1. Theories of unimolecular reactions
This course material is supported by the Higher Education Restructuring Fund
allocated to ELTE by the Hungarian Government
Theories of unimolecular reactions
Ernő Keszei Eötvös Loránd University, Institute of Chemistry

2. 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

3. Introduction Unimolecular reaction – formal description A  products
Typical unimolecular reactions
– photochemical (laser induced)
– isomerisations
– gas phase reactions
– very low pressure reactions (mass spectrometer, upper atmosphere)

4. 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 infrared 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

5. 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

6. Lindemann theory (1922) A + M 𝑘 1 A* + M activation (via collisions)
A*+ M 𝑘 − A + M deactivation (via collisions)
A* 𝑘 P (products) decomposition (spontaneous)
M: any molecule (A, P or inert gas molecules added)
Rate equation for the energised molecules:
formation
removal
Considering A* as a steady-state component:

7. Lindemann theory In accordance with experiments
A + M 𝑘 A* + M activation
A*+ M 𝑘 − A + M deactivation
A* 𝑘 P (products) decomposition
Low pressure limit:
2nd order
High pressure limit:
1st order
In accordance with experiments

8. Lindemann theory A + M 𝑘 1 A* + M activation
A*+ M 𝑘 − A + M deactivation
A* 𝑘 P (products) decomposition
Low pressure:
2nd order
: accumulation of energised molecules
Activation is the rate determining step
High pressure:
1st order
: fast removal of energised molecules via collisions
Product formation is the rate determining step, but 𝑘 uni ≠ 𝑘 2

9. kuni extrapolated to infinite pressure
Comparison with experiment
lg k
kuni extrapolated to infinite pressure
lg (kuni / s–1)
lg (k / 2)
[M]1/2 : where kuni = k / 2
lg ([M]1/2)
from experiment: k and [M]1/2
from collision theory: 𝑘 1 =𝑍 𝑒 − 𝐸 𝑅𝑇
Calculated [M] 1/2 = 𝑘 ∞ 𝑘 1
lg ([M] / mol dm–3
We can compare [M]1/2 and [M]1/2

10. Comparison with experiment
from experiment
calculated
Reaction
T / K
Z / s–1
E / kJmol–1
[M]1/2
760
3·1015
276
3·10–4
2·104
720
4·1015
267
1·10–5
670
256
1·10–6
1·104
MeNC → MeCN
500
4·1013
161
4·10–3
2·102
EtNC → EtCN
6·1013
160
4·10–5
4·102
N2O → N2 + O
890
8·1011
8·10–1 4 2 +
Pilling & Seakins 1995
Many orders of magnitude differences!

11. Comparison with experiment
should be linear — it is not
Pilling & Seakins 1995

12. 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
neglecting internal vibrational relaxation in product formation

13. 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)

14. 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)

15. quantum number of the lowest excited state
Hinschelwood theory
Considering vibrational excitation in Lidemann mechanism
Supposing strong collisions:
states A and A* are independent
minimal energy of activation
Boltzmann distribution
vibrational excitation energy
quantum number of the lowest excited state
vibrational partition function
multiplicity

16. 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
0,0
1,0
0,1
2,0
1,1
0,2
3,0
2,1
1,2
0,3 4 3 2 1
v1,v2 : vibrational quantum numbers
Degeneration of s oscillators with the same frequency:

17. Degeneration of s oscillators with the
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)

18. 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
Definitions: number of states between 0 and E’ :
density of states at energy E :

19. Density of states at energy E
1D translation
a : 1 D box length
3D translation
V : volume
rotation (linear molecule)
B : rotational constant
Ba , Bc , Bd : rotational constants
rotation (3D molecule)
general expression:
s : number of degrees of freedom

20. Vibrational density of states at energy E
Classical harmonic oscillators:
underestimation by
several orders of magnitudes
Semiclassical harmonic oscillators (no quantum effects but zero-point energy):
more moderate underestimation

