ChE 452 Lecture 23
Transition State Theory, Continued
1
Last Time Transition State Theory
Energy
Also called conventional transition state theory
(CTST)
A
‡
Barrier
Reactants
Products
Reaction Cordinate
Figure 7.5 Polanyi’s picture of
excited molecules.
†
†
T
†
†
E
 k BT  q
k BT
k A  BC = 
e
 h  q q
 p  A BC
(7.43)
2
Results From Last Time
F+H2HF+H
Table 7.C.3 A comparison of the preexponential calculated
by transition state theory and collision theory to the
experimental value.
ko Transition state theory with
adjusted transition state geometry
ko Transition state theory with exact
transition state geometry
ko Collision theory
ko Experiment
2.05 1014 Å3/mole-sec
1.65 1014 Å3/mole-sec
1.9 1014 Å3/mole-sec
2.3 1014 Å3/mole-sec
3
Comparison Of Transition State Theory And
Collision Theory
1.
2.
Transition state theory uses the
transition state diameter rather than
the collision diameter in the
calculation.
Transition state theory multiplies by
two extra terms: The ratio of the
vibrational partition function, and the
electronic partition function for the
transition state, and the reactants.
4
Key Prediction Of TST



Ea associated with barrier
Lots of parameters (i.e. qs) to fit to data.
Predicts
k BT
=5x1012 /sec at 300 K, 1x1013 /sec at 600 K
hP
0.1 <
‡
qT
q AqBC
< 10Å3 / molecule
Preexponentials
3
3
Å
Å
5x1011
< k o < 1x1014
molecule sec
molecule sec
5
Is The Saddle Point Energy A Good
Approximation To The Activation Energy?
Table 7.4 A comparison of the saddle point energy from
accurate ab-initio calculations and the activation barrier
measured experimentally for a number of reactions.
Reaction
H + H2  H2 + H
D + H2  DH + H
H + CH4  H2 + CH3
H + CH3CH3  H2 +
C2H5
F + H2  HF + H
H + CH3CH2CH3H2
+ C3H7
H + CH3OH 
CH2OH + H
CH3 +CF3I  CH3I +
CF3
Saddle Point Energy
kcal/mole
9.60
9.60
13.5
11.8
Ea
kcal/mole
8.0
7.78
11.9
9.7
5.8
10.4
1.7
8.2
9.8
9.0
4.95
7.5
6
Difference Between EA And Saddle Point
Energy Due To:


Dynamic effects and quantum effects
Tunneling
Tail of
Wavefunction
Extends
through
barrier
Center of
Wavefunction
before barrier
B
A
Wavefunction
Barrier
Figure 7.8 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.
7
Accuracy of TST Only Fair
Very hard to get better than 1 order of
magnitude
5
Experiment
2
1
0.5
0.2
0.1
0.05
0.01 0.02 0.05 0.1 0.2
0.5 1
2
5
10
Transition State Theory
Figure 7.9 A comparison of the hp/kBT measured experimentally for the
reactions in Table 9.4 to the transition state approximation of the same
quantities.
8
Why Was Transition State So Popular?






Gave a way of thinking about activation
barriers.
Lots of parameters to fit data.
Eyring 500 papers.
People thought it was right (except for
tunneling).
Only in 1998 did people realize that TST with
tunneling is not accurate.
Still thought to be a useful way of thinking
about reactions.
9
One Application Of TST Is To Estimate Ko
‡
Approximate Evaluation of
q
qq
A
B
Earlier in this chapter, we said that typically
q
is within an order of magnitude or two
‡
qq
A
B
of unity. The object of this problem is to
q
evaluate
using the approximations in
‡
qq
A
B
Chapter 6 for a case where A, B and C are
atoms, not molecular ligands.
10
Solution Step 1: Count The Modes


Consider A+BCABC‡AB+C with a linear transition state.
In lecture 14 we found that molecules have 3 vibrational
modes, 2 or 3 rotational modes and 5Na total modes
Molecule
A
BC ABC‡
Atoms
1
2
3
Total Modes
3
6
9
Translations
3
3
3
Rotations
0
2
2
Vibrations
0
1
4
11
Step 2: Find The Partition Functions
A:
q q
A
3
t

