ChE 553 Lecture 25 Theory Of Activation

Barriers

1

Objective

• Describe more quantitative models of activation barriers– Marcus– Blowers-Masel

• Show one can get reasonable estimates of activation barriers

2

Review: Barriers To Reaction Are Caused By

• Uphill reactions

• Bond stretching and distortion.

• Orbital distortion due to Pauli repulsions.

• Quantum effects.

• Special reactivity of excited states.

3

Polayni’s Model

• Assumptions– Bond stretching dominates– Ignore Pauli repulsions, quantum effects– Linearize energy vs bond stretch

• Result Ea varies linearly with heat of reaction

– Only fair approximation to data

4

E = E + Ha ao

P r

-2.4 -4.8 -7.2 -9.6

-6

-4

-2

0

2

Lo

g kac

, kcal/molerH

MarcusEquation

Seminov Approximation: Use Multiple Lines

5

-40 -20 0 20 400

0

10

20

30

40

Heat Of Reaction, kcal/mole

Act

ivat

ion

Bar

rier,

kcal

/mol

e Seminov

-100 -50 0 50 1000-20

0

20

40

60

80

100

Heat Of Reaction, kcal/moleA

ctiv

atio

n B

arrie

r, kc

al/m

ole Seminov

Figure 11.11 A comparison of the activation energies of a number of hydrogen transfer reactions to those predicted by the Seminov relationships, equations (11.33) and (11.34)

 

Figure 11.12 A comparison of the activation energies of 482 hydrogen transfer reactions to those predicted by the Seminov relationships, over a wider range of energies.

 

Marcus Model

• Assumptions– Bond stretching dominates– Ignore Pauli repulsions, quantum effects– Parabolic energy vs bond stretch

• Prediction

6

E 1H

4EEA

r

a0

2

a0

-2.4 -4.8 -7.2 -9.6

-6

-4

-2

0

2

Log kac

, kcal/molerH

MarcusEquation

Derivation: Fit Potentials To Parabolas Not Lines

7

Figure 10.13 An approximation to the change in the potential energy surface which occurs when Hr changes.

Ener

gy

rX

E a

ApproximationActual

Potential

r1

r2

E1

E2

Eleft E

right

1

Derivation

8

Figure 10.13 An approximation to the change in the potential energy surface which occurs when Hr changes.

Ene

rgy

rX

E a

ApproximationActual

Potential

r1

r2

E1

E2

Eleft

Eright

1

E (r ) SS (r r ) Eleft X 1 X 1

21

(10.21)

E (r ) SS (r r ) H Eright X 2 X 22

r 2

(10.22)

Solving

9

r22

2

‡

212

1‡

1H+E+)r(rSS=E+)r(rSS

SS SS1 2

Result

E 1H

4EE wA

r

a0

2

a0

r

(10.23)

(10.24)

(10.31)

E 1H

4EEA

r

a0

2

a0

(10.33)

Qualitative Features Of The Marcus Equation

10

-2.4 -4.8 -7.2 -9.6

-6

-4

-2

0

2L

og k

ac

, kcal/molerH

MarcusEquation

Figure 10.11 A Polanyi plot for the enolization of NO2(C6H4)O(CH2)2COCH3. Data of Hupke and Wu[1977]. Note Ln (kac) is

proportional to Ea.

Marcus Is Great For Unimolecular Reactions

11

0 50 100 150 200 250

rate

, sec-1

MarcusEquation

Data

105

106

107

108

109

1010

1011

H , kcal/moler

Figure 10.17 The activation energy for intramolecular electron transfer across a spacer molecule plotted as a function of the heat of reaction. Data of Miller et. al., J. Am. Chem. Soc., 106 (1984) 3047.

Marcus Not As Good For Bimolecular Reactions

12

-25 0 25 50 75 100 125

rate

, lit

mol s

ec

-1

MarcusEquation

Data

105

106

107

108

109

1010

1011

H , kcal/moler

-1

Figure 10.18 The activation energy for florescence quenching of a series of molecules in acetonitrite plotted as a function of the heat of reaction. Data of Rehm et. al., Israel J. Chem. 8 (1970) 259.

