[PDF] [PDF] Lecture 9: Phase Transitions

[PDF] Water Phase Diagram

Under the singular conditions of temperature and pressure where liquid water, gaseous water and hexagonal ice stably coexist, there is a 'triple point' where both 

[PDF] Phase Diagrams and the Triple Point - Employees Csbsju

In other words, we find that at any given temperature, the pressure of water vapor in phase equilibrium with liquid water will have some definite, characteristic 

[PDF] Phase diagram of water Note: for H O melting point decreases with

Water has a high boiling point and high freezing point If water were “normal” it would be a gas at room temperature CH 4 , 

[PDF] Triple Point Water

Discussion: On a phase diagram, the point at which all three curves intersect is known as the triple point At this temperature and pressure, all three phases are in equilibrium The triple point of water is 273 16 K and 4 58 torr

[PDF] Lecture 14 Equilibrium between phases (Ch5) - Rutgers Physics

at the triple point – in this case, all three phases coexist at (T tr , P tr ) The shape of coexistence curves on the P-T phase diagram is determined by the condition: ( ) ( ) at the boiling point, the values of G for liquid water and water vapor

[PDF] Phase Diagrams

Pressure-Temperature Phase Diagram: Solid-gas coexistence curve ends at the triple point and the liquid-gas curve Example – Phase diagram of water

[PDF] Lecture 9: Phase Transitions

Here are some example phase diagrams for carbon dioxide, argon and water Figure 4 Phase At the triple point, three phases are in equilibrium together:

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MatthewSchwartz

StatisticalMechanics,Spring2019

Lecture9:PhaseTransitions

1Introduction

Figure1.Twoofthephasesofsolidcarbon

•A⎷haseisauniformstateofmatter.

wantittobe.Wewant"gas"tobethephase.

Amoretechnicallyprecisedefinitionis

ofP;V;Tetc). importantareaofphysics. 1 ofthephasetransformationarerelated.

2Solids,liquidsandgases

T=¡1

V@V @P T (1) gaslawPV=NkBT,weseethatforagasT=∞

P=/0.Compressibilityisanexampleofanorder

=G

Aiscalled

thesurfacetension:2Section2

Atohavinga

surface,with thesurfacetension. (e.g. water=73mN mcomparedtosay,

CO2=17mN

m),butnotthelargest(mercuryhas

Hg=486mN

m). ofmetal. temperatureandpressure. k BT3 fora

2mkBp;foramoregeneralideal

gas=kBTlnP k BT equilibriumthatwearediscussinghere). energyG.Recallthat dG=VdP¡SdT+1dN1+2dN2(2) gas,ittakestheform=kBTlnN V

Melting:transitionfromsolidtoliquid.

Condensation:transitionfromgastoliquid.

Sublimation:transitionfromsolidtogas.

Nfixed.Then,=G

Nandso@

@T P =1 N@G @T P =-S N(4) infinitesimallybelowit∂G ∂T P ∂T P =-Sgas.Inapure solid=liquid=gas.

3Phaseboundaries

dominantes,andNisfixed(dN=0).SinceG

N=,thendG=NdandfromEq.(2),dG=

VdP-SdT,wefind

d=V NdP-S

NdT(5)

whichgives V 1

N1dP-S1

N1dT=V2

N2dP-S2

N2dT(6)

Thatis

dP dT=S N V

N(7)Phaseboundaries5

3.1Latentheat

N.RecallthatG=H-TSso

μ=G

N=H N-TS

N=·H

Natthephaseboundary.Sowecan

canalsowritetheClapeyronequationas dP dT=∞ TL

·¡1

n(8) wheren=N

Visthenumberdensityand

L=·H

N (9) iscalledthelatentheat.

