1

SF Chemical Kinetics

Lecture 5Microscopic theory of chemical

reaction kinetics

Microscopic theories of chemical reaction kinetics

bull A basic aim is to calculate the rate constant for a chemical reaction from first principles using fundamental physics

bull Any microscopic level theory of chemical reaction kinetics must result in the derivation of an expression for the rate constant that is consistent with the empirical Arrhenius equation

bull A microscopic model should furthermore provide a reasonable interpretation of the pre-exponential factor A and the activation energy EA in the Arrhenius equation

bull We will examine two microscopic models for chemical reactions We will examine two microscopic models for chemical reactions ndash The collision theoryndash The activated complex theory

bull The main emphasis will be on gas phase bimolecular reactions since reactions in the gas phase are the most simple reaction types

2

References for Microscopic Theory of Reaction Rates

bull Effect of temperature on reaction rateBurrows et al Chemistry3 Section 8 7 pp 383 389ndash Burrows et al Chemistry3 Section 87 pp383-389

bull Collision Theory Activated Complex Theoryndash Burrows et al Chemistry3 Section 88 pp390-395ndash Atkins de Paula Physical Chemistry 9th edition

Chapter 22 Reaction Dynamics Section 221 pp832-838pp

ndash Atkins de Paula Physical Chemistry 9th edition Chapter 22 Section224-225 pp 843-850

Collision theory of bimolecular gas phase reactions

bull We focus attention on gas phase reactions and assume that chemical reactivity is due to collisions between moleculesthat chemical reactivity is due to collisions between molecules

bull The theoretical approach is based on the kinetic theory of gasesbull Molecules are assumed to be hard structureless spheres Hence themodel neglects the discrete chemical structure of an individualmolecule This assumption is unrealistic

bull We also assume that no interaction between molecules until contactbull Molecular spheres maintain size and shape on collision Hence thecentres cannot come closer than a distance δ given by the sum ofthe molecular radii

bull The reaction rate will depend on two factors

bull the number of collisions per unit time (the collision frequency)bull the fraction of collisions having an energy greater than a certain

threshold energy E

3

Simple collision theory quantitative aspects

Products)()( ⎯rarr⎯+ kgBgA

Hard sphere reactantsMolecular structure anddetails of internal motionsuch as vibrations and rotationssuch as vibrations and rotationsignored

Two basic requirements dictate a collision event

bull One must have an AB encounter over a sufficiently short distance to allowreaction to occur

bull Colliding molecules must have sufficient energy of the correct type to overcome the energy barrier for reaction A threshold energy E is requiredovercome the energy barrier for reaction A threshold energy E is required

Two basic quantities are evaluated using the Kinetic Theory of gases the collision frequency and the fraction of collisions that activate moleculesfor reactionTo evaluate the collision frequency we need a mathematical way to definewhether or not a collision occurs

A

Ar

Brδ

B

( )22BA rr +== ππδσ

The collision cross section σdefines when a collision occurs

Effective collisiondiameter

BA rr +=δArea = σ

The collision cross section for two molecules can be regarded to be the area within which the center of theprojectile molecule A must enter around the target molecule Bin order for a collision to occur

BA rr +δ

4

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=Tk

mvTk

mvvFBB 2

exp2

4)(223

2

ππ

Maxwell-Boltzmann velocity distribution function

bull The velocity distribution curvehas a very characteristic shape

bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediate

JC Maxwell1831-1879

)(vFspeeds

bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed

bull The shape of the Gaussian distribution curve changes as the temperature is raised

bull The maximum of the curve shifts tohigher speeds with increasing temperature and the curve becomes

Gas molecules exhibita spread or distributionof speeds

v

temperature and the curve becomesbroader as the temperatureincreases

bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature

The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB

rBAAB vnnZ 2δπ= Mean relative velocityUnits m2s-1

nj = number density of molecule j (units m-3)

Mean relative velocity evaluated via kinetic theory

( )= intinfin

dvvFvv

Average velocity of a gas moleculeMB distribution of velocitiesenables us to statistically estimate the spread of

l l l iti i

BA rr +=δ

( )

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=

int

Tkmv

TkmvvF

BB 2exp

24)(

2232

0

ππ

Maxwell-Boltzmann velocityDistribution function m

Tkv B

π8

= Mass ofmolecule

Some maths

molecular velocities in a gas

5

We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across

22BAr vvv +=

Bj m

Tkvπ8

=Tkv B8Hence the collision numberjmπ

μπv B

r =

Reduced mass

BA

BA

mmmm+

=μ

Hence the collision numberbetween unlike moleculescan be evaluated

rBAAB vnnZ 2δπ=

⎫⎧21 For collisions between like molecules vv 2=

BA

BBAAB

nZn

TknnZ

=⎭⎬⎫

⎩⎨⎧

=21

8μπ

σ

Collision frequencyfactor

vvr 2The number of collisions per unittime between a single A molecule and other AMolecules

2182

⎭⎬⎫

⎩⎨⎧

=A

BAA m

TknZπ

σ

Total number of collisionsbetween like molecules 21

2 822

2 ⎭⎬⎫

⎩⎨⎧

==A

BA

AAAA m

TknnZZπ

σWe divide by 2 to ensureThat each AA encounterIs not counted twice

E

Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value

Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue ε

( ) ⎥⎦

⎤⎢⎣

⎡minus=gt

TkF

B

exp εεε

6

The simple collision theory expression for the reaction rate R between unlikemolecules