Comparison To Wider Bimolecular Data Set

13

-100 -50 0 50 1000-20

0

20

40

60

80

100

Heat Of Reaction, kcal/mole

Act

ivat

ion

Bar

rier,

kca

l/mol

e

Marcus

Blowers Masel

Figure 10.29 A comparison of the barriers computed from Blowers and Masel's model to barriers computed from the Marcus equation and to data for a series of reactions of the form R + HR1 RH + R1 with wO = 100 kcal/mole and = 10 kcal/mole.

EAO

Why Inverted Behavior?

14

rAB

rAB

rAB

E =0aEa

Medium distance Smaller Distance

InvertedRegion

Even Smaller Distance

At small distances bonds need to stretch to to TS.

Limitations Of The Marcus Equation

• Assumes that the bond distances in the molecule at the transition state does not change as the heat of reaction changes– Good assumption for

unimolecular reactions – In bimolecular reactions

bonds can stretch to lower barriers

15

rAB

rAB

rAB

E =0aEa

Medium distance Smaller Distance

InvertedRegion

Even Smaller Distance

Explains Why Marcus Better For Unimolecular Than Bimolecular

16

-25 0 25 50 75 100 125

rate

, lit

mol s

ec

-1

MarcusEquation

Data

105

106

107

108

109

1010

1011

H , kcal/moler

-1

0 50 100 150 200 250

rate

, sec-1

MarcusEquation

Data

105

106

107

108

109

1010

1011

H , kcal/moler

Blowers-Masel Equation

• Assumptions– Bimolecular reaction– Pauli repulsions important– Include bond stretching but ignore quantum

limitations

17

Blowers-Masel Derivation

• Fit potential energy surface to empirical equation.

• Solve for saddle point energy.

• Only valid for bimolecular reaction.

18

V E E VCC CH CC CC CH CH Pauli ,

(10.52)

V , w exp ( ) 1 1 w

w exp ( ) 1 1

V exp

CC CH CC CC CC CCe 2

CC

CH CH CH CHe 2

O CC CC CH CH

(10.56)

Blowers-Masel Derivation

Approximate VE by:

19

Plot Of Potential

20

C-C distance, Angstroms

C-C

dis

tan

ce,

An

gstr

om

s

 

Figure 10.28 A potential energy surface calculated from equation (10.59) with wCC = 95 kcal/mole, wCH = 104 kcal/mole,

VP = 300 kcal/mole, qCC = 0.7, qCH = 0.5.

Derive Analytical Expression

21

E

0 When H E

w + 0.5 H V - 2w H

V 4 w + HWhen -1 H E

H When H E

a

r Ao

O r P O r2

P O2

rr A

o

r r Ao

/

/

/

4 1

4 1

4 1

2 2

w w wO CC CH / 2

Bond energies(10.63)

(10.64)

V ww E

w EP O

O AO

O AO

2

(10.65)

Only parameter

Comparison To Experiments

22

-100 -50 0 50 1000-20

0

20

40

60

80

100

Heat Of Reaction, kcal/mole

Act

ivat

ion

Bar

rier,

kca

l/mol

e

Marcus

Blowers Masel

Figure 10.29 A comparison of the barriers computed from Blowers and Masel's model to barriers computed from the Marcus equation and to data for a series of reactions of the form R + HR1 RH + R1 with wO = 100 kcal/mole and = 10 kcal/mole.

EAO

Comparison Of Methods

23

-100 -50 0 50 1000

0

50

100A

ctiv

atio

n E

nerg

y ,

kca

l/mole

Heat of Reaction, Kcal/mole

Blowers Masel

Marcus

Polanyi

Blowers Masel

MarcusPolanyi

Figure 10.32 A comparison of the Marcus Equation, The Polanyi relationship and the Blowers Masel approximation for =9 kcal/mole and wO = 120 kcal/mole.

