Physical

Chemistry

Reaction Kinetics (5)

Xuan Cheng
Xiamen University

1
 Physical Reaction Kinetics
Chemistry

Key Words
 Pyrolysis  高温分解
 Acetaldehyde  乙醛
 Methane  甲烷
 Polymerization  聚合
 Monomer  单体
 Initiator  引发剂
 Relaxation  迟豫

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 Physical Reaction Kinetics
Chemistry Rice-Herzfeld Mechanisms
A chain reaction can lead to a simple rate law.
Pyrolysis of acetaldehyde 
CH 3CHO( g )  CH 4 ( g )  CO( g )
d [CH 4 ]
 k[CH 3CHO]3 / 2
dt
Simple rate laws can follow from quite complex chain mechanisms.
The Rice-Herzfeld mechanism for the pyrolysis of acetaldehyde is
(a) Initiation: CH 3CHO 
a k
CH 3  CHO r  k a [CH 3CHO]
(b) Propagation:
CH 3CHO  CH 3 
bk
CH 4  CH 3CO  r  kb [CH 3CHO ][CH 3 ]

(c) Retardation: CH 3CO  

c k
CH 3  CO r  kc [CH 3CO]

r  k d [CH 3 ]2
k
(d) Termination:  CH 3  CH 3 
d
CH 3CH 3

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 Physical Reaction Kinetics
Chemistry Rice-Herzfeld Mechanisms
CH 3CHO 
a k
CH 3  CHO r  k a [CH 3CHO]

CH 3CHO  CH 3 

b k
CH 4  CH 3CO  r  kb [CH 3CHO ][CH 3 ]

CH 3CO  
c k
CH 3  CO r  kc [CH 3CO]

r  k d [CH 3 ]2
k
 CH 3  CH 3 
d
CH 3CH 3

The net rates of the formation of the two intermediates are

d [CH 3 ]
 k a [CH 3CHO ]  kb [CH 3CHO ][CH 3 ]  kc [CH 3CO]  k d [CH 3 ]2  0
dt
d [CH 3CO]
 kb [CH 3CHO][CH 3 ]  kc [CH 3CO]  0
dt

The sum of the two equation is

k a [CH 3CHO]  k d [CH 3 ]2  0
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 Physical Reaction Kinetics
Chemistry Rice-Herzfeld Mechanisms
k a [CH 3CHO]  kd [CH 3 ]2  0

1/ 2
k 
[CH 3 ]   a  [CH 3CHO]1 / 2
 kd 

The rate of formation of CH4 is

1/ 2
d [CH 4 ] k 
 kb[CH 3CHO][CH 3 ]  kb  a  [CH 3CHO]3 / 2
dt  kd 

in agreement with the three-halves order observed experimentally.

However, the true mechanism is more complicated than R-H mechanism.
Other products (acetone, CH3COCH3, and propanaldehyde,
CH3CH2CHO) can be formed.

Prob. 17.81 5
 Physical Reaction Kinetics
Chemistry Free-Radical Polymerizations
Chain polymerization
Results in the rapid growth of an individual polymer chain for each
activated monomer, and often occurs by a radical chain process.
Let I and M stand for the initiator and monomer
k
(a) Initiation I 
i
2R 

k
R   M 
a
RM 
k
(b) Propagation M  M1  

p1
 M 2
k
M  M 2  

p2
 M 3

k
p , n 1
M  M n 1    M n

(c) Termination  M n  M m 

t k
M m n m  0,1,2,  , n  0,1,2, 
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 Physical Reaction Kinetics
Chemistry Free-Radical Polymerizations
(a) Initiation I 
ik
2R  r  ki [I ]
k
R   M 
a
RM  (fast)
The rate-determining step is the formation of the radicals R.
k p1
(b) Propagation M  M1    M 2
k
M  M 2  

p2
 M 3

k
p , n 1
M  M n 1    M n r  k p [ M ][M ]

