ChE 551 Lecture 19 Transition State Theory Revisited 1.

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ChE 551 Lecture 19 Transition State Theory Revisited 1

Transcript of ChE 551 Lecture 19 Transition State Theory Revisited 1.

Page 1: ChE 551 Lecture 19 Transition State Theory Revisited 1.

ChE 551 Lecture 19 Transition State Theory Revisited

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Page 2: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Last Time We Discussed Advanced Collision Theory:

Method Use molecular dynamics to

simulate the collisions Integrate using statistical

mechanics

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r r (v , E , , b , , v )

D v E , , b , , v dv dE d db d dvA BC A BC A BC BC A BC BC

A BC BC A BC BC BC A BC BC A BC BC BC

R

R R

(8.20)

Page 3: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Summary:

Key finding – we need energy and momentum to be correct to get reaction to happen.

Many molecules which are hot enough do not make it.

Today how does that affect transition state theory?

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Page 4: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Next: Review Arrhenius’ Model vs Transition State Theory

Arrhenius’ model: Consider two populations of molecules:

1. Hot molecules in the right configuration to react (i.e. molecules moving toward each other with the right velocity, impact parameter etc to react).

2. Molecules not hot enough or not moving together in the right configuration.

Assume concentration of hot reactive molecules in equilibrium with reactants.

Assume reaction occurs whenever hot molecules collide in right configuration to react.

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Page 5: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Can Derive Tolman’s Equation

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Where kABC is the rate constant for reaction (9.1), is Boltzmann’s constant; T is the absolute temperature; hP is Plank’s constant; qA is the microcanonical partition function per unit volume of the reactant A; qBC is the microcanonical partition function per unit volume of the reactant A; qBC is the microcanonical partition function per unit volume for the reactant BC; E+ is the average energy of the molecules which react and, q+ is the average partition function of the molecules which react, divided by the partition function for the translation of A toward BC.

Tolman’s Equation is almost exact.

+

+BA BC B

P A BC

k T qk = exp -E /k T

h q q

Page 6: ChE 551 Lecture 19 Transition State Theory Revisited 1.

A Comparison of Tolman’s Equation and Transition State Theory

Equation

Definitions

Tolman’s

E+=Average energy of the hot molecules before they react.q+=partition function for the hot molecules before they react. The partition function includes all of the normal modes except the mode bringing the species together.

TST E‡ = Saddle Point energy.

= partition function for molecules at the

saddle point in the potential energy

surface. The partition function includes all

of the normal modes of the AB-C complex

except the normal mode carrying the

species over the barrier.

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+ +B

A BCP A BC B

k T q Ek = exp -

h q q k T

‡ ‡B T

A BCP A BC B

k T q Ek = exp -

h q q k T

‡Tq

Page 7: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Approximate Derivation Of TST:

Let’s go back to the statistical mechanics definition of the partition functions.

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q

P gE E

T

VH ot

NH ot

NN

BN

exp

(9.40)

q

P gE E

T

VT S T

NT S T

NN

BN

e x p

(9.41)

Need to assume Phot = PTST (Distribution of hot molecules poised to react equals the distribution of states at the transition state).

Page 8: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Assumptions

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rBC

0.5

24 kcal/mole

Å

15 kcal/mole

rBC

prod

ucts

0.5

24 kcal/mole

Å

15 kcal/mole

9 kcal/mole

reactants

rABr

Ignores: Re-crossing trajectories Changes in q’s Dynamics

Figure 9.7 A recrossing trajectory.

Page 9: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Let Me Examine The Assumption That q’s Are Equal

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Rx

Ry

Figure 9.3 The minimum energy pathway for motion over the barrier as determined by the trajectory calculations in Chapter 8.

(9.38)

Assume! Vibration q = translation q

q q q qHotX Y everything else _

Page 10: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Additional Assumptions In Conventional State Theory

All the in equation (9.40) are 0 for states with energies below the barrier and unity for states with energy above the barrier so that .

Motion along the Rx direction is pure translation.

Motion perpendicular to the Rx direction is pure vibration.

The are determined only by the properties of the transition state; and independent of shape at barrier.

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PNHot

P PNHot

NTST

PNHot

HNC HCN(9.46)

Page 11: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Example HNCHCN

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H

/ \

C N

(9.47)

q q q q q q qHNC t3

r2

sym asym BendA BendB

(9.48)

q = q q q q qT‡

t‡ 3

r‡ 2

sym‡

asym‡

BendB‡

(9.50)

TST assume that the bending mode is pure translation, the other modes are pure vibration.