⎥⎦

⎤⎢⎣

⎡minus=minus=

TknZn

dtdnR

BBA

A exp ε21

8

⎭⎬⎫

⎩⎨⎧

=μπ

σ TkZ B

The rate expression for aThe rate expression for abimolecular reaction between A and B A

A BdcR kc cdt

= minus =

A

A BA B

A A

A AA

E Nn nc cN N

dn dcNdt dt

ε=

= =

=

We introduce molar variables

Hence the SCT rate expression

expAA A B

dc ER ZN c cdt RT

⎡ ⎤= minus = minus⎢ ⎥⎣ ⎦Avogadro constant dt dt

The bimolecular rate constant forcollisions between unlike molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTETkNk

AB

BA

exp

exp821

μπσ

constant

The rate constant for bimolecular collisionsbetween like molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTE

mTkNk

AA

BA

exp

exp221

πσBoth of these

expressions aresimilar to the Arrhenius equation

CollisionFrequencyfactor

7

We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor

⎥⎦⎤

⎢⎣⎡minus=

RTEAk A

obs exp

⎥⎤

⎢⎡

⎬⎫

⎨⎧

=ETkNk B exp8

21

σ⎤⎡⎬⎫

⎨⎧ ETkNk B 2

21

AB encounters

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣minus

⎭⎬

⎩⎨=

RTEz

RTNk

AB

A

exp

expμπ

σ

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬

⎩⎨=

RTEz

RTmNk

AA

BA

exp

exp2π

σ

AA encounters

μπσ B

A

ABobs

kNA

ATAzA

8

=

==

mkNA

ATAzA

BA

AAobs

πσ 82

=

==

bull SCT predictsPre-exponentialfactorμ

that the pre-exponentialfactor should depend on temperaturebull The threshold energy and theactivation energy can also becompared 2

lnRTE

dTkd A= 2

2lnRT

RTEdT

kd +=

2 RTEEA +=Arrhenius

SCT

EEA congbull Activation energy exhibits

a weak T dependence

SCT a summarybull The major problem with SCT is that the threshold energy E is very

difficult to evaluate from first principlesbull The predictions of the collision theory can be critically evaluated by

comparing the experimental pre-exponential factor with that computed using SCTusing SCT

bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors

bull We can incorporate P into the SCTexpression for the rate constant

bull For many gas phase reactionsP is considerably less than unity

bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of

calcAAP exp=

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus=

RTEPzk

RTEPzk

AA

AB

exp

exp

the chemical nature of the reactants and productsbull What has gone wrong The SCT assumption of hard sphere collision

neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction

bull We need a better theory that takes molecular structure into account The activated complex theory does just that

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

2

References for Microscopic Theory of Reaction Rates

bull Effect of temperature on reaction rateBurrows et al Chemistry3 Section 8 7 pp 383 389ndash Burrows et al Chemistry3 Section 87 pp383-389

bull Collision Theory Activated Complex Theoryndash Burrows et al Chemistry3 Section 88 pp390-395ndash Atkins de Paula Physical Chemistry 9th edition

Chapter 22 Reaction Dynamics Section 221 pp832-838pp

ndash Atkins de Paula Physical Chemistry 9th edition Chapter 22 Section224-225 pp 843-850

Collision theory of bimolecular gas phase reactions

bull We focus attention on gas phase reactions and assume that chemical reactivity is due to collisions between moleculesthat chemical reactivity is due to collisions between molecules

bull The theoretical approach is based on the kinetic theory of gasesbull Molecules are assumed to be hard structureless spheres Hence themodel neglects the discrete chemical structure of an individualmolecule This assumption is unrealistic

bull We also assume that no interaction between molecules until contactbull Molecular spheres maintain size and shape on collision Hence thecentres cannot come closer than a distance δ given by the sum ofthe molecular radii

bull The reaction rate will depend on two factors

bull the number of collisions per unit time (the collision frequency)bull the fraction of collisions having an energy greater than a certain

threshold energy E

3

Simple collision theory quantitative aspects

Products)()( ⎯rarr⎯+ kgBgA

Hard sphere reactantsMolecular structure anddetails of internal motionsuch as vibrations and rotationssuch as vibrations and rotationsignored

Two basic requirements dictate a collision event

bull One must have an AB encounter over a sufficiently short distance to allowreaction to occur

bull Colliding molecules must have sufficient energy of the correct type to overcome the energy barrier for reaction A threshold energy E is requiredovercome the energy barrier for reaction A threshold energy E is required

Two basic quantities are evaluated using the Kinetic Theory of gases the collision frequency and the fraction of collisions that activate moleculesfor reactionTo evaluate the collision frequency we need a mathematical way to definewhether or not a collision occurs

A

Ar

Brδ

B

( )22BA rr +== ππδσ

The collision cross section σdefines when a collision occurs

Effective collisiondiameter

BA rr +=δArea = σ

The collision cross section for two molecules can be regarded to be the area within which the center of theprojectile molecule A must enter around the target molecule Bin order for a collision to occur

BA rr +δ

4

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=Tk

mvTk

mvvFBB 2

exp2

4)(223

2

ππ

Maxwell-Boltzmann velocity distribution function

bull The velocity distribution curvehas a very characteristic shape

bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediate

JC Maxwell1831-1879

)(vFspeeds

bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed

bull The shape of the Gaussian distribution curve changes as the temperature is raised

bull The maximum of the curve shifts tohigher speeds with increasing temperature and the curve becomes

Gas molecules exhibita spread or distributionof speeds

v

temperature and the curve becomesbroader as the temperatureincreases

bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature

The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB

rBAAB vnnZ 2δπ= Mean relative velocityUnits m2s-1

nj = number density of molecule j (units m-3)

