Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy :...
Transcript of Entropy Rudolf Clausius - University of California, San Diegoruben.ucsd.edu/20/r05.pdf · Entropy :...
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
2
ThermodynamicsofLigandBindingandEfficiencyAlookatligandbindingthermodynamicsindrugdiscovery.ExpertOpinDrugDiscov.2017Claveria-Gimeno,VegaS,AbianO,Velazquez-CampoyA
Entropyasameasureofnumberofstates
Log(W)func,on(whereWisthenumberofstates)makesEntropyaddi,veandextensive
EntropicBotzmannbyLesDuHon,PhD
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
ClassicalEntropyandUnits
• UnitsofΔS:Joule/K(orcal/K)• Warnings:TheunitsofEntropyarethesameasHeatCapacity,howeverΔSandCaretotallydifferent:
TqS =Δ
TqCΔ
=
EntropyChangeofaReversibleProcess=SmallHeatoverAbsoluteTemperature
HeatCapacityofmaterial=HeatoverSmallTemperatureChange
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
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)
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
#
$%
&
'(
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
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
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)
EntropyandGrowingCrystals-I
HCl–hydrogenchloride• Lowentropy–perfectcrystals• Oncethe1stmoleculeisinplace,thereisonlyonewayto
putthecrystaltogether(Wtotalnumberofstates):W=1
• AtT=0,S=klnW=kln1=0
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)
EntropyChanges:Examples• Configura,onalEntropyofonemolecule(permole):
• Example:thenumberofstatesWinonemolecule,withbrotatablebondswith3equi-probablestates,n1is#statesin1molecule
Sm=Rln(n1)n1=3b
S=Rbln(3)
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
StandardEntropiesofForma,onΔSm=S2-S1=Rln(n2/n1)• Thenumberofvibra,onquantummicrostatesdependsontheatommasses
Solid:116
ΔS°rxn=S°products–S°reactants
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)
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
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)?
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
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