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Theories of unimolecular
reactions
Ernő Keszei
Eötvös Loránd University, Institute of Chemistry
Overview
– Introduction
– Historical overview
– Lindemann theory
– Some aspects of statistical physics
– RRK theory
– RRKM/QET theory
– TST and activation entropy
– IVR: internal vibrational relaxation
– (Some) experimental evidence of TST
– Recent developments in RRKM theory
Introduction
Unimolecular reaction – formal description
A  products

d [A]
 k [A]
dt
Typical unimolecular reactions
– photochemical (laser induced)
– isomerisations
– gas phase reactions
– very low pressure reactions (mass spectrometer, upper atmosphere)
Historical overview
– early 20th century:
Gas phase unimolecular reactions (mostly pyrolysis)
Many 1st order reactions were thought to be unimolecular
 Problem: how molecules acquire activation energy?
– Perrin’s suggestion (1919)
molecules excited by radiation – opposed by many chemists
– the rate should depend on surface/volume ratio
– it should not depend on the pressure
– there should be a difference between dark / sunlight conditions
Perrin’s idea has a revival recently:
 IR laser activation
 Fourier-transform mass spectrometry
Historical overview
”inexplicable” experimental evidence
Decomposition of propionaldehyde and ethers (Hinshelwood)
Decomposition of azo-methane
1927: experiments of Ramsperger
high pressure: first order kinetics
low pressure: second order kinetics
The rate increases
with increasing temperature,
but first order kinetics is not in
accordance with collisonal activation
Pilling & Seakins 1995
Lindemann theory (1922)
A+M
A*+ M
A*
𝑘1
𝑘−1
𝑘2
A* + M
activation (via collisions)
A +M
deactivation (via collisions)
P (products)
decomposition (spontaneous)
M: any molecule (A, P or inert gas molecules added)
Rate equation for the energised molecules:
d [A*]
 k1[A][M ]  k 1[A*][M ]  k 2 [A*]
dt
formation
removal
Considering A* as a steady-state component:
k1[A][M ]  k 1[A][M ]  k 2 [A*]
k1[A][M ]
[A*] 
k 1[M ]  k 2
Lindemann theory
A+M
A*+ M
A*
𝑘1
𝑘−1
𝑘2
A* + M
activation
A +M
deactivation
P (products)
decomposition
k k [M ][A]
k uni [A]  2 1
k 1[M ]  k 2
k uni
k 2 k1[M ]

k 1[M ]  k 2
Low pressure limit: k 1 M   k 2
M  
k2
k 1
High pressure limit: k 1 M   k 2
M  
k2
k 1
k uni  k1 M 
k uni
k 2 k1

k 1
2nd order
1st order
In accordance with experiments
Lindemann theory
A+M
A*+ M
A*
𝑘1
𝑘−1
𝑘2
A* + M
activation
A +M
deactivation
P (products)
decomposition
Low pressure: k uni  k1 M  2nd order
k1M  k2 : accumulation of energised molecules
Activation is the rate determining step
High pressure: k uni 
k 2 k1
k 1
1st order
k1M  k2 : fast removal of energised molecules via collisions
Product formation is the rate determining step, but 𝑘uni ≠ 𝑘2
lg (kuni / s–1)
Comparison with experiment
lg k
k 
k 2 k1
k 1
kuni extrapolated to infinite pressure
k
k 2 k1
kk