Mean relative velocity evaluated via kinetic theory

( )= intinfin

dvvFvv

Average velocity of a gas moleculeMB distribution of velocitiesenables us to statistically estimate the spread of

l l l iti i

BA rr +=δ

( )

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=

int

Tkmv

TkmvvF

BB 2exp

24)(

2232

0

ππ

Maxwell-Boltzmann velocityDistribution function m

Tkv B

π8

= Mass ofmolecule

Some maths

molecular velocities in a gas

5

We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across

22BAr vvv +=

Bj m

Tkvπ8

=Tkv B8Hence the collision numberjmπ

μπv B

r =

Reduced mass

BA

BA

mmmm+

=μ

Hence the collision numberbetween unlike moleculescan be evaluated

rBAAB vnnZ 2δπ=

⎫⎧21 For collisions between like molecules vv 2=

BA

BBAAB

nZn

TknnZ

=⎭⎬⎫

⎩⎨⎧

=21

8μπ

σ

Collision frequencyfactor

vvr 2The number of collisions per unittime between a single A molecule and other AMolecules

2182

⎭⎬⎫

⎩⎨⎧

=A

BAA m

TknZπ

σ

Total number of collisionsbetween like molecules 21

2 822

2 ⎭⎬⎫

⎩⎨⎧

==A

BA

AAAA m

TknnZZπ

σWe divide by 2 to ensureThat each AA encounterIs not counted twice

E

Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value

Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue ε

( ) ⎥⎦

⎤⎢⎣

⎡minus=gt

TkF

B

exp εεε

6

The simple collision theory expression for the reaction rate R between unlikemolecules

⎥⎦

⎤⎢⎣

⎡minus=minus=

TknZn

dtdnR

BBA

A exp ε21

8

⎭⎬⎫

⎩⎨⎧

=μπ

σ TkZ B

The rate expression for aThe rate expression for abimolecular reaction between A and B A

A BdcR kc cdt

= minus =

A

A BA B

A A

A AA

E Nn nc cN N

dn dcNdt dt

ε=

= =

=

We introduce molar variables

Hence the SCT rate expression

expAA A B

dc ER ZN c cdt RT

⎡ ⎤= minus = minus⎢ ⎥⎣ ⎦Avogadro constant dt dt

The bimolecular rate constant forcollisions between unlike molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTETkNk

AB

BA

exp

exp821

μπσ

constant

The rate constant for bimolecular collisionsbetween like molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTE

mTkNk

AA

BA

exp

exp221

πσBoth of these

expressions aresimilar to the Arrhenius equation

CollisionFrequencyfactor

7

We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor

⎥⎦⎤

⎢⎣⎡minus=

RTEAk A

obs exp

⎥⎤

⎢⎡

⎬⎫

⎨⎧

=ETkNk B exp8

21

σ⎤⎡⎬⎫

⎨⎧ ETkNk B 2

21

AB encounters

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣minus

⎭⎬

⎩⎨=

RTEz

RTNk

AB

A

exp

expμπ

σ

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬

⎩⎨=

RTEz

RTmNk

AA

BA

exp

exp2π

σ

AA encounters

μπσ B

A

ABobs

kNA

ATAzA

8

=

==

mkNA

ATAzA

BA

AAobs

πσ 82

=

==

bull SCT predictsPre-exponentialfactorμ

that the pre-exponentialfactor should depend on temperaturebull The threshold energy and theactivation energy can also becompared 2

lnRTE

dTkd A= 2

2lnRT

RTEdT

kd +=

2 RTEEA +=Arrhenius

SCT

EEA congbull Activation energy exhibits

a weak T dependence

SCT a summarybull The major problem with SCT is that the threshold energy E is very

difficult to evaluate from first principlesbull The predictions of the collision theory can be critically evaluated by

comparing the experimental pre-exponential factor with that computed using SCTusing SCT

bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors

bull We can incorporate P into the SCTexpression for the rate constant

bull For many gas phase reactionsP is considerably less than unity

bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of

calcAAP exp=

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus=

RTEPzk

RTEPzk

AA

AB

exp

exp

the chemical nature of the reactants and productsbull What has gone wrong The SCT assumption of hard sphere collision

neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction

bull We need a better theory that takes molecular structure into account The activated complex theory does just that

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

3

Simple collision theory quantitative aspects

Products)()( ⎯rarr⎯+ kgBgA

Hard sphere reactantsMolecular structure anddetails of internal motionsuch as vibrations and rotationssuch as vibrations and rotationsignored

Two basic requirements dictate a collision event

bull One must have an AB encounter over a sufficiently short distance to allowreaction to occur

bull Colliding molecules must have sufficient energy of the correct type to overcome the energy barrier for reaction A threshold energy E is requiredovercome the energy barrier for reaction A threshold energy E is required

Two basic quantities are evaluated using the Kinetic Theory of gases the collision frequency and the fraction of collisions that activate moleculesfor reactionTo evaluate the collision frequency we need a mathematical way to definewhether or not a collision occurs

A

Ar

Brδ

B

( )22BA rr +== ππδσ

The collision cross section σdefines when a collision occurs

Effective collisiondiameter

BA rr +=δArea = σ

The collision cross section for two molecules can be regarded to be the area within which the center of theprojectile molecule A must enter around the target molecule Bin order for a collision to occur

BA rr +δ

4

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=Tk

mvTk

mvvFBB 2

exp2

4)(223

2

ππ

Maxwell-Boltzmann velocity distribution function

bull The velocity distribution curvehas a very characteristic shape

bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediate

JC Maxwell1831-1879

)(vFspeeds

bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed

bull The shape of the Gaussian distribution curve changes as the temperature is raised

bull The maximum of the curve shifts tohigher speeds with increasing temperature and the curve becomes

