CHBE 452 Lecture 15 Theory Of Activation Barriers

1

Last Time Found 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

2

Today Models

Polanyi Bond stretching + Linearization

Marcus Bond stretching + parabola

Blowers Masel Bond streching + Pauli repulsions

3

Polayni's Model

4

Ener

gy

E a

rBH

B RHr

BHr

RH

Figure 10.3 A diagram illustrating how an upward displacement of the B-H curve affects the activation energy when the B-R distance is fixed.

Ener

gy

rBH

B RHr

BHr

RH

Reactants

Products

Figure 10.4 A diagram illustrating a case where the activation energy is zero.

Derivation Of Polayni Equation

5

En

erg

y

rAB

Ea

Actual

PotentialLinear

Fit

Line 1

Line 2r

1r

2

BCAB

Figure 10.6 A linear approximation to the Polanyi diagram used to derive equation (10.11).

  

E E + Sl r - r (reactants)1 Reactant 1 ABC 1

E E + Sl r - r (products)2 product 2 2 ABC

(10.9)

(10.10)

Case Where Polayni Works (Over A Limited Range Of ΔH)

6

-20 -15 -10 -5 0 5 10 15 20

-20 -15 -10 -5 0 5 10 15 20

0

5

10

15

20

25

, kcal/molerH

, k

cal/

mo

lea

E

Figure 10.7 A plot of the activation barriers for the reaction R + H R RH + R with R, R = H, CH3, OH plotted as a function of the heat of reaction Hr.

Equation Does Not Work Over Wide Range Of H

7

-2.4 -4.8 -7.2 -9.6

-6

-4

-2

0

2

Log kac

, 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.

Seminov Approximation: Use Multiple Lines

8

-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.

 

Equation Does Not Work Over A Wide Data Set

9

-30 -20 -10 0 10 20 300

-30 -20 -10 0 10 20 300

0

5

10

15

20

25

30

, kcal/molerH

, kca

l/m

ole

aE

Figure 10.10 A Polanyi relationship for a series of reactions of the form RH + R R + HR. Data from Roberts and Steel[1994].

Key Prediction Stronger Bonds Are Harder To Break

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En

erg

y

rAB

Weak Bond

rAB

EaEa

Bond E

nerg

y

Strong Bond

Figure 10.8 A schematic of the curve crossing during the destruction of a weak bond and a strong one for the reaction AB + C A + BC.

Experiments Do Not Follow Predicted Trend

11

50 60 70 80 90 100 110 12030

35

40

45

50

55

60

65

70

75

, k

cal/

mo

lea

E

Predicted Trend

Experimental Trend

Figure 10.9 The activation barrier for the reaction X- +CH3X XCH3 +

X-. The numbers are from the calculations of Glukhoustev, Pross and Radam[1995].

Marcus: Why Curvature?

12

-2.4 -4.8 -7.2 -9.6

-6

-4

-2

0

2

Log kac

, 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.

Derivation: Fit Potentials To Parabolas Not Lines

13

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

14

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

15

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

16

-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

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0 50 100 150 200 250

rate

, se

c-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

18

-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

19

-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?

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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

21

rAB

rAB

rAB

E =0aEa

Medium distance Smaller Distance

InvertedRegion

Even Smaller Distance

Explains Why Marcus Better For Unimolecular Than Bimolecular

22

-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 Derivation

Fit potential energy surface to empirical equation.

Solve for saddle point energy. Only valid for bimolecular reaction.

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Derive Analytical Expression

24

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

25

-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 Ea

o = 10 kcal/mole.

EAO

Comparison Of Methods

26

-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

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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:

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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

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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

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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:

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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)

Summary

Polayni: fair approximation – good over a limited range of H

Marcus: Great for unimolecular Blowers-Masel Great for bimolecular

Only small differences between methods unless magnitude of heat of reaction is larger than 3-4 times Ea

0

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Query

What did you learn new in this lecture?

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