Ea

o

Example: Activation Barriers Via The Blowers-Masel Approximation

24

Use a) the Blowers-Masel approximation b) the Marcus equation to estimate the activation barrier for the reaction H + CH3CH3 H2 + CH2CH3

Solution: Step 1 Calculate VP

VP is given by:

25

V ww E

w EP o

o A

o A

2

0

0

(11.F.2)

Where EA0 is the intrinsic barrier given in

Table 11.3. According to table 11.3 EA0=10

kcal/mole. One could calculate a value of wo from data in the CRC. However, I decided to assume a typical value for a C-H bond, i.e. 100 kcal/mole. I choose H r=-2 kcal/mole from example 11.C

Step 1 Continued

Substituting into equation (11.F.2) shows

26

V kcal molekcal mole kcal mole

kcal mole kcal molekcal moleP

2 100

100 10

100 10244 4/

/ /

/ /. /

(11.F.3)

Step 2: Plug Into Equation 11.F.1

27

E mole mole

mole mole mole

w Hmolea

p o r

100kcal 0.5 2kcal

244.4kcal 2 100kcal 2kcal

49.0Kcal

2

2 2 2/ * // / /

(V )/

(11.F.1)

E w HV w H

V w Ha o r

p o r

p o r

0 5

2

4

2

2 2 2. *

( )

By comparison the experimental value from Westley is 9.6 kcal/mole.

Solve Via The Marcus Equation:

28

E EH

Ea A

A

0

0

2

14

From the data above EA0 =10 kcal/mole,

H r =-2 kcal/mole. Plugging into equation (11.F.4)

E kcal mole

kcal mole

kcal molekcal molea

10 1

2

4 109 0

2

//

/. /

which is the same as the Blowers-Masel approximation.

(11.F.4)

Limitation Of The Model: Quantum Effects:

29

 

Figure 10.39 A hypothetical four-centered mechanism for H2/D2 exchange. The

dotted lines in the figure denotes mirror planes which are preserved during the reaction (see the text). This reaction is symmetry forbidden. 

Note one electron has to be spin up and spin down at the same time.

H H

D D

H H

D D

H H

D D

1

2

1

2

1

2

Consider H2 + D2 2 HD

30

H2

D2

HD HD

Can Reaction Occur?

31

No Net Force To Distort Orbitals

Net Force, but product is HD + H + D (i.e. two atoms)Such a reaction is 104 kcal/mole endothermic

Conservation Of Orbital Symmetry

• Quantum effect leading to activation barriers

• Orbital symmetry/sign conserved during concerted reactions– Sometimes bonds must break before new

bonds can form

32

Orbital Diagram

33

H H H H H H H H

D D D D D D D D

* * *

1

2

1

2

1

2

1

2

*

 

Figure 10.40 A schematic of the key molecular orbitals for the transition state of reaction (10.92). Positive atomic orbitals are depicted as open circles, negative orbitals are depicted as shaded circles. Symmetry forbidden reactions Woodward hoffman rules

Summary

• Bonds need to stretch or distort during reaction. It costs energy to stretch or distort bonds. Bond stretching and distortion is one of the major causes of barriers to reaction.

• In order to get molecules close enough to react, the molecules need to overcome Pauli repulsions (i.e., electron electron repulsions) and other steric effects. The Pauli repulsions are another major cause of barriers to reaction.

34

Summary Continued

• In certain special reactions, there are quantum effects which prevent the bonds in the reactants from converting smoothly from the reactants to the products. Quantum effects can produce extra barriers to reaction.

• There are also a few special cases where the reactants need to be promoted into an excited state before a reaction can occur. The excitation energy provides an additional barrier to reaction.

35

Summary Continued

Polayni: linear potential

Marcus: parabolic potential

Blowers-Masel – size of TST varies

Fails with quantum effects ( varies)

36

oAE