Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy :...

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Entropy CLASSICAL THERMODYNAMICS study of macroscopic/thermodynamic properties of systems: U, T , V , P , … STATISTICAL THERMODYNAMICS establishing relationships between microstates and macrostates Rudolf Clausius Ludvig Boltzmann 1844-1906 ΔS = q rev T A new State Func,on: 1 st law: ΔU = q – w Hear capacity: C=q/ΔΤ S = k ln N

Transcript of Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy :...

Page 1: Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy : counng microstates • the observed macrostate looks like a typical microstate

Entropy

CLASSICAL THERMODYNAMICS study of macroscopic/thermodynamic properties of systems: U, T, V, P, …

STATISTICAL THERMODYNAMICS establishing relationships between

microstates and macrostates

RudolfClausius

LudvigBoltzmann1844-1906

ΔS =qrevT

AnewStateFunc,on:

1st law: ΔU= q – w Hear capacity: C=q/ΔΤ

S = k lnN

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2

ThermodynamicsofLigandBindingandEfficiencyAlookatligandbindingthermodynamicsindrugdiscovery.ExpertOpinDrugDiscov.2017Claveria-Gimeno,VegaS,AbianO,Velazquez-CampoyA

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Entropyasameasureofnumberofstates

Log(W)func,on(whereWisthenumberofstates)makesEntropyaddi,veandextensive

EntropicBotzmannbyLesDuHon,PhD

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Entropy:coun9ngmicrostates•  theobservedmacrostatelookslikeatypicalmicrostatewiththesameU

•  Amacro-state(egphase)consistsofNmicrostates

•  Microstatesarecombinedfrommolecularmicrostates(nmicro)

•  PhaseBnmicro=10

PhaseAnmicro=1

1 molecule

S = kB lnNmicrostates

N forNAmolecules= nNA

lnax = x lnaSmole = kB lnN = kBNA lnn = R lnn

1 molecule

ΔSABmole = R ln10

1 molecule

1 molecule

1 molecule

1 molecule

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ClassicalEntropyandUnits

•  UnitsofΔS:Joule/K(orcal/K)•  Warnings:TheunitsofEntropyarethesameasHeatCapacity,howeverΔSandCaretotallydifferent:

TqS =Δ

TqCΔ

=

EntropyChangeofaReversibleProcess=SmallHeatoverAbsoluteTemperature

HeatCapacityofmaterial=HeatoverSmallTemperatureChange

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EntropyChangeinanIrreversibleProcess

•  Foranirreversibletransi,onfromstateAtostateB:•  Irreversible⇔spontaneous⇔noworkrequired⇔ΔU=qClausius:“Noprocessispossiblewhosesoleresultisthetransferof

heatfromabodyoflowertemperaturetoabodyofhighertemperature”

Kelvin:“Noprocessispossibleinwhichthesoleresultistheabsorp,onofheatfromareservoiranditscompleteconversionintowork”

dqTA

B

∫ <dqrevTA

B

∫ = ΔS

the Second Law of Thermodynamics ΔS ≥ 0 : in an isolated system S is increasing as it is

reaching its equilibrium maximum value

Page 7: Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy : counng microstates • the observed macrostate looks like a typical microstate

EntropyChangesinSpecificProcesses

•  Changesinvolumeorpressure(e.g.inisothermalexpansionofidealgas).Largervolume,morespacemicrostates,N~V,S~lnV

•  Changesintemperature(illustratedbyisobarichea,ngofidealgas)PV=nRT.Moremicrostatesforenergyvalues~T.N~TS~lnT

•  Phasechanges(atthetransi,ontemperature)•  Entropyofmixing(biggervolumeforeachmolecule)

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Entropy.Gas.Classicalmethod:S(Volume)atT=const

ClassicalDeriva,on(op,onal)•  ConsideridealgasisothermallyexpandingfromVAtoVB.ΔU=0

becausetheinternalenergyforidealgasonlydependsonthetemperature•  AsΔSdoesnotdependonpath,chooseareversiblepath•  Entropychange:

Increase in entropy

VA VB

B

A

B

A

B

A

B

A

B

A

VV

V

V

V

V

V

Vrev

V

V

rev VnRdVVnRPdV

Tdq

TTdqS ∫∫∫∫ =====Δ )ln(11

qrev = w = nRT ⋅ lnVBVA

"

#$

%

&' ⇒ ΔST = nR ⋅ ln

VBVA

"

#$

%

&'= -nR ⋅ ln PB

PA

"

#$

%

&'

T=constΔU=0,w=q S = nR ⋅ lnV

ΔST = nR ⋅ lnVBVA

#

$%

&

'(

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Entropy.Gas.S(Temperature),P=const

•  ConsideridealgasheaYngfromTAtoTBatconstantpressureoranyothersystemwhereCpdoesnotchangebetweenTAandTB

•  AsΔSdoesnotdependonpath,chooseareversiblepath:dqrev=CPdT

•  Entropychange:

•  IfCP~constant•  Notefornmoles

xxdx

TdTC

TdqS

B

A

B

A

T

TP

T

T

rev ln: ===Δ ∫∫∫ note

A

BP

T

TPP T

TCTdTCS

B

A

ln==Δ ∫

Increase in

entropy TA TB

S =CP ⋅ lnT

ΔST =CP ⋅ lnTBTA

#

$%

&

'(

ClassicalDeriva,on(op,onal)

CP = nCmP

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KirchhoffrulesextendedtoΔS•  NowwecancalculatebothHandSfromCpatT2

•  (Kirchhoff)

)()()( 1212 TTCTHTH P −+≈

⎟⎟⎠

⎞⎜⎜⎝

⎛+≈

1

212 ln)()(

TTCTSTS P

)(

ln

12

1

22

1

2

1

TTCH

TTC

TdTC

TdqS

P

P

T

T

PT

T

rev

−≈Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛≈==Δ ∫∫

Thesameformulaisusedtomeasureentropyinacalorimeter

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+≈

1

1212 )()(

TTTCTSTS P

ForΔT<<T1

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EntropyinSta9s9calThermodynamics•  Entropyisameasureofdisorder,orrandomness

•  S=–kΣpilnpi,where:–  piistheprobabilityofthemicrostatei–  k(orkB)istheBoltzmannconstant

•  Ifthemicrostatesareequi-probable:

S = k ln WwhereWisthenumberofmicrostatesfortheenYresystem(disYnguishablewaysthesystemcanbeputtogetherwithgivenUandV),Wmole=n1Na

•  Smole=Rln(n1 ) ifnisthenumberofmicrostatesforONEMOLECULE.

•  ΔSm aèb = Sb-Sa=R ln (na/nb)

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EntropyandGrowingCrystals-I

HCl–hydrogenchloride•  Lowentropy–perfectcrystals•  Oncethe1stmoleculeisinplace,thereisonlyonewayto

putthecrystaltogether(Wtotalnumberofstates):W=1

•  AtT=0,S=klnW=kln1=0

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EntropyandGrowingCrystals-IICO–carbonoxide•  Higherentropy–imperfect(rota,onally

disordered)crystals•  Oncethe1stmoleculeisinplace,there

aretwowaystoposi,onthe2ndmolecule,foreachofthese,twowaystoposi,onthe3rdmoleculeetc:NumberofstatesforNmoleculesW=2N

•  AtT=0,S=klnW=kln2N=Nkln2

•  ForamoleofCO,S=NAkln2=Rln2

(R–gasconstant)

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EntropyChanges:Examples•  Configura,onalEntropyofonemolecule(permole):

•  Example:thenumberofstatesWinonemolecule,withbrotatablebondswith3equi-probablestates,n1is#statesin1molecule

Sm=Rln(n1)n1=3b

S=Rbln(3)

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EntropyisAddi,ve•  Entropy(disorder)oftwoparts,AandBSA=klnnA,SB=klnnB,

•  Entropyofbothparts,A+B:Thenumberofstates:nAB=nA•nBSAB=kln(nA•nB)=klnnA+klnnB=SA+SBEntropyisADDITIVE,sincelnXY=lnX+lnY

•  FortheAvogadronumberofmoleculeswithn1states:Smole=NAklnn1=Rlnn1

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StandardEntropiesofForma,onΔSm=S2-S1=Rln(n2/n1)•  Thenumberofvibra,onquantummicrostatesdependsontheatommasses

Solid:116

ΔS°rxn=S°products–S°reactants

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EntropySummary

•  S=klnNforanynumberofmolecules,Nisthetotalnumberofcombina,onsofstatesforallmolecules.

