Lecture7Thermodynamics:TheSecondandThirdLaws

HOT COLDq

DIRECTIONOFSPONTANEOUSCHANGES

Thedirectionofchangeisrelatedtothedistributionofenergy:spontaneouschangesarealwaysaccompaniedbyareductionofthe“quality”ofenergyfromamoreorganizedtoamoredispersed,chaoticform.

SPONTANEOUSCHANGE-CHAOTICDISPERSALOFTHETOTALENERGY

TOWARDSGREATERDISORDER

ANEWFUNCTIONOFSTATE:ENTROPY

ThedispersalofenergycanberelatedtotheheattransferredinaprocessTHERMODYNAMICDEFINITION:

ΔS=qrev/T(Clausius)

ThedispersalofenergycanbecalculatedSTATISTICALDEFINITION:S=kBlnW(Boltzmann)

SPONTANEOUS PROCESSES CAUSE AN INCREASE IN THE ENTROPYOFTHEUNIVERSEIRREVERSBLEPROCESSESARESPONTANEOUSIRREVERSIBLEPROCESSESGENERATEENTROPYREVERSIBLEPROCESSESDONOTGENERATEENTROPY(BUTTHEYMAYTRANSFERITFROMONEPARTOFTHEUNIVERSETOANOTHER)

THEQUANTITATIVEDEFINITIONOFENTROPY1)Thermodynamic(Clausius)

Sisastatefunctiondqrevisnotanexactdifferentialdqrev/Tisanexactdifferential

1/Tistheintegratingfactorofdqrev

ReversibleIsothermalExpansion

ForagasbehavingperfectlyPV=nRT

IrreversiblevsReversibleExpansion

PathA PathB

qirrev<qrev

CLAUSIUSINEQUALITY

Foranisolatedsystem:ΔS≥0

ΔSuniverse≥0

SecondLaw:TheentropyofanisolatedsystemincreasesInthecourseofaspontaneouschange.

ΔSuniverse= Δssystem+Δssurroundings≥0Δssystem≥-ΔSsurroundings

Example1

Overallentropychangeforanisothermalreversibleexpansion:

PathB

Overallentropychangeforanirreversiblefreeexpansion:

Example2

Heatflowfromahottoacoldbody

qTh Tc

THEENTROPYCHANGEWHENASYSTEMISHEATED

Inaphasetransition,heatistransferredreversiblybetweenthesystemanditssurroundings:

AbsoluteEntropy

Unlikeenthalpy-wehaveanabsolutescaleforentropy.ThirdLawofThermodynamic:IftheentropyofeveryelementinitsstablestateatT=0Kistakenaszero,everysubstancehasapositiveentropywhichatT=0becomeszero forallperfectcrystallinesubstances,includingcompounds.

Unlikeenthalpy -wehaveanabsolute scale for entropy.Third Law of Thermodynamic: If the entropy of everyelementinitsstablestateatT=0Kistakenaszero,everysubstancehasapositiveentropywhichatT=0becomeszero for all perfect crystalline substances, includingcompounds.

• Entropyisameasureofdisorder:– Inaphasechange:

• Solid Liquid Gas• Highlyordered Lessordered Verydisordered

Weseewhysolidsgliquidggasphase@roomtemp

• Sgas>>Sliquid>Ssolid

RelativeEntropy

THEGIBBSFREEENERGY

P,T=CONSTANT

Thisresultsuggeststhatwedefineanewstatefunction,theGibbsFreeEnergy:

G=H-TS

Thedirectionofspontaneouschangeisthedirectioninwhichthefreeenergydecreases.

ΔG:themaximumnon-p,VworkatT,p=const.

F=U-TS

THEHELMHOLTZFREEENERGY

V,T=CONSTANT

ΔF:themaximumworkatT,V=const.

THEQUANTITATIVEDEFINITIONOFENTROPY2)Statistical(Boltzmann)

Distributions,microstatesanddisorder

state

1

2

1

Wi=#configurations=#microstates

0.25

0.25

0.50

1

2

3

Pi =wi

wii=1

n

∑

1.00.80.60.40.20.0

WE

IGH

T2.01.51.00.50.0

(N–NT)

N = 2 1:2:1

Nstates=3=N+1Nmicrostates=4=22=2N

N=2

1.00.80.60.40.20.0

WE

IGH

T3.02.52.01.51.00.50.0

(N–NT)

N = 3 1:3:3:1

Nstates=4=N+1Nmicrostates=8=23=2N

N=3

1.00.80.60.40.20.0

WE

IGH

T

43210(N–NT)

N =4 1:4:6:4:1

Nmicrostates=16=24=2N

Nstates=5=N+1

N=4

1.00.80.60.40.20.0

WE

IGH

T543210

(N–NT)

N = 5 1:5:10:10:5:1

Nstates=6=N+1Nmicrostates=32=25=2N

1.00.80.60.40.20.0

WE

IGH

T

76543210(N–NT)

N = 7

Nmicrostates=128=27

1.00.80.60.40.20.0

WE

IGH

T

50403020100(N–NT)

N = 50

1.00.80.60.40.20.0

WE

IGH

T

6543210(N–NT)

N = 6

Nmicrostates=64=26

N=1.1258999068x1015

1

2

3

4

L RDistributions,microstatesanddisorder

1.00.80.60.40.20.0

WE

IGH

T

43210(N–NT)

N =4 1:4:6:4:1

1.00.80.60.40.20.0

WE

IGH

T

43210(N–NT)

