Chapter 5 - Collision Theory.pdf

Colli

1. Collision theory
Thesimplestquantitativeaccountofreactionratesisintermsof
collisiontheory,whichcanbeusedonlyforthediscussionof
reactionsbetweensimplespeciesinthegasphase.
■Collisiontheory:
Weshallconsiderthebimolecularelementaryreaction;
A+B→P v=k
2[A][B]
wherePdenotesproducts,andaimtocalculatethesecond-order
rateconstantk
2.
Weexpecttheratevtobeproportionaltotherateofcollisions,and
thereforetothemeanspeedofthemolecules,c∝(T/M)
1/2
whereM
isthemolarmassofthemolecules,theircollisioncross-section,σ,
andthenumberdensitiesN
AandN
BofAandB.
v ∝σ(T/M)
1/2
N
AN
B∝σ (T/M)
1/2
[A][B]

2. Collision theory
However,acollisionwillbesuccessfulonlyifthekineticenergy
exceedsaminimumvalue,theactivationenergy,E
a,ofthereaction.
Thisrequirementsuggeststhattherateconstantshouldalsobe
proportionaltoaBoltzmannfactoroftheforme
−Ea/RT
.Sowecan
anticipate,bywritingthereactionrateintheform;
k
2∝σ (T/M)
1/2
e
−Ea /RT
Noteverycollisionwillleadtoreactioneveniftheenergy
requirementissatisfied,becausethereactantsmayneedtocollide
inacertainrelativeorientation.This‘stericrequirement’suggests
thatafurtherfactor,P,shouldbeintroduced,andthat
k
2∝σ P (T/M)
1/2
e
−Ea /RT
Thisexpressionhastheformpredictedbycollisiontheory.
k
2∝stericrequirement×encounterrate×minimumenergy
requirement

3. Collision theory
■(a)Collisionratesingases:
Wehaveanticipatedthatthereactionrate,andhencek
2,depends
onthefrequencywithwhichmoleculescollide.Thecollision
density,Z
AB,isthenumberof(A,B)collisionsinaregionofthe
sampleinanintervaloftimedividedbythevolumeoftheregionand
thedurationoftheinterval.
Whereσisthecollisioncross-sectionandμisthereducedmass,
Similarly,thecollisiondensityforlikemoleculesatamolar
concentration[A]is

4. Collision theory
■Thecollisioncross-sectionfortwomoleculescanberegarded
tobetheareawithinwhichtheprojectilemolecule(A)mustenter
aroundthetargetmolecule(B)inorderforacollisiontooccur.Ifthe
diametersofthetwomoleculesared
Aandd
B,theradiusofthetarget
areaisd=1/2(d
A+d
B)andthecross-sectionisπd
2
.

5. Collision theory
■Theenergyrequirement:
Accordingtocollisiontheory,therateofchangeinthemolar
concentrationofAmoleculesistheproductofthecollisiondensity
andtheprobabilitythatacollisionoccurswithsufficientenergy.
Collisiondensitycanbeincorporatedbywritingthecollisioncross-
sectionasafunctionofthekineticenergyofapproachofthetwo
collidingspecies,andsettingthecross-section,σ(ε),equaltozeroif
thekineticenergyofapproachisbelowacertainthresholdvalue,ε
a.
Later,weshallidentifyN
Aε
aasE
a,the(molar)activationenergyof
thereaction.Then,foracollisionwithaspecificrelativespeedof
approachv
rel(not,atthisstage,ameanvalue),

6. Collision theory
■Thestericrequirement:
Wecanaccommodatethedisagreementbetweenexperimentand
theorybyintroducingastericfactor,P,andexpressingthereactive
cross-section,σ*,asamultipleofthecollisioncross-section,σ*=
Pσ.Thentherateconstantbecomes
Thecollisioncross-sectionisthetarget
areathatresultsinsimpledeflectionof
theprojectilemolecule;thereaction
cross-sectionisthecorrespondingarea
forchemicalchangetooccuron
collision.

7. Collision theory
■Diffusion-controlledreactions:
Encountersbetweenreactantsinsolutionoccurinaverydifferent
mannerfromencountersingases.Reactantmoleculeshavetojostle
theirwaythroughthesolvent,sotheirencounterfrequencyis
considerablylessthaninagas.However,becauseamoleculealso
migratesonlyslowlyawayfromalocation,tworeactantmolecules
thatencountereachotherstayneareachotherformuchlongerthan
inagas.Thislingeringofonemoleculenearanotheronaccountof
thehinderingpresenceofsolventmoleculesiscalledthecageeffect.
Suchanencounterpairmayaccumulateenoughenergytoreact
eventhoughitdoesnothaveenoughenergytodosowhenitfirst
forms.