•  N=nNafor1mole(Na)ofmolecules,nisthenumberofstatesperMOLECULE

•  Sforonemole:Sm=klnN=kNaln(n)=Rln(n)•  SformmolesismSm(entropyisaddi,ve)•  ΔSm1->2=Rln(n2/n1)

Page 18: Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy : counng microstates • the observed macrostate looks like a typical microstate

Es,ma,ngln(1+x)andmore•  Inmanyproblemsyouneed

toes,mateexpressionslookinglike

ln(300K/280.)≈ln(1.07)or1/1.2,etc.

•  ToevaluatethemuseasimpletechniquebasedontheTaylorexpansion:

)(''

2'

!..

2)( n

a

n

aaa fnxfxxffxaf ++++=+

7.0)07.1ln(..21)1(211)1(

111

2,21,1:

1)1(1)1ln(

2

+≈+

+≈+

−≈+

−=

+≈+

+≈

≈+

gExx

xx

xx

nExamples

nxxxexx

n

x

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GibbsFreeEnergy

G = H - TS

ΔG = ΔH – TΔS(FreeEnergyChangeinaTransi,on)•  Gdefinesthedirec,onoftransforma,onsandreac,ons•  Chemicalreac,onsarespontaneousinthedirec,onof

decreasingG,i.e.dGT,P≤0•  InchemicalorphaseequilibriumG1=G2orΔG=0•  Gdefinesmaximaluseful(non-expansion)workthatcanbe

extractedfromthesystem

1839-1903,Americantheore,calphysicist

WhatfuncYondefinesthedirecYonofprocessesatconstanttemperatureandpressure(biology)?

Page 20: Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy : counng microstates • the observed macrostate looks like a typical microstate

Themeaningoflife:minimizingGloomMinimizeGbyshopping,bondingandpartying

•  Enthalpy(HforHording)isunrealizedpoten,al.ReducingHbyspendingmakesyouhappy(e.g.byformingstrongbonds)

•  Entropy(S)isfreedom,ln(States),abilitytohavemanyop,ons,personalSpace.Itsimportance,-TS,growswithtemperature.

•  Temperatureistherela,veimportanceoffreedom,Temperament

•  GistheGloomfunc,on(minimizingGvaluesmakeyouhappy).

G = H – T S •  Goal:MinimizeGloom(maximize

happiness)byspending/minimizingG.•  G,H,TSmeasuredintheJ,kJ,orcal/kcal

-TS

0 +

+

+

-

-

-H

G=H-TSHappy

Bonded/restricted

Bondedandhappy

Gloomy

Freeandhappy

EntropyactsasaCounterbalance

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Review•  HessrulesforH•  Entropy

•  EntropyfromCpatT2

•  Entropyofgasfrom(p1,v1)to(p2,v2):

•  Smole=kBln(Ntotal)=Rln(nstates_of_one_molecule)ΔS= nR ln(n1/n2)

•  Statefunc,ons(variables) •  1stLaw,work ΔU=q-w•  Enthalpy•  Calorimetry•  Heatcapacity, C≡q/ΔT •  CvandCp•  MolarC,specificheat•  Transi,ons,standardstates

Pp T

HC ⎟⎠

⎞⎜⎝

⎛∂

∂≡

PVUH +=

Vv T

UC ⎟⎠

⎞⎜⎝

⎛∂

∂≡

PreviousReview

)()()( 1212 TTCTHTH P −+≈

⎟⎟⎠

⎞⎜⎜⎝

⎛+≈

1

212 ln)()(

TTCTSTS P

∫=Δ=B

A

revrevTdqS

TdqdS or

ΔST= nR ln(v1/v2) ΔST= -nR ln(p1/p2)

Gibbs Free Energy (G)

G=H-TS G1 = G2 G→minimum