The chain of reactions propagates quickly,

d [M ]
 2 fki [ I ] (17.99)
dt
f is the yield of the initiation step, the fraction of radicals that R
successfully initiate a chain. 0.3  f  0.8
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 Physical Reaction Kinetics
Chemistry Free-Radical Polymerizations
(c) Termination k
 M n  M m 
t
M m n m  0,1,2,  , n  0,1,2, 

Assume that the rate of termination is independent of the length of

the chain,
r  kt [M ]2

the rate of change of radical concentration by this process is

d [M ]
 2kt [M ]2 (17.101)
dt

The total radical concentration is approximately constant throughout

the main part of the polymerization.

(the rate at which radicals are formed by initiation 

the rate at which they are removed by termination)
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 Physical Reaction Kinetics
Chemistry Free-Radical Polymerizations
Applying the steady-state approximation
d [M ]
 2 fki [ I ]  2kt [M ]2  0
dt

The steady-state concentration of radical chains

1/ 2
 fk 
[M ]   i  [ I ]1 / 2 (17.102)
 kt 

The rate of propagation of the chains (the monomer is consumed)

d[M ]
 k p [M ][ M ]
dt
1/ 2
d[M ]  fki 
 k p   [ I ]1 / 2 [ M ] (17.103)
dt  kt 

Centr The rate of polymerization is proportional to

al the square root of the initiator concentration. 9
 Physical Reaction Kinetics
Chemistry
Free-Radical Polymerizations
The degree of polymerization (DP)
The number of monomers in the polymer
 d [ M ]  d [ M ] / dt
 DP   (17.104)
d [ Ptot ] d [ Ptot ] / dt

1 d [ Rtot ]
d [ Ptot ] / dt    kt [ Rtot ]2  fki [ I ] (17.105)
2 dt

1/ 2
d[M ]  fk 
 k p  i  [ I ]1 / 2 [ M ] (17.103)
dt  kt 

k p [M ]
 DP  for termination by combination (17.104)
( fki kt )1 / 2 [ I ]1 / 2

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 Physical Reaction Kinetics
Chemistry Fast Reactions
Experimental methods for fast reactions
Rapid-flow method Movable
spectrometer
Pistons Mixing
chamber Disadvantage:
A large volume of
reactant solution

The reactants are mixed as they flow together in a chamber. The

reaction continues as the thoroughly mixed solutions flow through
the outlet tube, and observation of the composition at different
positions along the tube is equivalent to the observation of reactant
mixture at different times after mixing.
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 Physical Reaction Kinetics
Chemistry Fast Reactions
Experimental methods for fast reactions
Stopped-flow method Movable
spectrometer
Pistons Mixing Stopping Disadvantage:
chamber syringe
A large volume of
reactant solution

Small samples
The two solutions are mixed very rapidly by injecting them into a
tangential mixing chamber. Beyond the mixing chamber there is
an observation cell fitted with a stopping syringe, when a required
volume (1 mL) has been injected. The reaction continues in the
thoroughly mixed solution and is monitored.
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 Physical Reaction Kinetics
Chemistry Fast Reactions
Experimental methods for fast reactions
Flash photolysis method
The gaseous or liquid sample is exposed to a brief photolytic
flash of light and then the contents of the reaction chamber are
monitored. Both emission and adsorption spectroscopy may be
used to monitor the reaction, and the spectra are observed
electrochemically or photographically at a series of times
following the flash.