 2 1
2
k 1  k 2 M 1 2
2k 1
lg (k / 2)
[M]1/ 2 
k 2 k

k 1 k1
[M]1/2 : where kuni = k / 2
lg ([M]1/2)
from experiment: k and [M]1/2
from collision theory: 𝑘1 = 𝑍𝑒
lg ([M] / mol
[M]1/ 2 
k
k1
dm–3
Calculated [M]1/2 =
𝐸
− 𝑅𝑇
𝑘∞
𝑘1
We can compare [M]1/2 and [M]1/2
Comparison with experiment
from
experiment
Reaction
calculated
T/K
Z / s–1
E / kJmol–1
[M]1/2
[M]1/2
760
3·1015
276
3·10–4
2·104
720
4·1015
267
1·10–5
2·104
670
3·1015
256
1·10–6
1·104
MeNC → MeCN
500
4·1013
161
4·10–3
2·102
EtNC → EtCN
500
6·1013
160
4·10–5
4·102
N 2 O → N2 + O
890
8·1011
256
8·10–1
4
2
+
Pilling & Seakins 1995
Many orders of magnitude differences!
Comparison with experiment
k
k 2 k1 M 
k 1 M   k 2
1 k 1 M   k 2

k
k 2 k1 M 
1 k 1
1
1
1




k k1k 2 k1 M  k k1 M 
1
1
1 1


k k k1 M 
1
1
𝑣𝑠.
𝑘
M
should be linear — it is not
Pilling & Seakins 1995
Lindemann theory — summary
• Two-step procedure with collisional activation
• Steady-state approximation for the energised species
Advantage:
• A simple and qualitatively fairly correct picture
Disadvantages:
• underestimation of k1: neglecting molecular degrees
of freedom storing energy
• neglecting internal vibrational relaxation
in product formation
Improvements to Lindemann theory
Avoiding underestimation of k1
— vibrational excitation processes are more effective
supposing strong collisions (independent states before and after)
equivalent harmonic oscillators for vibrations (Hinshelwood, 1927)
anharmonic oscillators, quantum mechanical description
(Marcus, 1952)
— stepwise energy uptake replaces strong collisions
 ”master equations” (Tardy, Rabinovitch, Schlag, Hoare, 1966)
Improvements to Lindemann theory
Considering the role of IVR in decomposition
— equivalent oscillators: RRK theory (Rice, Ramsperger, Kassel, 1928)
semi-classical treatment of oscillators, plus
role of rotation: RRKM theory (Marcus, 1951)
— more precise calculation of density of states & partition functions
for vibrational states (Whitten-Rabinovich, 1959)
direct counting of vibrational states
(Current, Rabinovich 1963; Beyer, Swinehart, 1973)
Hinschelwood theory
Considering vibrational excitation in Lidemann mechanism
Supposing strong collisions:
states A and A* are independent
 E 

k1  Z  exp 
 k BT 
minimal energy of activation
 Ev 
gv

k1  Z   exp 
v m Q
 k BT 
quantum number of the
lowest excited state
multiplicity
Boltzmann distribution
vibrational
excitation energy
vibrational
partition function
Density of states
Discrete states: the number of quantum states at the same energy
(also called degeneration)
e. g. Two oscillators with the same frequency: Ev=(v+1/2)h
Ev
3,0
2,1
1,2
2,0
1,1
0,2
1,0
0,1
0,3
0,0
4
3
2
1
v1,v2 : vibrational quantum numbers
Degeneration of s oscillators with the same frequency:
(v  s  1)!
gv 
v!( s  1)!
Details of counting states
Let us distribute v quanta of vibration among s oscillators:
→ equivalent to the distribution of v pebbles among s boxes
equivalent to the distribution of v pebbles and s – 1 bars
Degeneration of s oscillators with the
same frequency:
(number of permutations with repetition)
 v  s  1 v  s  1!
 