BC:
q q q q
3
BC
A
(7.B.1)
t
v
2
r

BC
(7.B.2)
Transition state:
*
3 4 2
t v r TST
q =(q q q )
(7.B.3)
12
Transition State Approximation; Treat One Of The
Vibration Modes As A Translational Mode
*
A
3 3 2
t v r ABC TST
q =(q q q q
)
(7.B.4)
Where qAAC is the partition function for
the motion of A toward
BC. In transition
→
state theory, we want q‡ not q*. By
definition:
‡
q=
*
q
q ABC
3 3
t v
2
r TST
=(q q q )
(7.B.5)
13
Combining Equations 7.B.1 Through
7.B.5 Yields

qqq 
q
 3
q A q BC (q t ) A (q q q )
‡
3
v
2
r
3
t TST
3
2
t v r BC
(7.B.6)
14
Approximation: For Today’s Example
(qt)TST  (qt)A  (qt)BC = qt
(7.B.7)
(qr)TST  (qr)BC
(7.B.8)
(qv)TST  (qv)BC = qv
(7.B.9)
Substituting (7.B.7), (7.B.8) and (7.B.9) into
equation (7.B.6) yields
‡
q
q

q Aq B q
2
v
3
t
Take 2 translations
Convert to vibrations
q
qq
A
(7.B.10)
15
Solution Continued
According to Table 6.6; qv1, qt 3/Å.
Substituting into equation (7.B.10)
 12 
q‡
Å3
=
=0.037
3 
q A q B  (3/Å) 
molecule
(7.B.11)
16
Solution For Nonlinear TST
One can do the same analysis for a
nonlinear transition state. The result
is:
q v q r 1x50
q‡
Å3
= 3
=1.85
3
q A q B q t (3/Å)
molecule
(7.B.12)
17
Estimate ko From Above



Showed 0.04<q‡/qAqBC<2
kBT/h 5x1012/sec
Hence 1011 and 1013/sec
18
Comparison To Experiments
Table 7.2 A selection of the preexponentials reported by Wesley[1980].
Reaction
Preexponential
Reaction
Å3/molecule Sec
Preexponential
Å3/molecule Sec
H+C2H6C2H5+H2
1.6  1014
O+C2H6OH+C2H5
2.5  1013
H+CHH2+C
1.1  1012
O+C3H8(CH3)2CH+OH
1.4  1010
H+CH4H2+CH3
1  1014
O2+HOH+O
1.5  1014
O+H2OH+H
1.8  1013
OH+OHH2O+O
1  1013
O+OHO2+H
2.3  1013
OH+CH4H2O+CH3
5  1013
O+CH4CH3+OH
2.1  1013
OH+H2COH2O+HCO
5  1013
O+CH3H+CH3O
5  1013
OH+CH3H+CH3O
1  1013
O+HCOH+CO2
5  1012
OH+CH3H2O+CH2
1  1013
19
Not Accurate Compared To
Experiments/Detailed



Transition state larger than reactants
Rotational partition function larger
Vibration larger too
20
Strengths Of Transition State




Transition state theory allows you to calculate bcoll in
equation (7.16) rather than guess it. In Chapter 9, we will
show that for a simple collision bcoll is just the distance
between the reactants at the transition state.
Transition state theory suggests that Preactions can be less
than exp(E‡T /k BT) . For example, in the reaction:
D+CHC+HD
The incoming deuterium has to hit the hydrogen atom for
reaction to occur. Transition state theory accounts for that
by saying that the transition state must be a linear CHD
molecule.
Transition state theory also predicts that small changes in
the shape of the potential energy surface produce large
changes in q ‡ , which in turn changes the rate.
T
21
Accuracy Of TST Only Fair
Very hard to get better than 1 order of
magnitude
5
Experiment
2
1
0.5
0.2
0.1
0.05
0.01 0.02 0.05 0.1 0.2
0.5 1
2
5
10
Transition State Theory
Figure 7.9 A comparison of the khp/kBT measured experimentally for the
reactions in Table 9.4 to the transition state approximation of the same
quantities.
22
Next: Relationship Between TST And
Collision Theory

TST & collision theory very similar



Similar predictions
In fact, TST goes to collision theory if
one ignores vibration and rotation of
BC.
Book shows that TST goes to collision
theory for structureless molecules.
23
Transition State Theory Differs From
Collision Theory In Two Key Ways:


One can actually make a calculation.
Collision theory bcoll unknown (7.16).



Transition state theory replaces bcoll with dcoll. The
distance between reactants at the transition state
geometry. dcoll can be calculated exactly.
Transition state theory allows you to consider
reactions like reaction (7.27) where only special
configurations lead to the desired products.
Partition function for TST gives many
parameters
24
Summary
Transition state theory Extension of collision theory (used
collision theory to derive)
 Identifies bcoll as dcoll
 Identifies S as S of saddle point
 Slightly, more accurate that Collision
theory
 Still only accurate to order of
magnitude
25
Question

What did you learn new in this lecture?
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