T.Since·S=H

Twe putinpermoleculetochangethephase. -286kJ molandforwatervaporis·fH=-242kJ ofvaporizationofwaterat1atm:Lvap=44kJ

Notethat44kJ

molK, molofenergy.That L fuse=6.0kJ thatLfuse>0sinceittakesheattomeltice. L densethantheirliquidforms.So·¡1 n&0.ThereforebyEq.(8),dP dTisgenerallyverylargeand 2

Figure5.Phasediagramforwater.

dT<0.SinceL>0andT>0this mustmean·-1 n n=∞ n gas¡∞ n liquid∞ n gas=Vgas N gas=RT P(10) dP dT=PL

RT2(11)

PdP=L

RT2dT(12)

wecanintegratebothsidestogive

P=Cexp

¡L RT (13) wethenhave

P=Pexp

¡L

R∞

T¡∞

T (14)Phaseboundaries7 molandR=8.3J mol, weget T=1

T¡R

LlnP

P¡1

=370.1K(15) whichisthreedegreeslower.

3.2Vaporpressure

T =373K,P=1atmandL=42kJ w(P0;T0)=gas(P0;T0)(16) wmix(P;T)=w(P;T)¡kBTNs

Nw(17)

P

0bywritingP=P0+Pweget

wmixed(P;T0)=w(P0+P;T0)¡kBT0Ns

Nw=w(P0;T0)+P@w

@P T

¡kBT0Ns

Nw(18)8Section3

Similarly,

gas(P;T0)=gas(P0;T0)+P@gas @P T (19) Now,@ @P T =V PVw

Nw¡Vgas

Ngas =kBT0Ns

Nw(20)

Vismuchlarger),

sowecandropVw N wcomparedtoVgas N gas.Usingtheidealgaslaw,Vgas N gas=kBT0 P

0wethenhave

(P¡P0)

¡kBT0

P0 =kBT0Ns

Nw(21)

orP=¡Ns

NwP0(22)

atalowervaporpressure.

ClapeyronequationdP

dT=PL

RT2,Eq.(11).RecallthatdP

dTistheslopeofthephaseboundary.For smallPandTwecanusedP dT=P pressure,so

T=¡PRT2

P 0L=Ns

NwRT02

L(23) vaporizationofwaterL=42kJ mol,about1/7thofthelatentheatof vaporization.Then

T=¡8.7

558.3J

molK(273K)2 6.0kJ mol=¡16K(24)

3.3Chemicalpotentialphasediagrams

whichissometimesuseful. athighT.Wealsoknowthat@ @T P =¡S

N<0(25)

solidlines. bothexpansions.Tryityourself!10Section3 then@ @P T =V N=1 n>0(26) inqualitativeagreementwithEq.(14).

Here"sanotherexample

Figure8.Phasediagramforsubatomicmatter

4Generalphasetransitions

∂T P changesdiscontinuouslyatthe

Firstorderphasetransition:∂G

∂T P changesdiscontinuouslyatthephaseboundary ∂T P tobecontinuous, nthorderphasetransition:∂nG ∂Tn P intoone,andthelatentheatvanishes. theentropyS=¡@G @T P Vis equivalenttousingV=@F @P T

4.1Paramagnetism

anexampleofaphasetransition. phase. =2N F magentic=E¡TS=¡N"(27) andthefreeenergyofthedisorderedstateis F stateoccursat k BTc" ln2(29) T cisknownastheCurietemperature. transitionissecondorder. magneticfieldisavectorM,andwecandefineM=

M.AboveTc,M=0exactly.AsTis

symmetryisspontaneouslybroken.

4.2Criticalphenomena

properties.Generalphasetransitions13

N.Thegreenregionhas

liquidandgas. ∂v=0.Afterthephasetransitionis coexistenceregion. volume,v=1 ∂v? T =0.Atthecriticalpoint,thelengthofthe ∂v2? T =0aswell:thecritical14Section4 thederivativesofPvanish,∂nP ∂vn T

It"snotjustthederivatives∂nP

∂vn T diagramintheT-vplane:

Figure11.TVdiagramforwater

∂vn P =0,soT(v)isanon-analytic function. ∂vn T =∂nT ∂vn P =0,allthedimensionfulphysical andeverything surfacetension thecriticalvalues: T ^=T T c;P^=P P c;v^=v v c;n^=1 v^=n n c= c(30) n^?(T^)=1+3

4(1-T^)+7

4(1-T^)1/3;n^g(T^)=1+3

4(1-T^)-7

4(1-T^)1/3(32)

pointn^=1,T^(k)(1)=0.16Section4quotesdbs_dbs14.pdfusesText_20