g v  
 v  v !s  1!
Density of states
Continuous approximation
Continuous states: the density of continuous states at the same energy
notation: ρ(E )
Equivalent to the number of states between energies E and E + dE
E'
Definitions: number of states between 0 and E’ :
N ( E ' )    ( E )dE
0
density of states at energy E :
 (E) 
dN ( E )
dE
Density of states at energy E
1D translation
2ma 2
 (E) 
Eh 2
a : 1D box length
3D translation
4 32
 (E)  3 m 2E V
h
V : volume
rotation (linear molecule)
 (E) 
1
B
B : rotational constant
E
Ba Bb Bc
rotation (3D molecule)
 (E)  2
general expression:
 ( E )  E ( s / 21)
Ba , Bc , Bd :
rotational constants
s : number of
degrees of freedom
Vibrational density of states at energy E
Classical harmonic oscillators:
 E  
E s 1
s
( s  1)! h i
N E  
Es
s
s! h i
underestimation by
several orders of magnitudes
i 1
i 1
z
Semiclassical harmonic oscillators
(no quantum effects but zero-point energy):
 E  
( E  E z ) s 1
s
( s  1)! h i
i 1
N E  
( E  Ez ) s
s
s! h i
i 1
Ez 
 h
i 1
i
2
more moderate
underestimation
Hinschelwood theory
 Ev 
Z

k1    g v exp 
Q vm
 k BT 
 v  s  1

g v  
 v 

 h  
 
Q  1  exp 

k
T
 B 

s
Ev  v  12 h
Using continuous approximation (equivalent to h  k BT ):

 E 
Z
Z  E0 
 dE 


k1    N E  exp 
Q E0
s  1!  k BT 
 k BT 
s 1
 E0 

exp 
 k BT 
Hinschelwood theory
 E 

k L  Z  exp 
 k BT 
Z  E0 


k1 
s  1!  kBT 
k1
1  E0 



k L s  1!  k BT 
As E0  k BT
s 1
 E0 

exp 
 k BT 
s 1
1
E0
 1
k BT
Even so, an unrealistically large s should be given for large molecules
Vibrational density of states at energy E
Anharmonic quantum oscillators:
no closed (analytical) expression for the number of states
Solution: direct counting of states
(I)=[1,0,0,0….0]
FOR J=1 TO s
FOR I=(J) TO M
(I)=(I)+(I - (J))
NEXT I
NEXT J
-
counting array initiation
s: number of oscillators
M = maximum of energy
(J) - oscillator frequency, cm–1
loop for energy
loop for the oscillators
No serious underestimation —
the problem concerning k1 is more or less solved
RRK theory (Rice, Ramsperger and Kassel; 1928)
For the energised molecule to dissociate, excess energy should be
concentrated in the critical vibrational modes.
New mechanism:
A+M
A*+ M
A*
A‡
𝑘1
𝑘−1
𝑘∗
𝑘‡
A* + M
activation (via collisions)
A +M
deactivation (via collisions)
A‡
formation of the transition state (IVR)
P (products) decomposition (spontaneous)
IVR is much slower than the decomposition:
k* << k‡
Considering A‡ as a SS component: 𝑘 ∗ A∗ = 𝑘 ‡ A‡
‡
A
𝑘∗ = 𝑘‡ ∗
A
RRK theory
Underlying conditions: oscillators of the same frequency
equal probability of all (vibrational) states
The transition state is formed,
if there is enough energy concentrated
on the critical vibrational mode(s)
which results in the product formation
The critical number m of vibrational quanta
should be accumulated in the critical
vibrational mode
Pilling & Seakins 1995
RRK theory
𝑘∗
=
A‡
‡
𝑘 ∗
A
= 𝑘 ‡ 𝑃𝑚𝑣
𝑃𝑚𝑣 is the probability that
m quanta out of v are on the critical mode
 v  s  1 v  s  1!
 
