Chain reactions
Tamás Turányi
Institute of Chemistry
Eötvös University (ELTE)
Budapest, Hungary
Max Bodenstein (German, 1871-1942)
Investigated the H2Cl2 photochemical reaction
and observed that single photon  several million HCl product species
Explanation of Bodenstein (1913):
Primary reaction:
Absorption of a single photon 
single active molecule (maybe Cl2+ ???)
Secondary reactions:
Single active molecule 
several million product species
The origin of term ‘chain reactions’ :
the gold watch chain of Bodenstein
This term was printed for the first time in 1921 in the PhD thesis of
Jens Anton Christiansen (Danish, 1988-1969)
Bodenstein and Lind investigated (1907)
the production of hydrogen bromide in a thermal reaction: H2  Br2  2 HBr
d [HBr]
k H 2 Br2 

.
Br2   k [HBr]
dt
3/ 2
Empirical rate equation:
Bodenstein could not explain the origin of this equation.
The proper mechanism was suggested (1919)
independently from each other by
Jens A. Christiansen, Karl F. Herzfeld and Michael Polanyi :
Karl F. Herzfeld (Austrian, 1892-1978)
theory of reaction rates, chain reactions
Br2  M  2 Br  M
Br  H2  HBr  H
H  Br2  HBr  Br
H  HBr  H2  Br
2 Br  M  Br2  M
Michael Polanyi (Hungarian, 1891-1976)
first potential-energy surface, transition-state theory, sociology
Chain reactions
Chain carriers (also called chain centres, i.e. reactive intermediates)
are generated in the initiation steps.
In the chain propagation steps the chain carriers react with the reactants,
produce products and regenerate the chain carriers.
In the inhibition step the chain carriers react with the product,
reactants are reformed, and there is no reduction
in the number of chain carriers.
In the branching step two or more chain carriers are produced
from a single chain carrier.
In the termination steps the chain carriers are consumed.
Mechanism of the H2Br2 reaction
(a) initiation:
1
Br2  M  2 Br  M
v1  k1Br2 M
(b) propagation:
2
3
Br  H2  HBr  H
H  Br2  HBr  Br
v2  k2 [Br][H 2 ]
v3  k3[H][Br 2 ]
(c) inhibition:
4
H  HBr  H2  Br
v4  k 4[H][HBr]
(d) termination:
5
2 Br  M  Br2  M
v5  k5 [Br] 2 M 
Calculation of the concentrationtime profiles
dH 2 
 v2  v4  k 2 [Br][H 2 ]  k 4 [H][HBr]
dt
dBr2 
 v1  v3  v5  k1 Br2 M   k3 [H][Br 2 ]  k5 [Br] 2
dt
dH 
 v2  v3  v4  k 2 [Br][H 2 ]  k3 [H][Br 2 ]  k 4 [H][HBr]
dt
1
2
3
4
5
Br2  M  2 Br  M
Br  H2  HBr  H
H  Br2  HBr  Br
H  HBr  H2  Br
2 Br  M  Br2  M
dBr 
 2v1  v2  v3  v4  2v5  2k1 Br2 [ M ]  k 2 [Br][H 2 ]  k3[H][Br 2 ]  k 4 [H][ H Br]  2k5 [Br] 2 [ M ]
dt
dHBr 
 v2  v3  v4  k 2 [Br][H 2 ]  k3 [H][Br 2 ]  k 4 H [H Br]
dt
concentrationtime profiles of the H2Br2 reaction
(stoichiometric mixture, T= 600 K, p= 1 atm)
Relative rates at t = 1 second
(all rates are normed with respect to v1)
rates of reaction steps
rates of R1 and R5 << rates of R2 and R3
1.0 rate of R1 = rate of R5
R1 Br2+M2 Br+M
R2 Br+H2HBr+H
100.2 In the case of small [HBr] :
100.1 rate of R2 = rate of R3
R3 H+Br2HBr+Br
R4 H+HBrH2+Br
0.1
R5 2 Br+M  Br2+M
1.0
production rates
d[H2]/dt
-100.1
d[Br2]/dt
-100.1
d[HBr]/dt
+200.2
d[H]/dt
+0.0014
d[Br]/dt
+0.0026
Relation of reaction rates and production rates
dHBr 
 v2
dt
 v3
 v4
200.2 = +100.2 +100.1
–0.1
dH 
 v2
dt
 v4
 v3
0.0014 = +100.2 –100.1
dBr 
 2v1
dt
 v2
1
2
3
4
5
–0.1
 v3
 v4
 2v5
0.0026 = 2.0 – 100.2 + 100.1 + 0.1 – 2.0
Br2  M  2 Br  M
Br  H2  HBr  H
H  Br2  HBr  Br
H  HBr  H2  Br
2 Br  M  Br2  M
Calculation of [Br]
dH 