21. Hinshelwood theory
Using continuous approximation (equivalent to ):

22. Hinschelwood theory As
Even so, an unrealistically large s should be given for large molecules

23. 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] - counting array initiation
FOR J=1 TO s - s: number of oscillators
FOR I=(j) TO M - M = maximum of energy
(I)=(I)+(I - (J)) - (J) - oscillator frequency, cm–1
NEXT I - loop for energy
NEXT J - loop for the oscillators
No serious underestimation —
the problem concerning k1 is more or less solved

24. 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 𝑘 A* + M activation (via collisions)
A*+ M 𝑘 − A + M deactivation (via collisions)
A* 𝑘∗ A‡ formation of the transition state (IVR)
A‡ 𝑘 ‡ P (products) decomposition (spontaneous)
k* << k‡
IVR is much slower than the decomposition:
𝑘 ∗ = 𝑘 ‡ A ‡ A ∗
Considering A‡ as a SS component:
𝑘 ∗ A ∗ = 𝑘 ‡ A ‡

25. 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 modes
which result in the product formation
The critical number m of vibrational quanta
should be accumulated in the critical
vibrational mode
Pilling & Seakins 1995

26. RRK theory Recall: v – m m 𝑘 ∗ = 𝑘 ‡ A ‡ A ∗ = 𝑘 ‡ 𝑃 𝑚 𝑣 𝑃 𝑚 𝑣
is the probability that m quanta out of v are on the critical mode
Recall:
v – m m
v, m and v–m >> s

27. RRK theory 𝑘 2 𝐸 = 𝑘 ‡ 1− 𝐸 0 𝐸 𝑠−1 Recall: 𝑃 𝐸 0 𝐸 = A ‡ A ∗ and
𝑘 ∗ = 𝑘 ‡ 𝑃 𝑚 𝑣 = 𝑘 ‡ 1− 𝑚 𝑣 𝑠−1
Rewriting
for continuous energy:
𝑘 ∗ = 𝑘 ‡ 1− 𝐸 0 𝐸 𝑠−1
𝑃 𝐸 0 𝐸 = 1− 𝐸 0 𝐸 𝑠−1
𝑘 ∗ = 𝑘 ‡ A ‡ A ∗
Recall:
𝑃 𝐸 0 𝐸 = A ‡ A ∗
and
𝑘 ∗ = 𝑘 ‡ 𝑃 𝐸 0 𝐸
Resulting expression for k2 (product formation):
𝑘 2 𝐸 = 𝑘 ‡ 1− 𝐸 0 𝐸 𝑠−1
The reaction rate is energy dependent; it increases with increasing energy

28. RRK theory 𝑬 𝑬 𝟎 𝑘 2 𝐸 𝑘 ‡ Limit case (Lindemann theory)
𝑘 2 𝐸 𝑘 ‡
𝑬 𝑬 𝟎
Pilling & Seakins 1995

29. RRK theory — summary of results
𝑘 1 𝐸 = 𝑍 𝑁 𝐸 𝑄 𝑒 − 𝐸 𝑘 B 𝑇
𝑘 2 𝐸 = 𝑘 ∗ (𝐸)= 𝑘 ‡ 1− 𝐸 0 𝐸 𝑠−1
Energy / cm–1
compared to the Lindemann expression:
Reaction rate (arbitrary units)
Pilling & Seakins 1995

30. 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

31. 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 𝑘 A*(E ) + M activation (via collisions)
A*(E ) + M 𝑘 − A + M deactivation (via collisions)
A*(E ) 𝑘∗( 𝐸 ′ ) A‡ formation of the transition state (IVR)
A‡ 𝑘 ‡ P (products) decomposition (spontaneous)
E: total energy
E’: variable energy

32. RRKM theory — energy landscape
total energy
E0 ”activation energy”
stable molecule
E total energy
Er rotational energy
Ev energy of vibrations and internal rotations
transition state
E‡ max. available energy
𝐸 𝑟 ‡ rotational energy
𝐸 𝑣 ‡ energy of vibrations and internal rotations
𝐸 𝑡 ‡ rel. translational energy
𝐸 𝑟 ‡ Er
𝐸 𝑡 ‡ E‡
𝐸 𝑣 ‡ E Ev
zero point energy of transition state E0
zero point energy of stable A