Recall: g v  
s  1!v !
 v 
v–m
m
g vm
 v  m  s  1 v  m  s  1!
 
 
 v  m  s  1!v  m !
v, m and v–m >> s

g vm
v!
v  m  s  1 ! v  m 
P 


v  s  1! v  m !
gv
v s 1
s 1
v
m
 m
 1  
v

s 1
RRK theory
Rewriting 𝑘 = 𝑘
∗
‡
∗
‡
𝑃𝐸𝐸0 =
A‡
𝑘 =𝑘
Recall:
𝑃𝑚𝑣 =
A∗
𝑘
1−
‡
1−
𝑚 𝑠−1
𝑣
𝐸0 𝑠−1
𝐸
for continuous energy:
𝑃𝐸𝐸0 =
and 𝑘 ∗ = 𝑘 ‡
1−
𝐸0 𝑠−1
𝐸
A‡
A∗
𝑘 ∗ = 𝑘 ‡ 𝑃𝐸𝐸0
Resulting expression for k2 (product formation):
𝑘2 𝐸 = 𝑘 ‡
𝐸0
1−
𝐸
𝑠−1
The reaction rate is energy dependent;
it increases with increasing energy
RRK theory
𝑘2 𝐸
𝑘‡
Limit case
(Lindemann theory)
𝑬 𝑬𝟎
Pilling & Seakins 1995
RRK theory — summary of results
𝐸
𝑘B 𝑇
𝑘2 𝐸 = 𝑘 ∗ (𝐸) = 𝑘 ‡

k uni 

E0
𝐸0
1−
𝐸
𝑠−1
k1 ( E ) k 2 ( E )M 
dE
k 1 M   k 2 ( E )
Energy / cm–1
𝑍𝑁 𝐸 −
𝑘1 𝐸 =
𝑒
𝑄
compared to the Lindemann expression:
k uni
k 2 k1 M 

k 1 M   k 2
Reaction rate (arbitrary units)
Pilling & Seakins 1995
RRK theory — summary of concepts
• Two-step procedure with collisional activation
• Steady-state approximation for both the energised species
and the transition state
• k1 estimated by distributing energy over vibrational modes
(presupposing strong collisions)
• k2 estimated by the probability of TS formation with IVR
(fast flow of energy between vibrational modes)
Approximations used:
oscillators of the same frequency
continuous energy distribution
RRKM theory (Marcus; 1952)
k1 calculated using quantum statistical method
Proper (different!) frequencies of A* and A‡ used
Rotational modes of TS are properly considered
Introduction of constant and variable energy:
constant energy: zero point energy and translational energy
variable energy participates only in IVR
New mechanism:
A+M
A*(E ) + M
A*(E )
A‡
𝑘1
𝑘−1
𝑘∗(𝐸 ′ )
𝑘‡
E: total energy
A*(E ) + M
activation (via collisions)
A +M
deactivation (via collisions)
A‡
formation of the transition state (IVR)
P (products)
decomposition (spontaneous)
E’: variable energy
RRKM theory — energy landscape
total energy
𝐸𝑟‡
Er
E0
”activation energy”
stable molecule
𝐸𝑡‡
E‡
E
Ev
𝐸𝑣‡
zero point energy
of transition state
E0
zero point energy
of stable A
E
Er
Ev
total energy
rotational energy
energy of vibrations
and internal rotations
transition state
E‡
𝐸𝑟‡
𝐸𝑣‡
max. available energy
𝐸𝑡‡
rel. translational energy
rotational energy
energy of vibrations
and internal rotations
RRKM theory — role of rotation
Conservation laws:
both energy and momentum are conserved
𝐸𝑟‡
Er
E‡
Angular momentum: L = Iω
E
1
Rotational energy: 𝐸𝑟 =
𝐼𝜔2
2
I : moment of inertia; ω: angular velocity
𝐸𝑡‡
𝐸𝑣‡
Ev
E0
Formation of TS: I increases → Er decreases
I << I ‡ → 𝐸𝑟 − 𝐸𝑟‡ = 𝐸𝑟 1 −
𝐼
𝐼𝑟‡
Excess 𝐸𝑟 − 𝐸𝑟‡ becomes vibrational energy Ev
Loosening of internal bonds result in increase of vibrational energy
RRKM / QET theory
E = Ev + Er
Iternal degrees of freedom:
- active :
- adiabatic:
free energy flow (vibration and internal rotation)
rotational quantum state cannot change
(rotation of the whole molecule)
Expression of the rate constant:
#
#
#
#
#