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 Physical Reaction Kinetics
Chemistry Fast Reactions
Temperature-jump relaxation methods [A]
Relaxation T2
The return of a system to equilibrium
Temperature jump Exponential
T1 relaxation
Consider the reversible reaction
kf r f  k f [ A][ B ]
A  B 
 C Time, t
kb rb  kb [C ]

dA
For all times after the T jump   k f [ A][ B]  kb [C ] (17.107)
dt
Equilibrium concentrations at T2 [ A]eq , [ B ]eq , [C ]eq

d [ A] dx
let x  [ A]eq  [ A] x  [ B ]eq  [ B ]  x  [C ]eq  [C ] 
dt dt
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 Physical Reaction Kinetics
Chemistry Fast Reactions
Temperature-jump relaxation methods
[A]
dA
  k f [ A][ B]  kb [C ] (17.107)
dt T2
dx
   k f ([ A]eq  x)([ B ]eq  x)  kb ([C ]eq  x)
dt Exponential
dx T1 relaxation
 k f [ A]eq [ B ]eq  kb [C ]eq  xk f ([ A]eq  [ B ]eq
dt
 kb k f 1  x)
Time, t
(17.108)
d [ A]
At equilibrium 0 k f [ A]eq [ B]eq  kb [C ]eq  0 (17.109)
dt

The perturbation is small x  [ A]eq  [ B ]eq

dx
dt
  1x  
  k f ([ A]eq  [ B ]eq  kb 1 (17.110)
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 Physical Reaction Kinetics
Chemistry Fast Reactions
Temperature-jump relaxation methods

dx
dt
  1x  
  k f ([ A]eq  [ B ]eq  kb 1 (17.110)


x  x0e t
  
  k f ([ A]eq  [ B]eq )  kb 1

Where x is the departure from equilibrium at the new temperature and

x0 is the departure from equilibrium immediately after the
temperature jump.
t
x  [ A]eq  [ A]  [C ]  [C ]eq 
[ A]  [ A]eq  ([ A]0  [ A]eq )e 

The concentration of A (and of B) relaxes into the new equilibrium

at a rate determined by the sum of the two new rate constants.
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 Physical Reaction Kinetics
Chemistry
Fast Reactions
Analyzing a temperature-jump experiment
The H2O(l) H+(aq) + OH-(aq) equilibrium relaxes in 37 s at 298K
and pH7, pKw=14.01. Calculate the rate constants for the forward
and reverse reactions. r  k [ H O] f f 2
 
H 2O(l )  H (aq)  OH (aq)
rr  kb [ H  ][OH  ]
1
 k f  kb ([ H  ]  [OH  ])


The equilibrium condition is k f [ H 2O]eq  kb [ H  ]eq [OH  ]eq

kf [ H  ][OH  ] kw k Kw
   w molL1 K  1.8  10 16
kb [ H 2O ] [ H 2O] 55.6 55.6

1
 kb ( K  [ H  ]  [OH  ])  kb ( K  K 1w/ 2  K 1w/ 2 )  (2.0 10 7 )  kb molL1

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 Physical Reaction Kinetics
Chemistry Fast Reactions
Analyzing a temperature-jump experiment
The H2O(l) H+(aq) + OH-(aq) equilibrium relaxes in 37 s at
298K and pH7, pKw=14.01. Calculate the rate constants for the
forward and reverse reactions.
1
 kb ( K  [ H  ]  [OH  ])  kb ( K  K 1w/ 2  K 1w/ 2 )  (2.0 10 7 )  kb molL1

1
kb   1.4 1011 Lmol 1s 1
(37 10 6 s)  (2.0 10  7 molL1 )

k f  kb K  2.4  10 5 s 1

K and Kw are dimensionless

kf and kb are expressed in different units

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 Physical Reaction Kinetics
Chemistry Reactions in Liquid Solutions
Solvent Effects on Rate Constants
gas-phase reaction
solvent
liquid-phase reaction
Ionic Reactions
solvation H o 
gas-phase reaction
solvent
liquid-phase reaction ions G o 

Encounters, Collisions, and the Cage Effect

gas-phase reaction
Molecules are far apart and move freely between collisions
liquid-phase reaction
Little empty space between molecules and can’t move freely
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 Physical Reaction Kinetics
Chemistry Reactions in Liquid Solutions
Encounters, Collisions, and the Cage Effect

C
C

B B

Cage effect for B and C

Encounters
A process in which B and C diffuse together to become neighbors
Collisions
Each encounter in solution involves many collisions between B and C