Gas molecules exhibita spread or distributionof speeds

v

temperature and the curve becomesbroader as the temperatureincreases

bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature

The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB

rBAAB vnnZ 2δπ= Mean relative velocityUnits m2s-1

nj = number density of molecule j (units m-3)

Mean relative velocity evaluated via kinetic theory

( )= intinfin

dvvFvv

Average velocity of a gas moleculeMB distribution of velocitiesenables us to statistically estimate the spread of

l l l iti i

BA rr +=δ

( )

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=

int

Tkmv

TkmvvF

BB 2exp

24)(

2232

0

ππ

Maxwell-Boltzmann velocityDistribution function m

Tkv B

π8

= Mass ofmolecule

Some maths

molecular velocities in a gas

5

We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across

22BAr vvv +=

Bj m

Tkvπ8

=Tkv B8Hence the collision numberjmπ

μπv B

r =

Reduced mass

BA

BA

mmmm+

=μ

Hence the collision numberbetween unlike moleculescan be evaluated

rBAAB vnnZ 2δπ=

⎫⎧21 For collisions between like molecules vv 2=

BA

BBAAB

nZn

TknnZ

=⎭⎬⎫

⎩⎨⎧

=21

8μπ

σ

Collision frequencyfactor

vvr 2The number of collisions per unittime between a single A molecule and other AMolecules

2182

⎭⎬⎫

⎩⎨⎧

=A

BAA m

TknZπ

σ

Total number of collisionsbetween like molecules 21

2 822

2 ⎭⎬⎫

⎩⎨⎧

==A

BA

AAAA m

TknnZZπ

σWe divide by 2 to ensureThat each AA encounterIs not counted twice

E

Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value

Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue ε

( ) ⎥⎦

⎤⎢⎣

⎡minus=gt

TkF

B

exp εεε

6

The simple collision theory expression for the reaction rate R between unlikemolecules

⎥⎦

⎤⎢⎣

⎡minus=minus=

TknZn

dtdnR

BBA

A exp ε21

8

⎭⎬⎫

⎩⎨⎧

=μπ

σ TkZ B

The rate expression for aThe rate expression for abimolecular reaction between A and B A

A BdcR kc cdt

= minus =

A

A BA B

A A

A AA

E Nn nc cN N

dn dcNdt dt

ε=

= =

=

We introduce molar variables

Hence the SCT rate expression

expAA A B

dc ER ZN c cdt RT

⎡ ⎤= minus = minus⎢ ⎥⎣ ⎦Avogadro constant dt dt

The bimolecular rate constant forcollisions between unlike molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTETkNk

AB

BA

exp

exp821

μπσ

constant

The rate constant for bimolecular collisionsbetween like molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTE

mTkNk

AA

BA

exp

exp221

πσBoth of these

expressions aresimilar to the Arrhenius equation

CollisionFrequencyfactor

7

We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor

⎥⎦⎤

⎢⎣⎡minus=

RTEAk A

obs exp

⎥⎤

⎢⎡

⎬⎫

⎨⎧

=ETkNk B exp8

21

σ⎤⎡⎬⎫

⎨⎧ ETkNk B 2

21

AB encounters

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣minus

⎭⎬

⎩⎨=

RTEz

RTNk

AB

A

exp

expμπ

σ

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬

⎩⎨=

RTEz

RTmNk

AA

BA

exp

exp2π

σ

AA encounters

μπσ B

A

ABobs

kNA

ATAzA

8

=

==

mkNA

ATAzA

BA

AAobs

πσ 82

=

==

bull SCT predictsPre-exponentialfactorμ

that the pre-exponentialfactor should depend on temperaturebull The threshold energy and theactivation energy can also becompared 2

lnRTE

dTkd A= 2

2lnRT

RTEdT

kd +=

2 RTEEA +=Arrhenius

SCT

EEA congbull Activation energy exhibits

a weak T dependence

SCT a summarybull The major problem with SCT is that the threshold energy E is very

difficult to evaluate from first principlesbull The predictions of the collision theory can be critically evaluated by

comparing the experimental pre-exponential factor with that computed using SCTusing SCT

bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors

bull We can incorporate P into the SCTexpression for the rate constant

bull For many gas phase reactionsP is considerably less than unity

bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of

calcAAP exp=

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus=

RTEPzk

RTEPzk

AA

AB

exp

exp

the chemical nature of the reactants and productsbull What has gone wrong The SCT assumption of hard sphere collision

neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction

bull We need a better theory that takes molecular structure into account The activated complex theory does just that

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

4

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=Tk

mvTk

mvvFBB 2

exp2

4)(223

2

ππ

Maxwell-Boltzmann velocity distribution function

bull The velocity distribution curvehas a very characteristic shape

bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediate

JC Maxwell1831-1879

)(vFspeeds

bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed

bull The shape of the Gaussian distribution curve changes as the temperature is raised

bull The maximum of the curve shifts tohigher speeds with increasing temperature and the curve becomes

Gas molecules exhibita spread or distributionof speeds

v

temperature and the curve becomesbroader as the temperatureincreases

bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature

The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB

rBAAB vnnZ 2δπ= Mean relative velocityUnits m2s-1

nj = number density of molecule j (units m-3)

Mean relative velocity evaluated via kinetic theory

( )= intinfin

dvvFvv

Average velocity of a gas moleculeMB distribution of velocitiesenables us to statistically estimate the spread of

l l l iti i

BA rr +=δ

( )

