.4.
SKETCH:
SOLUTION:
Sincethefractionofliquidinalloy1isgreaterthanthatofsolid,theliquidusistotheleftof0.3,andsincethefractionofsolidinalloy2isgreaterthanthatofliquid,thesolidusmustlietotherightofalloy2.Usingtheleverrule:
foralloy1
foralloy2
0.8XL-0.8Xs=Xs-0.3
0.4XL-0.4Xs=Xs-0.5
SolveforXsandXL,
0.8Xs-0.8XL=Xs-0.3
0.8Xs-0.8XL=2Xs-1.0
____________________
0=-Xs+0.7,or
Xs=0.7B
TofindXL,substituteXsintooneoftheequations:
0.4-0.56=-0.8XL
-0.16=-0.8XL,or
XL=0.2B
Asacheck,usetheotherequationtocalculatethefractionofliquidfromthecompositions.
16.FIND:
LabelallregionsofthephasediagramandtheboundariesofmonovariantequilibriumforthediagramshowninFigureHP7-3.
SKETCH/SOLUTION:
17.FIND:
Sketchanequilibriumcoolingcurvefromabovetheeutectictoroomtemperatureforanalloyofeutecticcomposition.
SKETCH/SOLUTION:
18.FIND:
Explainwhytheequilibriumcoolingcurvesforalloysoneithersideoftheeutecticcompositionwillbedifferentthantheequilibriumcoolingcurveforaeutecticalloy.
SOLUTION:
Attheeutectictemperature,liquidofcompositionXEisinequilibriumwithtwosolids,XαandXβ.Forahypoeutecticalloy(compositiontotheleftoftheeutectic),thefirstphasetoformattheliquidustemperatureofthealloyisα.Whentheeutectictemperatureisreachedtheliquidoftheeutecticcompositionisinequilibriumwiththetwosolids,oneofcompositionXαandtheotherXβ.Consequently,dependinguponcomposition,theclosertheoverallcompositionofthealloyistoXE,thelessproeutecticαandthemoreeutectic.Similarly,foralloystotheright,proeutecticβwillform.Intermsofthephaseruleatconstantpressure,fortheeutecticreaction
F=C-P+1=2-3+1=0.Theeutecticisinvariant,andsolidifiesunderequilibriumconditionsatonetemperature,TE.ForalloysoneithersideoftheeutecticcompositionbutbetweenXαandXβ,theproeutecticphaseformsandcoolingoccursasitdoesinanytwo-phases-lregion.Oncetheeutecticisothermisreached,theremainingliquidofeutecticcompositionsolidifiesisothermally.
19.FIND:
ThemaximumsolidsolubilityofBinAandofAinBinFigureHP7-3.
SOLUTION:
ThemaximumsolubilityofBinAoccursat0.15BandthemaximumsolubilityofAinBis0.1A.
20.FIND:
ForanalloyofeutecticcompositioninFigureHP7-3,determinethecompositionofthesolidphasesinequilibriumwiththeliquid.
SOLUTION:
ThecompositionofthetwosolidphaseαandβinequilibriumwithliquidatT_occursat0.15Band0.9B.
21.FIND:
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