8. Collision theory
■Diffusion-controlledreactions:
Thecomplicatedoverallprocesscanbedividedintosimplerpartsby
settingupasimplekineticscheme.Wesupposethattherateof
formationofanencounterpairABisfirst-orderineachofthe
reactantsAandB:
A + B → AB v = k
d[A][B]
k
d(wherethedsignifiesdiffusion)isdeterminedbythediffusional
characteristicsofAandB.Theencounterpaircanbreakupwithout
reactionoritcangoontoformproductsP.Ifwesupposethatboth
processesarepseudofirst-orderreactions(withthesolventperhaps
playingarole),thenwecanwrite
AB → A + B v = k
d′[AB] and
AB → P v = ka[AB]

9. Collision theory
TheconcentrationofABcannowbefoundfromtheequationforthe
netrateofchangeofconcentrationofAB:
Therateofformationofproductsistherefore;
Twolimitscannowbedistinguished.Iftherateofseparationofthe
unreactedencounterpairismuchslowerthantherateatwhichit
formsproducts,thenk
′
d<<k
aandtheeffectiverateconstantis
Inthisdiffusion-controlledlimit,therateofreactionisgovernedby
therateatwhichthereactantmoleculesdiffusethroughthesolvent.

10. Collision theory
Anactivation-controlledreactionariseswhenasubstantialactivation
energyisinvolvedinthereactionAB→P.
Thenk
a<<k
d′
whereKistheequilibriumconstantforA+B AB.Inthislimit,the
reactionproceedsattherateatwhichenergyaccumulatesinthe
encounterpairfromthesurroundingsolvent.
■Therateofadiffusion-controlledreactioniscalculatedby
consideringtherateatwhichthereactantsdiffusetogether.therate
constantforareactioninwhichthetworeactantmoleculesreactif
theycomewithinadistanceR*ofoneanotheris
k
d= 4πR*DNA
whereDisthesumofthediffusioncoefficientsthetworeactant
speciesinthesolution.

11. Transition state theory
Weknowthat,anactivatedcomplexformsbetweenreactantsas
theycollideandbegintoassumethenuclearandelectronic
configurationscharacteristicofproducts.
Wealsosawthatthechangeinpotentialenergyassociatedwith
formationoftheactivatedcomplexaccountsfortheactivation
energyofthereaction.
Wenowconsideramoredetailedcalculationofrateconstants
usingtransitionstatetheory(alsowidelyreferredtoasactivated
complextheory).
Transitionstatetheoryisanattempttoidentifytheprincipal
featuresgoverningthesizeofarateconstantintermsofamodel
oftheeventsthattakeplaceduringthereaction.

12. ■TransitionstatetheorypicturesareactionbetweenAandBas
proceedingthroughtheformationofanactivatedcomplex,C
‡
,ina
rapidpre-equilibrium.
Whenweexpressthepartialpressures,p
J,intermsofthemolar
concentrations,[J],byusingp
J=RT[J],theconcentrationof
activatedcomplexisrelatedtothe(dimensionless)equilibrium
constantby
Theactivatedcomplexfallsapartbyunimoleculardecayinto
products,P,witharateconstantk
‡
:
The Eyringequation

13. The Eyringequation
Itfollowsthat,
Ourtaskistocalculatetheunimolecularrateconstantk
‡
andthe
equilibriumconstantK
‡
.

14. The Eyringequation
(a)Therateofdecayoftheactivatedcomplex
Anactivatedcomplexcanformproductsifitpassesthroughthe
transitionstate,thearrangementtheatomsmustachieveinorder
toconverttoproducts.
Ifitsvibration-likemotionalongthereactioncoordinateoccurswitha
frequencyν,thenthefrequencywithwhichtheclusterofatoms
formingthecomplexapproachesthetransitionstateisalsoν.
Therefore,wesupposethattherateofpassageofthecomplex
throughthetransitionstateisproportionaltothevibrational
frequencyalongthereactioncoordinate,andwrite
Whereκisthetransmissioncoefficient.Intheabsenceof
informationtothecontrary,κisassumedtobeabout1.

15. The Eyringequation
(b)Theconcentrationoftheactivatedcomplex:
EquilibriumconstantK
‡
isgivenby;
The arethestandardmolarpartitionfunctionsandtheunitsofN
A
andthearemol
−1
,soK
‡
isdimensionless.
Accordingtothepartitionfunction,avibrationoftheactivated
complexC
‡
tipsitthroughthetransitionstate.Thepartitionfunction
forthisvibrationis;
where νis its frequency (the same frequency that determines k
‡
).

16. The Eyringequation
If theexponentialmaybeexpandedandthepartition
functionreducesto;
Wecanthereforewrite;
Wheredenotesthepartitionfunctionforalltheothermodesofthe
complex.TheconstantK
‡
istherefore;
withK
‡
akindofequilibriumconstant,butwithonevibrationalmode
ofC
‡
discarded.

17. The Eyringequation
(c)Therateconstant:
Wecannowcombineallthepartsofthecalculationinto
Atthisstagetheunknownfrequenciesνcanceland,afterwriting;
We obtain the Eyringequation:
intermsofthepartitionfunctionsofA,B,andC
‡
,
soinprinciplewenowhaveanexplicitexpressionforcalculatingthe
second-orderrateconstantforabimolecularreactionintermsofthe
molecularparametersforthereactantsandtheactivatedcomplexand
thequantityκ.

