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
6.7 1015 cm6/mole2 sec
1.57 1015 cm6/mole2 sec
H2O + X OH + H + X
2.1 1015 cm6/mole2 sec
3.5 1017 cm6/mole2 sec
5.1 1016 cm6/mole2 sec
HO2 + X O2 + H
+ X
1.5 1015 cm6/mole2 sec
3.2 1016 cm6/mole2 sec
2 1015 cm6/mole2 sec
H2 + X H + H + X
6.4 1017 cm6/mole2 sec
2.6 1015 cm6/mole2 sec
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
1.7 1014 cm6/mole2 sec
6.7 1015 cm6/mole2 sec
1.57 1015 cm6/mole2 sec
H2O + X OH + H + X
2.1 1015 cm6/mole2 sec
3.5 1017 cm6/mole2 sec
5.1 1016 cm6/mole2 sec
HO2 + X O2 + H
+ X
1.5 1015 cm6/mole2 sec
3.2 1016 cm6/mole2 sec
2 1015 cm6/mole2 sec
H2 + X H + H + X
6.4 1017 cm6/mole2 sec
2.6 1015 cm6/mole2 sec
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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