Download - CHBE 551 Lecture 20

CHBE 551 Lecture 20 Unimolecular Reactions

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Last Time 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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Transition State Theory Fails For Unimolecular Reactions

Table 9.8 The preexponential for a series of unimolecular reactions, as you change the collision partner. Data of Westley[1980].

reaction k0 when X =

Argon

k0 when X =

Water

k0 when X = N2

NO2 + X OH + H + X

1.7 1014 cm6/mole2 sec


1.57 1015 cm6/mole2 sec

H2O + X OH + H + X




HO2 + X O2 + H

+ X



2 1015 cm6/mole2 sec

H2 + X H + H + X



O2 + X 2O + X 1.9 1013 cm6/mole2 sec

1.0 1014

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Why Does Transition State Theory Fail?

Ignores the effect of energy transfer on the rate

Consider a stable molecule AB. How can AB A + B

If you start with a stable molecule, it does not have enough energy to react. Need a collision partner so AB can accumulate

enough energy to react. Energy accumulation ignored in TST

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Lindeman Approximation

Assume two step process First form a hot complex via collission Hot complex reacts

Steady State Approximation Yields

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A X A X

A B

2

1

3

BA

XAXA

3

1

2

rk k [A][X]

k k [X]B1 3

3 2

Comparison To Data For CH3NC CH3CN

[A]kk

[A][A]kkr

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31B

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Pressure, torr

Rate m

-mole/lit-sec

0.01 0.1 1 10 100 1000 10000

1E-8

1E-6

1E-4

1E-2

1E+0

1E+2

First Order

Second Order

Lindeman Model

Data

But Preexponentials For Unimolecular Reactions Too Big

Table 9.8 The preexponential for a series of unimolecular reactions, as you change the collision partner. Data of Westley[1980].

reaction k0 when X =

Argon

k0 when X =

Water

k0 when X = N2

NO2 + X OH + H + X



1.57 1015 cm6/mole2 sec

H2O + X OH + H + X




HO2 + X O2 + H

+ X



2 1015 cm6/mole2 sec

H2 + X H + H + X



O2 + X 2O + X 1.9 1013 cm6/mole2 sec

1.0 1014

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Why The Difference?

Bimolecular collision lasts ~10-13 sec Molecule must be in the right

configuration to react Hot unimolecular complex lasts

~10-8 sec Even if energy is put in the wrong

mode, the reaction still happens

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RRK Model

Assume correction to TST by

Qualitative, but not quantitative prediction

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Number of ways to put an Energy E in s modes

Number of ways to put an Energy of E in one mode.a

a

RRKM Model

Improvement to RRK model proposed by Rudy Marcus (ex UIUC prof).

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*AP2 q

q

h

1k

Derive Equation

Consider

Excite molecule to above the barrier then molecule falls apart

Derive Equation for reverse reaction

At Equilibrium

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CD CO h CD CO * CD CO *2 22

2

CD * CO * CD CO *22

2

k

kK

q q

q2

22eq CD CO2

Derivation Continued

From Tolman's equ

Pages Of Algebra

*AP2 q

q

h

1k

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COCD

BCAABCABCABC

2 qq

qqdv

2d

1k

2

Note

Reactants have a fixed energy ~laser energy

Products have a fixed energy too, but since they have translation, the products can have vibrational+ rotation energy between the top of the barrier and E*

)E(Ggq *

nn

13

)E(Ngq *

nn*A

Substituting, And Assuming Energy Transfer Fast

N(E*) E* is the number of vibrational modes of the reactants with an vibrational energy between E* and E* + E*

G+(E*) is the number of vibrational modes of the transition state with a vibrational energy between E‡ and E* independent of whether the mode directly couples to bond scission.

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Next Separate Vibration and Rotation

where GVT is the number of vibrational

states at the transition state, with an energy between E‡ and E*. NV(E*) is the number of vibrational states of the reactants with an energy between E* and E* +E ; qR

‡ is the rotational partition function for the transition state and qR* is the rotational partition function for the excited products.

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*EN

*EG

q

q

h

1*Ek

V

TV

*R

‡R

P2

Note

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G E * N (E*)dE *T T

E

E*

‡

Qualitative Results

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0

500

1000

1500

2000

2500

3000

Ene

rgy,

cm

-1

0 10 20 30 40 50 601E-2

1E-1

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

Energy, Kcal/mole

G

(E*)

0

0 5 10 15 200

5

10

15

20

25

30

Energy, Kcal/mole

G

(E*)

0

Gives Good Predictions for Long Lived Excited States

CD CO h CD CO * CD CO *2 22

2

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DataRRKM

K, /

sec

x 10

400

10

Energy above transition state, cm -1

0

5

0 200 300100

Tunneling

Ignores Quantum Effects

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Energy, Kcal/mole

Rat

e

Data

RRKM Trend

1.50

OCCHOCCH 13122

12132

Details Of Calculation

Program Beyer_SwinehartC! density of vibrational states by C! Beyer-Swinehart algorithm

implicit noneinteger(2), parameter :: MODES=15integer(2), parameter :: points=5000integer(2):: vibr_freq(MODES) integer(2):: vibr_degen(MODES)integer i, j integer(2):: start_frequency=0real(8) n(0:points)real(8) g(0:points), x, yreal :: energy_scale=2.

c!energy_scale equals spacing for energy bins IN cm-1data vibr_freq /111,409,851,1067,1099,1 1295,1527,1589,1618,1625,3123,2 3193,3229,3268,3373/data vibr_degen/ 15*1/do 5 i=1,MODESvibr_freq(i)=vibr_freq(i)/energy_scale

5 enddostart_frequency=start_frequency/

energy_scaleC! next initialize arrays

do 2 i=1,pointsn(i)=0g(i)=1

2 enddon(0)=1g(0)=1

c! count the number of modesdo 10 j=1,MODES do 9 i=vibr_freq(j),points n(i)=n(i)+n(i-

vibr_freq(j))*vibr_degen(j) g(i)=g(i)+g(i-

vibr_freq(j))*vibr_degen(j) if(mod(i,500).eq.0)write(*,*)i,n(i)

9 enddo 10 enddo

n(0)=0.c! next write data in format for microsoft Excel, lotus

open(unit=8,file="statedens.csv",status= "replace",action="write")

write(8,101)write(8,102)

101 format("'E', 'E','N(E)','G(E)'") 102 format("'cm-1/molecule','kcal/mole','/cm-1','dimensionless'")

do 20 I=start_frequency,points,100x=I*energy_scaley=x*2.859e-3n(i)=n(I)/energy_scaleg(i)=g(I)-1.0write(8,100)x,y,n(i),g(i)

20 enddo 100 format(f9.1,', ',f9.3,', ',e15.7,', ',e15.7)

stopend

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Does RRKM Always Work?

Assumes fast dynamics compared to time molecule stays excited

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RRKM

Data

300020001000

Energy, cm-1

Rat

e co

nsta

nt, /

nano

sec

20

30

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A comparison of the experimental rate of isomerization of stilbene (C6H5)C=C(C6H5) to the predictions of the RRKM model

Also Fails for Barrierless Reactions

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1 1.5 2 2.5 3-150

-100

-50

0

50

100

150

Distance

Ene

rgy,

Kca

l/mol

e

Summary

Unimolecular reactions have higher rates because configurations that do not immediately lead to products still eventually get to products

RRKM – rate enhanced by the number of extra states Close but not exact – still have dynamic

effects

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Query

What did you learn new in this lecture?

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