v2  v3  v4
0
dt
dBr 
 2v1  v2  v3  v4  2v5  0
+
dt
_________________________________________
2v1
 2v5  0
v1  v5
1
Br2  M  2 Br  M
5
2 Br  M  Br2  M
k1 Br2 M  k5 [Br] 2 M 
k1
Br   Br2 
k5
Calculation of [H]
dH 
 k 2 [Br][H 2 ]  k3 [H][Br 2 ]  k 4 [H][HBr]
dt
0  k2 [Br][H 2 ]  k3[H][Br 2 ]  k4 [H][HBr]
k1
Br   Br2 
k5
Equation for [Br] is inserted:
0  [H 2 ] k2
k1
Br2   k3[H][Br 2 ]  k4[H][HBr]
k5
k1
Br2 
[H 2 ] k 2
k5
H 
k 3 [Br2 ]  k 4 [HBr]
Algebraic equations for the calculation of [H] and [Br]:
Br   f1 Br2  , k1 , k5 
H  f 2 Br2  , H 2  , HBr  , k1, k2 , k3 , k4 , k5 
Calculation of the production rate of HBr
Br  
k1
Br2 
k5
k1
Br2 
k5
H 
k 3 [Br2 ]  k 4 [HBr]
[H 2 ] k 2
dHBr 
 v2  v3  v4  k 2 [Br][H 2 ]  k3 [H][Br 2 ]  k 4 H [H Br]
dt
After insertion of the equations
for [Br] and [H] and rearrangement:
3
k1
2 k2
[H 2 ]Br2 2
k5
dHBr 

k
dt
[Br2 ]  4 [HBr]
k3
This is identical to the empirical equation of
Bodenstein and Lind:
d [HBr]
k H 2 Br2 

.
Br2   k [HBr]
dt
3/ 2
[HBr] is almost zero at the beginning of the reaction:
1
dHBr 
k1
 2 k2
[H 2 ]Br2 2
dt
k5
Order for H2 and Br2 are 1 and 0.5, respectively.
The overall order of the reaction is 1.5
Chain length
Mean number of propagation steps which occur before termination =
consumption rate of the chain carrier in the propagation step

consumption rate of the chain carrier in the termination step
v 2 100.2


 50.1
2 v5
2
The chain length at t=1 s
in the H2Br2 reaction
at the defined conditions
The origin of explosions
Mixture H2+Br2 cannot explode at isothermal conditions.
Suggestion of Christiansen and Kramers (1923):
explosions are due to branching chain reactions
BUT: it was a pure speculation
First experimental proof:
Nikolay Nikolaevich Semenov (Russian, 1896-1986)
Investigation (1926) of the phosphorus vapouroxygen reacion.
Explosion occurs, if the partial pressure of O2 is
between two limits. Interpretation via a branching chain reaction.
Sir Cyril Norman Hinshelwood (English, 1897-1967)
Investigation (1927) of the H2O2 reaction:
discovery of the 1st and 2nd explosion limits
The Nobel Prize in Chemistry 1956: Semenov and Hinshelwood:
"for their researches into the mechanism of chemical reactions"
Explosion of hydrogenoxygen mixtures
2 H2 + O2  2 H2O
Observations
The 1st explosion limit depends on the size of the vessel and the quality of the wall.
The 2nd and 3rd limits do not depend on these
1
2
3
4
5
6
7
8
9
10
11
H2 + O2  .H + .HO2
.OH + H2  .H + H2O
.H + O2  .OH + O
O + H2  .OH + .H
.H + O2 + M  .HO2 + M
.H  wall
:O  wall
.OH  wall
.HO2 + H2  .H + H2O2
2 .HO2  H2O2 + O2
H2O2  2 .OH
initiation
propagation
branching
branching
termination*
termination
termination
termination
initiation *
termination
initiation
1
2
3
4
5
6
7
8
9
10
11
H2 + O2  .H + .HO2
.OH + H2  .H + H2O
.H + O2  .OH + O
O + H2  .OH + .H
.H + O2 + M  .HO2 + M
.H  wall
:O  wall
.OH  wall
.HO2 + H2  .H + H2O2
2 .HO2  H2O2 + O2
H2O2  2 .OH
initiation
propagation
branching
branching
termination*
termination
termination
termination
initiation *
termination
initiation
Below the 1st explosion limit:
domination of the termination reactions at the
wall
 no explosion

1
2
3
4
5
6
7
8
9
10
11
H2 + O2  .H + .HO2
.OH + H2  .H + H2O
.H + O2  .OH + O
O + H2  .OH + .H
.H + O2 + M  .HO2 + M
.H  wall
:O  wall
.OH  wall
.HO2 + H2  .H + H2O2
2 .HO2  H2O2 + O2
H2O2  2 .OH
Between the
1st
and the
2nd
initiation
propagation
branching
branching
termination*
termination
termination
termination
initiation *
termination
initiation