33. RRKM theory — role of rotation
Conservation laws:
both energy and momentum are conserved
𝐸 𝑟 ‡ Er
𝐸 𝑡 ‡ E‡
Angular momentum: L = Iω
𝐸 𝑣 ‡ E
𝐸 𝑟 = 𝐼 𝜔 2 Ev
Rotational energy: E0
I : moment of inertia; ω: angular velocity
Formation of TS: I increases → Er decreases
𝐸 𝑟 − 𝐸 𝑟 ‡ = 𝐸 𝑟 1− 𝐼 𝐼 𝑟 ‡
I << I ‡ →
Excess 𝐸 𝑟 − 𝐸 𝑟 ‡ becomes vibrational energy Ev
Loosening of internal bonds result in increase of vibrational energy

34. RRKM / QET theory E = Ev + Er Iternal degrees of freedom:
- active : free energy flow (vibration and internal rotation)
- adiabatic: rotational quantum state cannot change (rotation of the whole molecule)
Expression of the rate constant:
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:

35. We get (back) the RRK equation
Relation of RRKM to RRK theory
Number of states and density of states for classical harmonic oscillators:
(The n. d. f. is one less for the TS than for the reactant molecule.)
Substitute them into the RRKM equation:
We get (back) the RRK equation

36. The role of activation entropy (S#)
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
Example:
- Butyl benzene radical decomposition:
formation of propene and propyl radical
 MS detection: internal energy can be estimated from fragment ion ratio
Internal energy (eV)
Keszei 2015

37. Relation of microcanonical to canonical rate constant
canonical energy distribution:
k(E) – microcanonical
k(T) – canonical rate constant
High pressure: equilibrium distribution is maintained
 (E) density of states
kB Boltzmann konstant
E internal energy
T temperature

38. 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.

39. Exchange of energy between modes
Harmonic vs. anharmonic vibrations
Linear ABA molecule
harmonic oscillator
QR - symmetric mode
TS - asymmetric mode
No exchange of energy
anharmonic oscillator
Lissajous motion
Exchange of energy between modes
Pilling & Seakins 1995

40. 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.

41. Experimental testing of IVR
Rabinovitch et al., JPC 78, 2535, 1974
Ratio of the two products:
1:1 at low pressures: perfect IVR
With increasing pressure
the ratio increases
→ more frequent collisions
more effective deactivations
not enough time for IVR
dissociation at the addition site
at the other site
Model calculations:
IVR rate is ~ 1 ps

42. A flaw in RRKM theory Possible explanation:
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

43. Indirect evidence of the transition state
Moore et al., Science 256, (1992)
CH2CO  CH2 + CO
Dye laser excitation and detection
1. pulse → S1
fast ISC → T1
(above activation energy)
2. pulse: detection of CO
Energy / 103 cm–1
Reaction coordinate
Pilling & Seakins 1995
k (E) is expected to increase stepwise with increasing energy

44. Indirect evidence of the transition state
(a) disappearance of ketene
(b) production of CO measured
(c) production of CO calculated
(d) disappearance of ketene calculated
Measured intensity / arbitrary units
Good correlation between experiment and model calculations
Energy / cm–1
Pilling & Seakins 1995

45. 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)

46. 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 𝑍𝑃 𝑗,𝑖 A( j) + M activation
A( j) + M 𝑍𝑃 𝑖,𝑗 A(i) + M deactivation
A( j) 𝑘 𝑗 P (products) decomposition
𝑃 𝑗,𝑖 is the probability of transition from state i to state j
𝑃 𝑖 is the probability of finding the molecule in state i
Z is the collision number

47. Using the master equation
Equations to describe probabilities of transitions
k (i)
population at level j (probability: Pj)
decomposed population
population at level i (probability: Pi)
Both i and j can have any value between 0 and n
Transfer equation:

48. Master equation
The probability of being in any of the states:
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)

49. 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

50. Acknowledgements
Figures and tables marked as Pilling & Seakins 1995 are reproduced from
M. J. Pilling, P. W. Seakins: Reaction Kinetics, Oxford University Press, 1995

51. Thank you for your attention !
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