⎥⎦

⎤⎢⎣

⎡minus

⎭⎬⎫

⎩⎨⎧

=

int

Tkmv

TkmvvF

BB 2exp

24)(

2232

0

ππ

Maxwell-Boltzmann velocityDistribution function m

Tkv B

π8

= Mass ofmolecule

Some maths

molecular velocities in a gas

5

We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across

22BAr vvv +=

Bj m

Tkvπ8

=Tkv B8Hence the collision numberjmπ

μπv B

r =

Reduced mass

BA

BA

mmmm+

=μ

Hence the collision numberbetween unlike moleculescan be evaluated

rBAAB vnnZ 2δπ=

⎫⎧21 For collisions between like molecules vv 2=

BA

BBAAB

nZn

TknnZ

=⎭⎬⎫

⎩⎨⎧

=21

8μπ

σ

Collision frequencyfactor

vvr 2The number of collisions per unittime between a single A molecule and other AMolecules

2182

⎭⎬⎫

⎩⎨⎧

=A

BAA m

TknZπ

σ

Total number of collisionsbetween like molecules 21

2 822

2 ⎭⎬⎫

⎩⎨⎧

==A

BA

AAAA m

TknnZZπ

σWe divide by 2 to ensureThat each AA encounterIs not counted twice

E

Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value

Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue ε

( ) ⎥⎦

⎤⎢⎣

⎡minus=gt

TkF

B

exp εεε

6

The simple collision theory expression for the reaction rate R between unlikemolecules

⎥⎦

⎤⎢⎣

⎡minus=minus=

TknZn

dtdnR

BBA

A exp ε21

8

⎭⎬⎫

⎩⎨⎧

=μπ

σ TkZ B

The rate expression for aThe rate expression for abimolecular reaction between A and B A

A BdcR kc cdt

= minus =

A

A BA B

A A

A AA

E Nn nc cN N

dn dcNdt dt

ε=

= =

=

We introduce molar variables

Hence the SCT rate expression

expAA A B

dc ER ZN c cdt RT

⎡ ⎤= minus = minus⎢ ⎥⎣ ⎦Avogadro constant dt dt

The bimolecular rate constant forcollisions between unlike molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTETkNk

AB

BA

exp

exp821

μπσ

constant

The rate constant for bimolecular collisionsbetween like molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTE

mTkNk

AA

BA

exp

exp221

πσBoth of these

expressions aresimilar to the Arrhenius equation

CollisionFrequencyfactor

7

We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor

⎥⎦⎤

⎢⎣⎡minus=

RTEAk A

obs exp

⎥⎤

⎢⎡

⎬⎫

⎨⎧

=ETkNk B exp8

21

σ⎤⎡⎬⎫

⎨⎧ ETkNk B 2

21

AB encounters

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣minus

⎭⎬

⎩⎨=

RTEz

RTNk

AB

A

exp

expμπ

σ

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬

⎩⎨=

RTEz

RTmNk

AA

BA

exp

exp2π

σ

AA encounters

μπσ B

A

ABobs

kNA

ATAzA

8

=

==

mkNA

ATAzA

BA

AAobs

πσ 82

=

==

bull SCT predictsPre-exponentialfactorμ

that the pre-exponentialfactor should depend on temperaturebull The threshold energy and theactivation energy can also becompared 2

lnRTE

dTkd A= 2

2lnRT

RTEdT

kd +=

2 RTEEA +=Arrhenius

SCT

EEA congbull Activation energy exhibits

a weak T dependence

SCT a summarybull The major problem with SCT is that the threshold energy E is very

difficult to evaluate from first principlesbull The predictions of the collision theory can be critically evaluated by

comparing the experimental pre-exponential factor with that computed using SCTusing SCT

bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors

bull We can incorporate P into the SCTexpression for the rate constant

bull For many gas phase reactionsP is considerably less than unity

bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of

calcAAP exp=

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus=

RTEPzk

RTEPzk

AA

AB

exp

exp

the chemical nature of the reactants and productsbull What has gone wrong The SCT assumption of hard sphere collision

neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction

bull We need a better theory that takes molecular structure into account The activated complex theory does just that

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

5

We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across

22BAr vvv +=

Bj m

Tkvπ8

=Tkv B8Hence the collision numberjmπ

μπv B

r =

Reduced mass

BA

BA

mmmm+

=μ

Hence the collision numberbetween unlike moleculescan be evaluated

rBAAB vnnZ 2δπ=

⎫⎧21 For collisions between like molecules vv 2=

BA

BBAAB

nZn

TknnZ

=⎭⎬⎫

⎩⎨⎧

=21

8μπ

σ

Collision frequencyfactor

vvr 2The number of collisions per unittime between a single A molecule and other AMolecules

2182

⎭⎬⎫

⎩⎨⎧

=A

BAA m

TknZπ

σ

Total number of collisionsbetween like molecules 21

2 822

2 ⎭⎬⎫

⎩⎨⎧

==A

BA

AAAA m

TknnZZπ

σWe divide by 2 to ensureThat each AA encounterIs not counted twice

E

Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value

Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue ε

( ) ⎥⎦

⎤⎢⎣

⎡minus=gt

TkF

B

exp εεε

6

The simple collision theory expression for the reaction rate R between unlikemolecules

⎥⎦

⎤⎢⎣

⎡minus=minus=

TknZn

dtdnR

BBA

A exp ε21

8

⎭⎬⎫

⎩⎨⎧

=μπ

σ TkZ B

The rate expression for aThe rate expression for abimolecular reaction between A and B A