H.
H.
explosion limits:
H.
H.
Branching steps (2), (3) and (4).
3
H + O2  .OH + :O
2
.OH + H2  .H + H2O
4
:O + H2  .H + .OH
2
.OH + H2  .H + H2O
+ ____________________
.H + O2 + 3 H2  3 .H + 2 H2O
 explosion
H.
H.
H.
H.
H.
H.
H.
H.
H.
H2 + O2  .H + .HO2
.OH + H2  .H + H2O
.H + O2  .OH + O
O + H2  .OH + .H
.H + O2 + M  .HO2 + M
.H  wall
:O  wall
.OH  wall
.HO2 + H2  .H + H2O2
2 .HO2  H2O2 + O2
H2O2  2 .OH
1
2
3
4
5
6
7
8
9
10
11
initiation
propagation
branching
branching
termination*
termination
termination
termination
initiation *
termination
initiation
Between the 2nd and the 3rd explosion limits:
5
.H + O2 + M  .HO2 + M
 no explosion
termination*

1
2
3
4
5
6
7
8
9
10
11
H2 + O2  .H + .HO2
.OH + H2  .H + H2O
.H + O2  .OH + O
O + H2  .OH + .H
.H + O2 + M  .HO2 + M
.H  wall
:O  wall
.OH  wall
.HO2 + H2  .H + H2O2
2 .HO2  H2O2 + O2
H2O2  2 .OH
initiation
propagation
branching
branching
termination*
termination
termination
termination
initiation *
termination
initiation
above the 3rd explosion limit
Reactions (9), (10), and (11) become important
 explosion

The two basic types of chain reactions
Open chain reactions
Chain reactions without branching steps
Examples: H2 + Br2, reaction,,
alkane pyrolysis and polimerisation reactions
Branched chain reactions
Chain reactions that include branching reaction steps
Examples: H2+O2 reaction,
hydrocarbonair explosions and flames
Two types of explosions
Branched chain explosions:
rapid increase of the concentration of chain carriers leads to
the increase of reaction rate and finally to explosion
Another possibility:
(i) exothermic reaction,
(ii) hindered dissipation of heat and
(iii) increased reaction rate with raising temperature, then
higher temperature  faster reactions  increased heat production
 thermal explosion
Presence of a chain reaction is not needed for a thermal explosion.
Branched chain reactions are
• exothermic and fast
• dissipation of heat is frequently hindered
 most branched chain explosions are also thermal explosions
Temperature dependence of the rate coefficient
Van’t Hoff’s equations (1884): k  A e

E
RT
or
k  Ae

B  DT 2
RT
Theoretical considerations of Arrhenius (1889):
• equilibrium between the ‘normal’ and ‘active’ species
• activation energy E is T-independent in small temperature range
Arrhenius equation:
k  Ae

E
RT
Jacobus Henricus Van’t Hoff (Dutch, 1852-1911)
The first Nobel Prize in Chemistry (1901) „in recognition of the
extraordinary services he has rendered by the discovery of the
laws of chemical dynamics and osmotic pressure in solutions”
Svante August Arrhenius (Swedish, 1859-1927)
Nobel Prize in Chemistry (1903), electrolytic theory of dissociation
Arrhenius-plot
Arrhenius equation:
 Ea 
k  A exp 

 RT 
A
Ea
preexponential factor
activation energy
Arrhenius-plot:
or
Ea
ln k  ln A 
RT
Plotting ln k against 1/T gives a line
Slope: m = -Ea/R gives activation energy Ea
Reaction CH4+OH  CH3 + H2O
the most important methane consuming reaction in the troposphere
one of the most important reactions of methane combustion
Arrhenius-plot between 220 K (53 C )
and 320 K (+47 C)
Arrhenius-equation
is usually very accurate in a
narrow temperature range
(solution phase kinetics,
atmospheric chemistry).
Arrhenius-plot between 300 K
(27 C )
and 2200 K (1930 C)
Arrhenius-equation
is usually not applicable
in a wide temperature range
(combustion, explosions, pyrolysis).
Extended Arrhenius-equation
k  BT n e

C
RT
Note that if n0  AB and EaC
General definition of activation energy:
  ln k 

Ea   R
  1 T   p
Thank you all
for your attention
Literature used:
Michael J. Pilling – Paul W. Seakins
Reaction Kinetics
Oxford University Press, 1995
Keith J. Laidler
The World of Physical Chemistry
Oxford University Press, 1995
‘The Nobel Prize in Chemistry 1956’
Presentation speech by Professor A. Ölander
http://nobelprize.org/chemistry/laureates/1956/press.html
H2Br2 and H2O2 concentration-time profiles
were calculated by Dr. István Gy. Zsély
(Department of Physical Chemistry, Eötvös University, Budapest)
Comments of Dr. Judit Zádor, Mr. János Daru, and Dr.Thomas Condra
are gratefully acknowledged.
Special thank to Prof. Preben G. Sørensen (University of Copenhagen)
for the photo of J. A. Christiansen and
to Prof. Ronald Imbihl (Universität Hannover)
for the photo of the gold watch of Bodenstein