18. Thermodynamic aspects
Thestatisticalthermodynamicversionoftransitionstatetheoryrapidly
runsintodifficultiesbecauseonlyinsomecasesisanythingknown
aboutthestructureoftheactivatedcomplex.
However,theconceptsthatitintroduces,principallythatofan
equilibriumbetweenthereactantsandtheactivatedcomplex,have
motivatedamoregeneral,empiricalapproachinwhichtheactivation
processisexpressedintermsofthermodynamicfunctions.
(a)Activationparameters
Ifweacceptthatisanequilibriumconstant,wecanexpressitin
termsofaGibbsenergyofactivation,Δ
‡
G,throughthedefinition

19. Thermodynamic aspects
Thentherateconstantbecomes;
----------(1)
BecauseG=H−TS,theGibbsenergyofactivationcanbedivided
intoanentropyofactivation,Δ
‡
S,andanenthalpyofactivation,
Δ
‡
H,bywriting;Δ
‡
G=Δ
‡
H−TΔ
‡
S-----------(2)
Wheneqn(2)isusedineqn(1)andκisabsorbedintotheentropy
term,weobtain;
Theformaldefinitionofactivationenergy, then
gives;E
a=Δ
‡
H+2RT,so

20. ItfollowsthattheArrheniusfactorAcanbeidentifiedas;
Theentropyofactivationisnegativebecausetworeactantspecies
cometogethertoformonespecies.
wecanidentifythatadditionalreductioninentropy,Δ
‡
S
steric,asthe
originofthestericfactorofcollisiontheory,andwrite
Thus,themorecomplexthestericrequirementsoftheencounter,the
morenegativethevalueofΔ
‡
S
steric,andthesmallerthevalueofP.
Gibbsenergies,enthalpies,entropies,volumes,andheatcapacities
ofactivationarewidelyusedtoreportexperimentalreactionrates,
especiallyfororganicreactionsinsolution.
Thermodynamic aspects

21. ■Thethermodynamicversionoftransitionstatetheorysimplifies
thediscussionofreactionsinsolution.
■Thestatisticalthermodynamictheoryisverycomplicatedtoapply
becausethesolventplaysaroleintheactivatedcomplex.Inthe
thermodynamicapproachwecombinetheratelaw
withthethermodynamicequilibriumconstant.
Then,
Ifk
o
2istherateconstantwhentheactivitycoefficientsare1(thatis,
k
o
2=k
‡
K),wecanwrite;
Reactions between ions

22. Atlowconcentrationstheactivitycoefficientscanbeexpressedin
termsoftheionicstrength,I,ofthesolutionbyusingtheDebye–
Hückellimitinglawintheform;
withA=0.509inaqueoussolutionat298K.Then
--------(1)
ThechargenumbersofAandBarez
Aandz
B,sothechargenumber
oftheactivatedcomplexisz
A+z
B;thez
Jarepositiveforcationsand
negativeforanions.
Equation(1)expressesthekineticsalteffect,thevariationofthe
rateconstantofareactionbetweenionswiththeionicstrengthofthe
solution.
Reactions between ions

23. Reactions between ions
Ifthereactantionshavethesamesign(asinareactionbetween
cationsorbetweenanions),thenincreasingtheionicstrengthbythe
additionofinertionsincreasestherateconstant.

24. The dynamics of molecular collisions
■Reactivecollisions:
Molecularbeamsallowustostudycollisionsbetweenmoleculesin
preselectedenergystates,andcanbeusedtodeterminethestates
oftheproductsofareactivecollision.
Detailedexperimentalinformationabouttheintimateprocessesthat
occurduringreactiveencounterscomesfrommolecularbeams,
especiallycrossedmolecularbeams.
Itispossibletostudythedependence
ofthesuccessofcollisionsonthese
variablesandtostudyhowtheyaffect
thepropertiesoftheoutcoming
productmolecules.

25. The dynamics of molecular collisions
Chemiluminescence:
■Onemethodforexaminingtheenergydistributionintheproducts
isinfraredchemiluminescence,inwhichvibrationallyexcited
moleculesemitinfraredradiationastheyreturntotheirground
states.
■Bystudyingtheintensitiesoftheinfraredemissionspectrum,the
populationsofthevibrationalstatesmaybedetermined
Laser-inducedfluorescence:
■Anothermethodmakesuseoflaser-inducedfluorescence.Inthis
technique,alaserisusedtoexciteaproductmoleculefroma
specificvibration-rotationlevel;theintensityofthefluorescencefrom
theupperstateismonitoredandinterpretedintermsofthe
populationoftheinitialvibration-rotationstate.

26. (a)Therateofdecayoftheactivatedcomplex
Noteveryoscillationalongthereactioncoordinatetakesthecomplexthroughthe
transitionstateandcentrifugaleffectofrotationsmightalsobeanimportantcontribution
tothebreakupofthecomplex,andinsomecasesthecomplexmightberotatingtoo
slowly,orrotatingrapidlybutaboutthewrongaxis.