N =4 1:4:6:4:1

1

2

3

4

1

23

4

3

1

2

4

1

2

3

4

Distributions,microstatesanddisorder

1

2

3

4

1 2

3 4

12

3 4

1

2

3

4

12

34

1

2

3

4

Wmax = max # microstates

Nature evolves toward Wmax

ΔU = ΔH = 0

1.00.80.60.40.20.0

WE

IGH

T

43210(N–NT)

N =4 1:4:6:4:1

Positionaldisorder:crystals

Weaddenergy

Weremoveenergy

Energeticdisorder

EK = 3

2RT

n1 + n2 +…+ nm =N

εK = 3

2kBT

ε1n1 + ε2n2 +…+ εmnm = EK

n1 + n2 +…+ nm =N = 100

ε1n1 + ε2n2 +…+ εmnm = EK = 10000

Egalitariandistribution

Feudaldistribution

Boltzmanndistribution

n1 + n2 +…+ nm =N ε1n1 + ε2n2 +…+ εmnm = EK

n2

n1

= e−

ε2−ε1( )kbT

�

nin j

= exp −ε i −ε j

kBT

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

nin j

= exp − ΔεkBT

⎛

⎝ ⎜

⎞

⎠ ⎟

1.0

0.8

0.6

0.4

0.2

0.0

Ene

rget

ic L

evel

s

14121086420Occupation Probability

T1 T2 T3

T3 > T2 > T1

Wmax = max # microstates

Nature evolves toward Wmax

Wmax=max#microstates

NatureevolvestowardWmax

W=W1✕ W2

S=kBlnW=kB(lnW1+lnW2)

S=ClnW=C(lnW1+lnW2)

•  States,S.–Themicroscopicenergylevelsavailableinasystem.

•  Microstates,W.–  The particular way in which particles are distributed amongst the

states.Numberofmicrostates=W.•  TheBoltzmannconstant,kB=1.38✕ 10-23J/K.

–  Effectivelythegasconstantpermolecule=R/NA.

S=kB✕lnW

TheBoltzmannEquationforEntropy

ΔS=S2–S1=kB(lnW2–lnW1)=kBln2

ForoneparticleconsideranexpansionatT=cost

V2=2V1

W2=2W1

N =1ΔS=kBln2

ΔS=S2–S1=kB(lnW2–lnW1)=kBln22

FortwoparticlesconsiderthesameexpansionatT=costV2=2V1

W2/W1=2✕2(eachparticlemovesindependentlyoftheother)

N =2ΔS=kB✕2✕ln2

ForthreeparticlesconsiderthesameexpansionatT=costV2=2V1

W2/W1=2✕2✕2(eachparticlemovesindependentlyoftheother)

ΔS=S2–S1=kB(lnW2–lnW1)=kBln23

N =3ΔS=kB✕3✕ln2

ForNparticlesconsiderthesameexpansionatT=costV2=2V1

W2/W1=2N

ΔS=S2–S1=kB(lnW2–lnW1)=kBln2N

N =NΔS=kB✕N✕ln2

FromthethermodynamicdefinitionofΔS:

N =NΔS=kB✕N✕ln2=Rln2

LINKINGTHECHANGESINGIBBSFREEENERGYTOTHEEQUILIBRIUMCONSTANT

dU=δq+δw FirstLaw

Forareversiblechangeintheabsenceofnon-p,Vwork:δq=TdS(SecondLaw);δw=-pdVdU=TdS-PdVHowever,Uisastatefunction,dUisanexactdifferentialthisfundamentalequationholdsforanyreversibleorirreversiblechangeofaclosedsysteminvolvingnonon-p,Vwork.

FirstStep:CombiningtheFirstandSecondLaws

SecondStep:ManipulatingtheGibbsFunction

G=G(T,p)Thermodynamics

Mathematics

Compare

Similarly,onecanderive:GREAT

PHYSICISTS

HAVE

STUDIEDUNDER

VERY

FINETEACHERS

G=G(T,p)

ThirdStep:ThePressureDependenceoftheGibbsFunction

Foraperfectgas:

Byintegratingweobtain:

Letp’=p0=1bar�G=G0

ThemolarGibbsfunctionGm=G/nthusis:

Gm(p)iscalledtheCHEMICALPOTENTIALandisindicatedbythesymbolµ

Foraperfectgas andPVm=RTareequivalent

Proof:

FourthStep:AllowingForChangesInComposition

Ingeneral:G=G(p,T,n1,n2….)

CHEMICALPOTENTIAL

Forasinglecomponentµ=Gm=G/n

AtconstantT,p,ifweadddn1molesof1toamixture,GincreasesbydG=µ1dn1

Considerthesimplereaction:ADB

Foranamount-dnA=-dξofAthatisconsumed,anamountdnB=+dξforms.Atconstantp,T:

isthechangeintheGibbsfunctionpermoleofreactionatadefinedcompositionofthereactionmixture.

IfΔrG<0thereactionproceedsAgBIfΔrG>0thereactionproceedsAfBIfΔrG=0thereactionisatequilibrium

Fifthstep:linkingΔrGandtheequilibriumconstant

Atequilibrium:ΔrG=0andQ=Kp

A,Bperfectgases:

Theresultisvalidingeneral:aA+bBDcC+dD

Inthegasphase:

Insolution:

TemperaturedependenceofK

Gibbs-Helmholtzequation

Forachange(e.g.achangeofstate,achemicalreaction):

van’tHoffequation

LeChatelierprinciplemadequantitative!Exothermicreactions:increasedTfavoursthereactantsEndothermicreactions:increasedTfavourstheproducts

Integratingthevan’tHoffequation