A BdcR kc cdt

= minus =

A

A BA B

A A

A AA

E Nn nc cN N

dn dcNdt dt

ε=

= =

=

We introduce molar variables

Hence the SCT rate expression

expAA A B

dc ER ZN c cdt RT

⎡ ⎤= minus = minus⎢ ⎥⎣ ⎦Avogadro constant dt dt

The bimolecular rate constant forcollisions between unlike molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTETkNk

AB

BA

exp

exp821

μπσ

constant

The rate constant for bimolecular collisionsbetween like molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTE

mTkNk

AA

BA

exp

exp221

πσBoth of these

expressions aresimilar to the Arrhenius equation

CollisionFrequencyfactor

7

We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor

⎥⎦⎤

⎢⎣⎡minus=

RTEAk A

obs exp

⎥⎤

⎢⎡

⎬⎫

⎨⎧

=ETkNk B exp8

21

σ⎤⎡⎬⎫

⎨⎧ ETkNk B 2

21

AB encounters

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣minus

⎭⎬

⎩⎨=

RTEz

RTNk

AB

A

exp

expμπ

σ

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬

⎩⎨=

RTEz

RTmNk

AA

BA

exp

exp2π

σ

AA encounters

μπσ B

A

ABobs

kNA

ATAzA

8

=

==

mkNA

ATAzA

BA

AAobs

πσ 82

=

==

bull SCT predictsPre-exponentialfactorμ

that the pre-exponentialfactor should depend on temperaturebull The threshold energy and theactivation energy can also becompared 2

lnRTE

dTkd A= 2

2lnRT

RTEdT

kd +=

2 RTEEA +=Arrhenius

SCT

EEA congbull Activation energy exhibits

a weak T dependence

SCT a summarybull The major problem with SCT is that the threshold energy E is very

difficult to evaluate from first principlesbull The predictions of the collision theory can be critically evaluated by

comparing the experimental pre-exponential factor with that computed using SCTusing SCT

bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors

bull We can incorporate P into the SCTexpression for the rate constant

bull For many gas phase reactionsP is considerably less than unity

bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of

calcAAP exp=

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus=

RTEPzk

RTEPzk

AA

AB

exp

exp

the chemical nature of the reactants and productsbull What has gone wrong The SCT assumption of hard sphere collision

neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction

bull We need a better theory that takes molecular structure into account The activated complex theory does just that

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

6

The simple collision theory expression for the reaction rate R between unlikemolecules

⎥⎦

⎤⎢⎣

⎡minus=minus=

TknZn

dtdnR

BBA

A exp ε21

8

⎭⎬⎫

⎩⎨⎧

=μπ

σ TkZ B

The rate expression for aThe rate expression for abimolecular reaction between A and B A

A BdcR kc cdt

= minus =

A

A BA B

A A

A AA

E Nn nc cN N

dn dcNdt dt

ε=

= =

=

We introduce molar variables

Hence the SCT rate expression

expAA A B

dc ER ZN c cdt RT

⎡ ⎤= minus = minus⎢ ⎥⎣ ⎦Avogadro constant dt dt

The bimolecular rate constant forcollisions between unlike molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTETkNk

AB

BA

exp

exp821

μπσ

constant

The rate constant for bimolecular collisionsbetween like molecules

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=

RTEz

RTE

mTkNk

AA

BA

exp

exp221

πσBoth of these

expressions aresimilar to the Arrhenius equation

CollisionFrequencyfactor

7

We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor

⎥⎦⎤

⎢⎣⎡minus=

RTEAk A

obs exp

⎥⎤

⎢⎡

⎬⎫

⎨⎧

=ETkNk B exp8

21

σ⎤⎡⎬⎫

⎨⎧ ETkNk B 2

21

AB encounters

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣minus

⎭⎬

⎩⎨=

RTEz

RTNk

AB

A

exp

expμπ

σ

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬

⎩⎨=

RTEz

RTmNk

AA

BA

exp

exp2π

σ

AA encounters

μπσ B

A

ABobs

kNA

ATAzA

8

=

==

mkNA

ATAzA

BA

AAobs

πσ 82

=

==

bull SCT predictsPre-exponentialfactorμ

that the pre-exponentialfactor should depend on temperaturebull The threshold energy and theactivation energy can also becompared 2

lnRTE

dTkd A= 2

2lnRT

RTEdT

kd +=

2 RTEEA +=Arrhenius

SCT

EEA congbull Activation energy exhibits

a weak T dependence

SCT a summarybull The major problem with SCT is that the threshold energy E is very

difficult to evaluate from first principlesbull The predictions of the collision theory can be critically evaluated by

comparing the experimental pre-exponential factor with that computed using SCTusing SCT

bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors

bull We can incorporate P into the SCTexpression for the rate constant

bull For many gas phase reactionsP is considerably less than unity

bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of

calcAAP exp=

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus=

RTEPzk

RTEPzk

AA

AB

exp

exp

the chemical nature of the reactants and productsbull What has gone wrong The SCT assumption of hard sphere collision

neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction

bull We need a better theory that takes molecular structure into account The activated complex theory does just that