E

E



E
t
#
v
k Ev ,Et  

  Ev 
  Ev 
Number of states in the transition state
Number of states
for the active degrees of freedom

Integrating for all energies,
we get the rate constant:
k EV  
‡
#


E
  t d t


 EV 
 
N # E#

 EV 
Relation of RRKM to RRK theory
Number of states and density of states
for classical harmonic oscillators:
 E  
E s 1
( s  1)!(h )
s
N E  
Es
s!(h ) s
(The n. d. f. is one less for the TS than for the reactant molecule.)
Substitute them into the RRKM equation:
k E  
N # ( E  E0 )
 E 
( E  E0 ) s 1
1 ( s  1)!(hv) s 1
 E0 



1  
s 1
(E )
h
E

( s  1)!(hv) s
s 1
We get (back) the RRK equation
The role of activation entropy (S#)
q# U U
S  kb ln

q
T
#
#
 S
kbT
A
exp
 R
h

#




A – preexponential
q – partition function
U – zero point energy
T – temperature
S# > 0
fuzzy transition state
e.g. bond braking
S# < 0
“closed” transition state
e.g. rearrangement
12
Example:
10
8
- Butyl benzene radical decomposition:
 MS detection: internal energy can be
estimated from fragment ion ratio
lg(k)
formation of propene and
propyl radical
6
4
2
0
-2
-4
-6
2
4
6
8
Internal
energy(eV)
(eV)
Belsõ
energia
Keszei 2015
10
Relation of microcanonical to canonical
rate constant
k(E) – microcanonical
k(T) – canonical rate constant
High pressure: equilibrium
distribution is maintained
canonical energy distribution:
Pth ( E ) 
 ( E )e


  ( E )e
E
k bT

E
k bT

 ( E )e
0

k (T )   Pth ( E )k ( E )dE
0

k (T ) 

E0
 ( E )e
Q

E
k BT
density of states
Boltzmann konstant
internal energy
temperature
 (E)
kB
E
T