Cancellation of error.

Page 12: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Successes Of Transition State Theory

Transition state theory generally gives preexponentials of the correct order of magnitude.

Transition state theory is able to relate barriers to the saddle point energy in the potential energy surface;

Transition state theory is able to consider isotope effects;

Transition state theory is able to make useful prediction in parallel reactions like reactions (7.27) and (7.29).

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Page 13: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Key Prediction

Experimentally, if one lowers the energy of the transition state, one usually lowers the activation energy for the reaction.

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Page 14: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Example Of Isotope Effects

Table 9.5 The relative rates of a series of reactions at 300K.

Reaction Experimental Relative Rate

CTST Relative Rate

D+H2 DH+H 6 12.3

H+H2 H2+H 4 6.7

D+D2 D2+D 2 1.7

H+D2 HD+D 1 1

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Page 15: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Isotope Effects Continued

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Table 9.6 The various contributions to the relative rate of the reactions in Table 9.5at 300K.

Reaction Symmetric Stretching Frequency,

cm-1

Bending Frequen

cy cm-1

Zero Point

Energy kJ/Mole

Relative Rate 300K

D + H2 DH + H

1732 924 4.79 12

H + H2 H2

+ H 2012 965 6.96 6.7

D + D2 D2 + D

1423 683 7.60 1.7

H + D2 HD + D

1730 737 10.16 1

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Limitations Of TST

0.05

0.1

0.2

0.5

1

2

5

0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10

Exp

erim

ent

Transition State Theory

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Correction to collision theory.

Page 17: ChE 551 Lecture 19 Transition State Theory Revisited 1.

TST ignores Tunneling

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Barrier

B A

Tail of WavefunctionExtends through barrier

Center of Wavefunctionbefore barrier

Wavefunction

Figure 9.9 A diagram showing the extent of the wavefunction for a molecule. In A the molecule is by itself. In B the molecule is near a barrier. Notice that the wavefunction has a finite size (i.e. there is some uncertainty in the position of the molecule.) As a result, when a molecule approaches a barrier, there a component of the molecule on the other side of the barrier.

Page 18: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Tunneling Data For H+H2H2+H

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Rat

e

2 2.2 2.4 2.6 2.8 3 3.2 3.4

1000

10000

1000/T

300 K350 K400K450 K

Figure 9.13 An Arrhenius plot for the reaction H+H2

H2+H.

Page 19: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Modern Versions Of Transition State Theory

Variational transition state theory. Tunneling corrections.

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Page 20: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Variational Transition State Theory

Key idea: TST is saddle point in the free energy plot, not the energy plot

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Distance Over the Potential Energy Surface

En

erg

y

Zero Point Of Reactants

Zero Point Energy

Zero Point OfTransition state

 

Figure 9.4 The activation barriers for the reactions in Table 9.6.

Page 21: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Example: X+H+CH3HCH3+X

1000

3000

2000

2.0 2.5 3.0 3.5 4.01.5

C-H Distance, Angstroms

Ene

rgy,

kca

l/mol

e

-1

Vibrational Frequency

1.0 2.0 3.0 4.0-20

-10

0

10

20

The Potential

Zero point energy

Barrier

C-H Distance, Angstroms

Vib

ratio

nal F

requ

enci

es, c

m

Sum

Figure 9.17 The vibrational frequency, zero point energy potential energy and the sum of poential and zero point energy for reaction (9.51) as a function of the C-H bond length. After Hase [1997].

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Little improvement over TST except for methane.

Page 22: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Tunneling Corrections:

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Barrier

B A

Tail of WavefunctionExtends through barrier

Center of Wavefunctionbefore barrier

Wavefunction

Figure 9.9 A diagram showing the extent of the wavefunction for a molecule. In A the molecule is by itself. In B the molecule is near a barrier. Notice that the wavefunction has a finite size (i.e. there is some uncertainty in the position of the molecule.) As a result, when a molecule approaches a barrier, there a component of the molecule on the other side of the barrier.

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Barrier

HydrogenWavefunction

Figure 9.8 Tunneling through a barrier.

Tunneling through a barrier.