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 Physical Reaction Kinetics
Chemistry Reactions in Liquid Solutions
Diffusion-controlled Reactions

C
C

B B

Cage effect for B and C

Encounter pair
Gas-phase
More encounters, shorter time together
Liquid-phase
Less encounters but stay near each other for much longer than in a gas

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 Physical Reaction Kinetics
Chemistry Reactions in Liquid Solutions
Diffusion-controlled Reactions
Suppose the rate of formation of an encounter pair BC is
B  C 
d
BC
k r  k d [ B ][C ]

k d' r  k d' [ BC ]
BC  B  C

BC 
a
P
k r  k a [BC ]

The steady-state concentration of BC

d[ BC ]
 k d [B][C]  k d' [BC]  k a [BC]  0
dt

k [B][C]
[ BC ]  d
k a  kd'

The overall rate law for the formation of

products 22
 Physical Reaction Kinetics
Chemistry Reactions in Liquid Solutions
Diffusion-controlled Reactions k [B][C]
[ BC ]  d
k a  k d'
The overall rate law for the formation of products
d[ P] ka kd
 k a [BC]  k 2 [B][C] k2 
dt k a  kd'

(1) diffusion-controlled reaction

If the rate of separation of the unreacted encounter pair is much
slower than the rate at which it forms products kd
B  C  BC
k d'  k a k k k d'
k2  a d  kd BC  B  C
ka
k
BC 
a
P

The rate of reaction is governed by the rate at which the reactant

particles diffuse through the medium.
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 Physical Reaction Kinetics
Chemistry Reactions in Liquid Solutions
Diffusion-controlled Reactions k [B][C]
[ BC ]  d
k a  k d'
The overall rate law for the formation of products
d[ P] ka kd
 k a [BC]  k 2 [B][C] k2 
dt k a  kd'

(2) activation-controlled reaction

If the rate of separation of the unreacted encounter pair is much
faster than the rate at which it forms products B  C 
kd
BC k
K d
k a  k d' k k k d' k d'
k2  a d  ka K BC  B  C
k d'
k
BC 
a
P

The reaction proceeds at the rate at which energy accumulates in the

encounter pair from the surrounding solvent.
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 Physical Reaction Kinetics
Chemistry Reactions in Liquid Solutions
Diffusion-controlled Reactions
The rate of a diffusion-controlled reaction is calculated by
considering the rate at which the reactants diffuse together.
k
B  C 
d
P

k D  4N A (rB  rC )( DB  DC ) where B  C, nonionic (17.111)

k D  2N A (rB  rC )( DB  DC ) where B = C, nonionic (17.112)
W
k D  4N A (rB  rC )( DB  DC ) where B  C, ionic (17.113)
eW  1

z B zC e 2
W
4 0 r kT (rB  rC )

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 Physical Reaction Kinetics
Chemistry
Reactions in Liquid Solutions
Diffusion-controlled Reactions
When apply the Stokes-Einstein equation (16.37)
kT kT
DB 
6rB
DC 
6rC  Is the solvent’s viscosity

k D  4N A (rB  rC )( DB  DC ) where B  C, nonionic (17.111)

2 RT (rB  rC ) 2 2 RT  r r 
kD 
3 rB rC

3
 2  B  C 
rC rB 
where B  C, nonionic

8RT
if rB = rC where B  C, nonionic
3 (17.114)
kD 
4RT
3 where B = C, nonionic (17.115)

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 Physical Reaction Kinetics
ChemistryReactions in Liquid Solutions
Activation Energies
Gas-phase reactions: high temperature (up to 1500K)
Gas-phase reactions: activation energy range -3100 kcal/mol
Liquid-phase reactions: relatively lower temperature (up to 500K)
Liquid-phase reactions: activation energy range 235 kcal/mol

Home Work
17.67 17.70 17.77
17.83 17.87 17.89

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