N ( E  E0 )
1
dE   N # ( E  E0 )e
 E 
Q E0
#

E
k BT
dE
Q

E
k bT
Presuppositions of RRKM/QET theory
Random distribution of states
Necessary condition is that the IVR be much faster
compared to the decomposition rate
All TS configurations transform into products
A good approximation for relatively large molecules, and
relatively low energies above the activation threshold.
Reaction coordinate is orthogonal to other coordinates:
it is separable from them
It is fulfilled at low energies where vibrational modes can be
described by non-coupled normal modes. With increasing
energy, the probability of coupling increases.
Harmonic vs. anharmonic vibrations
harmonic
oscillator
Linear ABA molecule
QR - symmetric mode
TS - asymmetric mode
No exchange of energy
anharmonic
oscillator
Lissajous motion
Exchange of energy
between modes
Pilling & Seakins 1995
Role of IVR
(Intramolecular Vibrational Relaxation or Redistribution - IVR)
Excitation: typically one single vibrational mode
This localised energy should be redistributed
into other (critical) vibrational mode(s)
If IVR is fast compared to the reaction,
random energy redistribution can be applied
If the reaction is fast compared to IVR,
random energy redistribution cannot be applied.
State dependent rate constants should be calculated,
and all relevant states should be considered.
Experimental testing of IVR
Rabinovitch et al., JPC 78, 2535, 1974
Ratio of the two products:
CF2
CF
+ CH2
With increasing pressure
the ratio increases
CF2
→ more frequent collisions
more effective deactivations
not enough time for IVR
CF
CF
CD2
CF2
CH2
1
CF2
CF
CD2
IVR rate is ~ 1 ps
CF2
CD2
1:1 at low pressures: perfect IVR
Model calculations:
CF
2
CF
+
CF2
CH2
dissociation at
the addition site
CD2
CF
CF
+
CF2
CF2
CH2
at the
other site
A flaw in RRKM theory
Calculations are made using harmonic oscillators
In principle, no exchange of vibrational energy is possible
But for RRKM theory to work, fast IVR is needed
However, RRKM theory performs quite well!
Possible explanation:
Small deviations from harmonicity of the anharmonic oscillators
does not affect the precision of the results much but is just enough
to maintain fast energy transfer between vibrational modes
Indirect evidence of the transition state
Moore et al., Science 256, 1541
(1992)
Dye laser excitation
and detection
1. pulse → S1
fast ISC → T1
Energy / 103 cm–1
CH2CO  CH2 + CO
(above activation energy)
2. pulse: detection of CO
Reaction coordinate
Pilling & Seakins 1995
k (E) is expected to increase stepwise with increasing energy
Indirect evidence of the transition state
Measured intensity / arbitrary units
(a) disappearance of ketene
calculated
(b) production of CO
measured
(c) production of CO
calculated
(d) disappearance of ketene
measured
Good correlation
between experiment
and model calculations
Energy / cm–1
Pilling & Seakins 1995
Further improvements in RRKM theory
— correct account of anharmonicity
(especially for reactions with several minima in the PES)
— correct account of vibrational-rotational couplings
(especially important for smaller molecules)
— Variational Transition State Theory (VTST)
If no saddle point is found on the Potential Energy Surface,
reaction rate should be calculated at the minimum of N‡(E‡)
— Orbiting Phase Space Theory
For reactions, where the opposite reaction (recombination or
association) does not have an activation energy.
(There is no need to take into account a transition state.)
— Using master equations
(if the approximation of strong collisions does not apply)
Using the master equation
If collisions are not strong, we should consider all subsequent collisions
Unlike Lindemann theory, where only ground state and energised state
are considered, we should follow all vibrationally excited states
Formal mechanism used:
A(i) + M
A( j) + M
A( j)
𝑍𝑃𝑗,𝑖
𝑍𝑃𝑖,𝑗
𝑘𝑗
A( j) + M
activation
A(i) + M
deactivation
P (products)
decomposition
𝑃𝑗,𝑖 is the probability of transition from state i to state j
𝑃𝑖 is the probability of finding the molecule in state i
Z is the collision number
Using the master equation
Equations to describe probabilities of transitions
population at level j
(probability: Pj)
n
Z  Pi , j
j 0
k (i)
decomposed
population
n
Z  Pj ,i
j 0
population at level i
(probability: Pi)
Transfer equation:
Both i and j can have
any value between 0 and n
n
n
dPi (t )
 Z  Pi , j  Z  Pj ,i  k (i)
dt
j 0
j 0
Master equation
n
The probability of being in any of the states:
P
j 0
j ,i
1
 n

dPi (t )
 Z   Pi , j  1  k (i )
dt
 j 0

a homogeneous linear system of ordinary
differential equations:
one equation for each state i
It can be solved with any standard solution method:
(finding eigenvalues and eigenvectors is the usual one)
Further reading
– M. J. Pilling, P. W. Seakins: Reakciókinetika,
Nemzeti tankönyvkiadó, Budapest, 1997
– T. Baer , W. L. Hase: Unimolecular Reaction Dynamics,
Oxford University Press, 1996
– J. I. Steinfeld, J. S. Francisco, W. L. Hase:
Chemical Kinetics and Dynamics, Prentice Hall, 1989
– P. J. Robinson, K. A. Holbrook: Unimolecular Reactions, Wiley, 1972
Acknowledgements
Figures and tables marked as Pilling & Seakins 1995 are reproduced from
M. J. Pilling, P. W. Seakins: Reaction Kinetics, Oxford University Press, 1995
END of the lecture
Theory of unimolecular
reactions
Thank you for your attention !
This course material is supported by the Higher Education Restructuring Fund
allocated to ELTE by the Hungarian Government