Page 24: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Derive: An Equation For Tunneling

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xA0-de

Pot

enti

al E

nerg

y

0

VB

Figure 9.11 A plot of the square-well barrier.

(9.73)

Solve Shroedinger equation for Motion near a barrier.

2 2

B i i2i A

h d+V ψ =Eψ

2m dX

ħ

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Solution:

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P(-d ) =2

pd

2m (V Ee

i

i ie

i B i2

KK

exp

)2

(9.81)

PotentialBarrier

TotalWavefunction

IncidentWavefunction

ScatteredWavefunction

X A

Figure 9.12 The real part of the incident (i)

scattered (r) and total wavefunction (T) for the

square-well barrier.

ħ2

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Eckart Barriers Better Assumption

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V1

DistanceE

nerg

y

V0

Figure 9.14 The Eckart potential.

Significant improvement over TST.

‡ ‡B T

A BCP A BC B

k T q Ek = exp -

h q q k TTk

2

P i B‡

B

h ν k T1k(T)=1+ 1+

24 k T E

(9.63)

(9.89)

Page 27: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Example 9.A Tunneling Corrections Using The Eckart Barrier

In problem 7.C we calculated the preexponential for the reaction:

 F+H2 HF+H

(9.A.1) How much will the pre-exponential

change at 300 K if we consider tunneling?

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Page 28: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Solution

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From Equation 9.69

k TT

h

q

q qexp - E TA BC

B

p

A BC

‡B

/

(9.A.2)

We already evaluated the term in brackets in Equation (9.A.1) in Example 7.C. Substituting that result:

k T Å molecule-secA BC3

2 1014 / (9.A.3)

Bk TBk T

Page 29: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Next Evaluate К(T)

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 According to Equation (9.101) 

 

Where according to Table 7.C.1 =310 cm-1.

Th

Tp i

B

B‡

11

241

2

T

E

(9.A.4)

Bk T

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Solution Continued

Equation (7.C.17) says:

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h

T4.784 10 cm

300K

Tp i

B

-3

(9.A.5)

Substituting i = 310 cm-1 at T=300K into Equation (9.A.5) yields

h

Tcm cm

K

300Kp i

B

-1

310 4784 10

3001483. . .

(9.A.6)

(9.A.5)

Page 31: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Substituting Into Equation (9.A.4) Yields

Therefore the rate will only go up by 10% at 300K.

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T

1

1

24148 1

00198 300

561102.

. /

. /.

kcal mole K K

kcal mole

(9.A.7)

Page 32: ChE 551 Lecture 19 Transition State Theory Revisited 1.

At 100K

Substituting into Equation (9.A.9)

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hcm cm

300K

100Kp i -1

310 4784 10 4453. .

T

1

1

24445 1

00198 100

561852.

.

..

(9.A.8)

(9.A.9)

Page 33: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Note

The tunneling correction in this example is smaller than normal, because the barrier has such a small curvature. A typical number would be 1000 cm-1. Still it illustrates the point that tunneling becomes more important as the temperature drops.

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Page 34: ChE 551 Lecture 19 Transition State Theory Revisited 1.

If We Replace The Hydrogen With A Deuterium:

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F+DH FD+H

(9.A.10)

the curvature drops to 215 cm-1. In that case

h

Tcm cm

300K

100Kp i

B

-1

215 4784 10 3083. .

(9.A.11)

Th

T

T

E

p i

11

241 1 41

2

B

B‡ .

(9.A.12)

so the tunneling correction for deuterium is half that of hydrogen.

Bk T

Page 35: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Semiclassical Approximation For Tunneling

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P Sm V E

dSi B iSS

o

exp 2

2

(9.96)

Marcus-Coltrin Path

MinimumEnergy path

Figure 9.15 Some of the paths people assume to calculate tunneling rates.

ħ2

Page 36: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Exact Calculations

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rBC

0.5

24 kcal/mole

Å

15 kcal/mole

rBC

pro

du

cts

0.5

24 kcal/mole

Å

15 kcal/mole

9 kcal/mole

reactants

rAB

r

Figure 9.18 Some trajectories from the reactants to products.

Page 37: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Conclusions:

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Conclusions: Key assumptions in TST1) no recrossing trajectories2) qhot=qtst

3) no tunneling OK to factor of 20 for bimolecular reactionsHard to do better

Page 38: ChE 551 Lecture 19 Transition State Theory Revisited 1.

Question

What did you learn new in this lecture?

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