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

7

We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor

⎥⎦⎤

⎢⎣⎡minus=

RTEAk A

obs exp

⎥⎤

⎢⎡

⎬⎫

⎨⎧

=ETkNk B exp8

21

σ⎤⎡⎬⎫

⎨⎧ ETkNk B 2

21

AB encounters

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣minus

⎭⎬

⎩⎨=

RTEz

RTNk

AB

A

exp

expμπ

σ

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus

⎭⎬

⎩⎨=

RTEz

RTmNk

AA

BA

exp

exp2π

σ

AA encounters

μπσ B

A

ABobs

kNA

ATAzA

8

=

==

mkNA

ATAzA

BA

AAobs

πσ 82

=

==

bull SCT predictsPre-exponentialfactorμ

that the pre-exponentialfactor should depend on temperaturebull The threshold energy and theactivation energy can also becompared 2

lnRTE

dTkd A= 2

2lnRT

RTEdT

kd +=

2 RTEEA +=Arrhenius

SCT

EEA congbull Activation energy exhibits

a weak T dependence

SCT a summarybull The major problem with SCT is that the threshold energy E is very

difficult to evaluate from first principlesbull The predictions of the collision theory can be critically evaluated by

comparing the experimental pre-exponential factor with that computed using SCTusing SCT

bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors

bull We can incorporate P into the SCTexpression for the rate constant

bull For many gas phase reactionsP is considerably less than unity

bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of

calcAAP exp=

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎤

⎢⎣⎡minus=

RTEPzk

RTEPzk

AA

AB

exp

exp

the chemical nature of the reactants and productsbull What has gone wrong The SCT assumption of hard sphere collision

neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction

bull We need a better theory that takes molecular structure into account The activated complex theory does just that

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

8

⎥⎤

⎢⎡=

⎥⎦⎤

⎢⎣⎡minus=

EPzk

RTEPzk AB

exp

exp

⎥⎦⎤

⎢⎣⎡minus

⎭⎬⎫

⎩⎨⎧

=RTETkNk B

Aexp8

21

μπσ

AB encounters

Energy criterionTransportpropertySummary of SCT

⎥⎦⎢⎣minus=

RTPzk AA exp

⎥⎦⎤

⎢⎣⎡minus=

⎭⎩

RTEzAB

exp

⎥⎤

⎢⎡minus⎬

⎫⎨⎧

=ETkNk B

Aexp2

21

σ

AA encounters

Steric factor(Orientation requirement)

Weaknessesbull No way to compute P from molecularparameters

bull No way to compute E from first principlesTheory not quantitative or predictive

⎥⎦⎤

⎢⎣⎡minus=

⎥⎦⎢⎣⎭⎬

⎩⎨

RTEz

RTmNk

AA

A

exp

exp2π

σ h ory not quant tat or pr ct StrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates

bull Provides useful guide to upper limits for rateconstant k

Henry Eyring1901-1981

Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

9

Potential energy surfacebull Can be constructed from experimental measurements or from

Molecular Orbital calculations semi-empirical methodshelliphellip

Various trajectories through the potential energy surface

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

10

[ ] CABABCBCA +rarrrarr+

Potential energy hypersurface for chemical reaction between atom and diatomic molecule

Reading reaction progress on PE hypersurface

AEAE

00UΔ

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

11

Transition state theory (TST) or activated complex theory (ACT)

bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms

bull They form a loose structure ABDagger of hi h t ti l ll d th

Transition stateActivated complex

high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D

bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step

Ea

Ea

Ener

gybull The potential energy falls as the

atoms rearrange in the cluster and finally reaches the value for the products

bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step

Reaction coordinate

A + BC + D

Transition state theory

bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)

bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule

bull When far apart the potential energy is the sum of the values for H andF2

bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FFndash The FF bond starts to lengthen

bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerstronger and the F F bond becomes longer and weakerndash The atoms enter the region of the activated complex

bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

12

Thermodynamic approachbull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an

equilibrium constant designated KDagger and decomposes to products with rate constant kDagger

KDagger kDagger

A + B activated complex ABDagger products where KDagger][AB

=∓

A + B activated complex ABDagger products where KDagger

bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]

bull Compare this expression to the rate law rate of formation of products = kr [A][B]

bull Hence the rate constant kr = kDagger KDagger

bull The Gibbs energy for the process is given by ∆Dagger G = minusRTln (KDagger ) and so KDagger = exp(minus ∆Dagger GRT)

k kDagger ( (∆Dagger ∆Dagger ) )

[A][B]

bull Hence rate constant kr = kDagger exp (minus(∆Dagger H minus T∆Dagger S)RT) bull Hence kr = kDagger exp(∆Dagger SR) exp(minus ∆Dagger HRT)bull This expression has the same form as the Arrhenius expression

ndash The activation energy Ea relates to ∆Dagger Hndash Pre-exponential factor A = kDagger exp(∆Dagger SR) ndash The steric factor P can be related to the change in disorder at the transition state

Statistical thermodynamic approachbull The activated complex can form products if it passes though the

transition state ABDagger

bull The equilibrium constant KDagger can be derived from statistical mechanics )

RTΔE

-exp(q

=K 0AB∓Dagger

bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]

ndash q is the partition function for each speciesndash ∆E0 (kJ mol-1)is the difference in internal

energy between A B and ABDagger at T=0

bull It can be shown that the rate

)RT

p(qq BA

It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical

vibrational motion has been resolved out from quantities KDagger and qABDagger

ndash v cancels out from the equationndash k = Boltzmann constant h = Planckrsquos

constant

)RTΔE-

exp(qq

qh

kT = k Hence

Kh

kT= k

0

BA

ABr

r

∓

Dagger

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

13

Statistical thermodynamic approachbull Can determine partition functions qA and qB from spectroscopic

measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)N d t p stul t st u tu f th ti t d mpl x nd d t min bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + Hndash In more complex cases may use mixture of calculated and experimental

parametersbull Potential energy surface 3-D plot of the energy of all possible

arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition stateposition of the transition state

bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure

Example of a potential energy surfacebull Hydrogen atom exchange reaction

HA + HBHC HAHB + HC

bull Atoms constrained to be in a straight line (collinear) HA HB HC

P h C l h ll d

MolHAHB

bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley

bull Paths A and B go over much more difficult routes through regions of high energy

bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states

(translational and vibrational) lead to the most rapid reaction

Diagramwwwoupcoukpowerpointbtatkins

MolHBHC

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

14

Advantages of transition state theorybull Provides a complete description of the nature of the reaction including

ndash the changes in structure and the distribution of energy through the transition state

ndash the origin of the pre-exponential factor A with units t-1 that derive from frequency or velocity

ndash the meaning of the activation energy Eag gy abull Rather complex fundamental theory can be expressed in an easily

understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate

bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases

bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state

bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the

t iti t t ( l t ff t )transition state (eg solvent effects)Disadvantage

bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need

experimental values

A (g) + B(g) X ProductsKc

ν

Bimolecular reaction

Basic assumption activated complex X is treated as a thermodynamic quantity in equilibrium with the reactants

frequency of decomposition

Quantitative Activated Complex Theory (ACT)

of activated complexEquilibrium constant forreactanttransition statetransformation

Reaction rate R depends on the frequency ofdecomposition and concentration of activatedcomplexeslowastlowast= xR ν

cA B

xKc c

lowastlowast =

abKR clowastlowast=ν

A BR kc c=

bimolecular reaction

Reaction molecularity m number ofmolecules which come together to formthe activated complexm = 1 unimolecular reactionm = 2 bimolecular reaction

abKx clowastlowast =

lowastlowast= cKk νrateconstant

This fundamental expression must now beevaluated

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

15

ideal gas or for reactions in solution 0 )( nC cKK Δlowastlowast =

1 molL-1

Change in molesfor reactant TStransformation

thermodynamicequilibrium constant

We assume that the TS

Can be proved rigorouslyFrom Statistical Mechanics

121 minus=minus=Δ lowastn0c

KKC

lowastlowast =

⎟⎟⎞

⎜⎜⎛

==lowast

lowastlowastlowast0

KKk C νν

decomposes with a frequency given by

hTkB=lowastν

h = Planckrsquos constant = 663x10-34JskB = Boltzmann constant = lowast

lowastlowastlowast⎟⎟⎞

⎜⎜⎛ KKk νκνκ

k has units Lmol-1s-1

⎟⎠

⎜⎝

0cCB138x10-23 JK-1

For T = 298 K112106 minuslowast timescong sν

This is of the correctorder of magnitude fora molecular vibrationfrequency

Not all transition states go and form products

lowast

⎭⎬⎫

⎩⎨⎧=

⎟⎟⎠

⎜⎜⎝

==

Khc

Tk

cKk

B

C

0

0

κ

νκνκ

κ = transmission coefficient 0 lt κ lt 1

We relate K to the Gibbs energy of activation ΔG lowastlowastminus=Δ KRTG ln0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus=

lowastlowast

RTGK

0

exp⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δminus

⎭⎬⎫

⎩⎨⎧=

lowast

RTG

hcTkk B

0

0 expκ⎥⎦⎢⎣ RT⎥⎦⎢⎣⎭⎩ RThc

Eyring equation fundamental ACT result lowastlowastlowastΔminusΔ=Δ 000 STHG

enthalpy ofactivation

entropy ofactivation

⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

⎥⎦⎤

⎢⎣⎡minus=

RTEAk Aexp We obtain a useful

interpretation foractivation energy andpre-exponential factor

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln

16

⎟⎠⎞

⎜⎝⎛=

dTkdRTEA

ln2⎥⎦

⎤⎢⎣

⎡ Δminus

⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ⎭⎬⎫

⎩⎨⎧=

lowastlowast

RTH

RS

hcTkk B

00

0 expexpκ

Relating ACT parameters and Arrhenius Parameters

⎠⎝ ⎦⎣⎥⎦⎢⎣⎭⎩

lowastlowastlowast

lowast

Δ+Δ=Δ

+Δ=000

0

VPUHRTUEA

internal energy of activation

volume of activationcondensed phases

m = 2 bimolecular reaction

⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎠⎞

⎜⎝⎛=

lowast

RTE

RS

hcTkk AB exp2exp

0

0κ

bimolecular gasphase reaction

pre-exponential factor A

( ) RTRTmRTnVPV

minus=minus=Δ=Δ

congΔlowastlowast

lowast

10

0

0

ideal gases reaction molecularity

m = 2 bimolecular reaction

mRTEH A minus=Δ lowast0

m = 1 condensed phases unimoleculargas phase reactionsm = 2 bimolecular gas phase reactions

p

( ) ⎥⎦⎤

⎢⎣⎡minus⎥

⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RTE

RSm

chTkk A

mB

m expexp0

10κ

ΔS0 explained in f h i l i l

m = molecularity

ACT interpretation of Arrhenius Equation

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast

minus RSm

chTkA m

B0

10expκ R

ES Aminus=

kln

1Pre-exponential factor related to entropy

terms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS

Tof activation (difference in entropy betweenreactants and activated complex

( ) ⎥⎦

⎤⎢⎣

⎡ Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛==

lowast

minus RSm

chTkPZA m

B0

10expκ

collision theory positive1Pnegative1

01

0

0

0

lowast

lowast

lowast

Δgt

Δlt

congΔcong

SSPSP

steric factor TS less ordered thanreactants

TS more orderedthan reactants

Aln