Physical ChemistryLecture 9

Kinetics of Complex Reactions.

The kinetics of complex reactions

• Chain reactions• Polymerization reactions• Enzyme catalyzed reactions• Photochemistry• Oscillating reaction and chaos

Chain Reactions• Chain reaction: reaction where an intermediate produced in

one step can generate another intermediate in the next step. These intermediates are called chain carriers.

Example: Thermal decomposition of acetaldehyde (ethanal)

3/ 23 4 3( ) ( ) [ ]CH CHO g CH g CO v k CH CHO⎯⎯→ + =

3 3 3

3 3 4 3 3 3

3 3 3

3 3 3 3 3

Initiation: ( ) [ ]

Propagation: [ ][ ]

Propagation: [ ]

Termination: [ ]

i

p

p

t

CH CHO CH g CHO v k CH CHO

CH CHO CH CH CH CO v k CH CHO CH

CH CO CH CO v k CH CO

CH CH CH CH v k CH

⎯⎯→• +• =

+ • ⎯⎯→ + • = •

•⎯⎯→• + = •

• + • ⎯⎯→ = •

Rice-Herzfeld mechanism Chain carriers

In steady-state approximation, [●CH3]=const and [CH3CO●]=const

233 3 3 3 3

[ ] [ ] [ ][ ] [ ] 2 [ ] 0i p p td CH k CH CHO k CH CHO CH k CH CO k CH

dt ′•

= − • + • − • =

33 3 3

[ ] [ ][ ] [ ] 0p pd CH CO k CH CHO CH k CH CO

dt ′•= • − • =

Chain Reactions23

3 3 3 3 3[ ] [ ] [ ][ ] [ ] 2 [ ] 0i p p t

d CH k CH CHO k CH CHO CH k CH CO k CHdt ′•

= − • + • − • =

33 3 3

[ ] [ ][ ] [ ] 0p pd CH CO k CH CHO CH k CH CO

dt ′•= • − • =

+

2 1/ 233 3 3 3

3/ 243 3 3

[ ] [ ] 2 [ ] 0 [ ] [ ]2

[ ] [ ][ ] [ ]2

ii t

t

ip p

t

d CH kk CH CHO k CH CH CH CHOdt k

kd CH k CH CH CHO k CH CHOdt k

•= − • = • =

= • =

Chain Reactions• Example: Hydrogen-Bromine reaction

2 23/ 2

2 2

2

( ) ( ) 2 ( )

[ ][ ][ ] [ ]

H g Br g HBr g

k H BrvBr k HBr

+ ⎯⎯→

=′+

2 2

2 2

2 2

2

2

Initiation: [ ][ ]

Propagation: [ ][ ]

Propagation: [ ][ ]

Retardation: [ ][ ]

Termination: [ ]

i

p

p

r

t

Br M Br Br M v k Br M

Br H HBr H v k Br H

H Br HBr Br v k H Br

H HBr H Br v k H HBr

Br Br M Br M v k Br

′

+ ⎯⎯→ •+ •+ =

•+ ⎯⎯→ + • = •

•+ ⎯⎯→ + • = •

•+ ⎯⎯→ + • = •

•+ •+ ⎯⎯→ + = • 2[ ]M

collision with Br2 or H2.

Chain Reactions

2 2[ ] [ ][ ] [ ][ ] [ ][ ]p p r

d HBr k Br H k H Br k H HBrdt ′= • + • − •

2 2

2 2

2 2

2

2

Initiation: [ ][ ]

Propagation: [ ][ ]

Propagation: [ ][ ]

Retardation: [ ][ ]

Termination: [ ]

i

p

p

r

t

Br M Br Br M v k Br M

Br H HBr H v k Br H

H Br HBr Br v k H Br

H HBr H Br v k H HBr

Br Br M Br M v k Br

′

+ ⎯⎯→ •+ •+ =

•+ ⎯⎯→ + • = •

•+ ⎯⎯→ + • = •

•+ ⎯⎯→ + • = •

•+ •+ ⎯⎯→ + = • 2[ ]M

2 2[ ] [ ][ ] [ ][ ] [ ][ ] 0p p r

d H k Br H k H Br k H HBrdt ′•= • − • − • =

2 2 2 2[ ] 2 [ ][ ] [ ][ ] [ ][ ] [ ][ ] 2 [ ][ ] 0i p p r t

d Br k Br M k Br H k H Br k H HBr k Br Mdt ′•= − • + • + • − =

Steady-state approximation:

( )

1/ 2 1 22 21/ 2

22

1 22 2

2

[ ][ ][ ] [ ] [ ]

[ ] [ ]

2 [ ][ ][ ][ ] [ ]

p i ti

t p r

p i t

r p

k k k H BrkBr Br Hk k Br k HBr

k k k H Brd HBrdt Br k k HBr

′

′

⎛ ⎞• = • =⎜ ⎟ +⎝ ⎠

=+

Explosions

• Thermal explosion is caused by a very rapid reaction arising from a rapid increase of reaction rate with temperature

• Chain-branching explosion occurs when number of chain centers grows exponentially

ExplosionsExample: reaction of oxygen and hydrogen 2 2 22 ( ) ( ) 2 ( )H g O g H O g+ ⎯⎯→

2

2 2 2

2 2

2 2

2

2 2 2

Initiation:

Propagation: [ ][ ]

Branching: [ ][ ]

[ ][ ]1Termination: [ ]2

[ ][ ]

p

b

b

t

t

H H H v const

H OH H H O v k H OH

O H O OH v k O H

O H OH H v k O H

H wall H v k H

H O M HO M v k H O

⎯⎯→ •+ • =

+ • ⎯⎯→ •+ = •

•+ •⎯⎯→• •+ • = • • •

′• • + ⎯⎯→• + • = • •

•+ ⎯⎯→ = •

′• + + ⎯⎯→ •+ = • [ ]M

Recombination on the walls is dominant

Three-particle collisions are important

2 2 2 2Regeneration: HO H H H O•+ ⎯⎯→ •+

Above 3rd explosion limit

Polymerization Kinetics

• Stepwise polymerization:any two monomers can link together, growth of polymer is not confined to already formed chains. Average mass increasing with time

• Chain polymerization:an activated monomer attacks another monomer to form growing chain. Yield, not the average mass, increasing with time.

Stepwise Chain

Polymerization Kinetics• Stepwise polymerization: formation of

– polyamides (nylon-66)2 2 6 2 2 4 2 2 6 2 4 2

2 6 2 4

( ) ( ) ( ) ( )

[ ( ) ( ) ]n

H N CH NH HOOC CH COOH H N CH NHCO CH COOH H O

H HN CH NHCO CH CO OH

+ ⎯⎯→ +

⎯⎯→ − −

Condensation reaction should be overall second rate:

2 0

0

[ ][ ][ ] [ ] [ ]1 [ ]

AdA k OH A k A Adt kt A

= − = − =+

0 0

0 0

[ ] [ ] [ ][ ] 1 [ ]

A A kt ApA kt A−

= =+

Fraction of condensed groups:

Degree of polymerization: 00

[ ] 1 1 [ ][ ] 1An kt AA p

= = = +−

The average length grows linearly with time

HO M COOH− −– polyester: stepwise condensation of hydroxyacid

Polymerization Kinetics• Chain polymerization occurs by addition of monomers to a growing

polymer, as in polymerization of ethene, methyl methacrylate and styrene:

2 2 2 2CH CHX CH CHX CH CHXCH CHX− •+ = ⎯⎯→− •

• Rate of polymerization is proportional to the square root of the initiator concentration:

1

1 2

2 3

1

Initiation: [ ]

( )

Propagation:

[ ][ ]

i

n n p

I R R v k I

M R M fast

M M M

M M M

M M M v k M M+

⎯⎯→ •+ • =

+ •⎯⎯→•

+• ⎯⎯→•

+• ⎯⎯→•

+• ⎯⎯→• = •

Total concentration of radicals will be determined by the initiation step (rate-determining process)

[ ] 2 [ ]iproduction

d M fk Idt•⎛ ⎞ =⎜ ⎟

⎝ ⎠

Polymerization Kinetics

Assuming only mutual termination occurs:

2[ ] [ ]tdepletion

d M k Mdt•⎛ ⎞ = •⎜ ⎟

⎝ ⎠

Termination: (mutual termination)

(disproportionation)

(chain transfer)

n m n m

n m n m

n n

M M M

M M M M

M M M M

+• + • ⎯⎯→

• +• ⎯⎯→ +

+• ⎯⎯→• +

In a steady-state approximation: 1 2

2 1 2[ ] 2 [ ] [ ] 0 [ ] [ ]ii t

t

fkd M fk I k M M Idt k

⎛ ⎞•⎛ ⎞ = − • = • = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

The propagation rate is the negative of the monomer consumption rate:

1 21 2[ ][ ] [ ] [ ]i

p p pt

fkv k M M k I Mk

⎛ ⎞= • = ⎜ ⎟

⎝ ⎠

1Propagation: [ ][ ]n n pM M M v k M M++ • ⎯⎯→• = •

Polymerization KineticsKinetic chain length:

In chains are terminated by mutual termination, than the average chain length is the length of two chains combining:

The slower initiation rate the higher the average mass for polymers

number of monomer units consumed rate of propagation of the chainsnumber of activated centers produced rate of production of radicals

v = =

( )

2

1 2 1 2

[ ][ ] [ ]2 [ ] 2 [ ]

[ ]

p p

t t

p i t

k M M k Mv

k M k M

v k fk k I− −

•= =

• •

=

( ) 1 2 1 22 2 [ ][ ]p i tn v k fk k M I− −= =

Homogeneous Catalysis• Catalyst is a substance that accelerates a reaction

but undergoes no net chemical change. Catalyst lowers the activation energy by providing and alternative path.

• Enzymes are biological catalysts• Homogeneous catalyst is a catalyst in the same

phase as the reaction mixture• Heterogeneous catalyst is a catalyst in a different

phase

Example: decomposition of hydrogen peroxide

Activation energy Rate increase at 298KNon-catalyzed: Ea=76 kJ/mol 1Iodide catalyzed Ea=57 kJ/mol 2000Enzyme catalase Ea=8 kJ/mol 1015.

Homogeneous Catalysis

2 2 2 22 ( ) 2 ( ) ( )BrH O aq H O l H g−

⎯⎯→ +

3 23 2 2 3 2 2

2 2 3

3 2 2 3 2

2 2 3 2

[ ]( )[ ][ ]

[ ][ ]

( )

H OH O H O aq H O H O KH O H O

H O Br HOBr H O v k H O Br

HOBr H O H O O Br fast

++ +

+

+ − + −

+ −

+ ←⎯→ + =

+ ⎯⎯→ + =

+ ⎯⎯→ + +

Pre-equilibrium

22 2 3

[ ] [ ][ ][ ]d O kK H O H O Brdt

+ −=

In acid catalysis the critical step is the transfer of a proton to the substrate:

X HA HX A HX products+ − ++ ⎯⎯→ + ⎯⎯→

In base catalysis the critical step is the transfer of a proton from the substrate to a base:

HX B X BH X products− + −+ ⎯⎯→ + ⎯⎯→

Enzymes• Enzymes are special proteins or nucleic

acids that contain an active site responsible for binding the substrate and processing it into products

• Enzyme-catalysed reactions can be inhibited by molecules that interfere with the formation of products

Principal features of many enzyme-catalysed reactions are:

1. For a given initial concentrationof substrate [S]0 the initial rate of product formation is proportional to the enzyme concentration [E]0.

2. For a given [E]0 and low values of [S]0 the rate of the product formation is proportional to [S]0

3. For a given [E]0 and high values of [S]0 the rate of the product formation becomes independent of [S]0reaching value of maximum velocity

Enzymes• Michaelis-Menten mechanism

,a a

b

E S ES k k

ES P k

′+ ←⎯→

⎯⎯→

[ ][ ] [ ][ ] [ ] [ ] 0

[ ] [ ][ ]

b

a a b

a

a b

v k ESd ES k E S k ES k ES

dtkES E S

k k

=

= − − =

⎛ ⎞= ⎜ ⎟′ +⎝ ⎠

Steady-state approximation

As 0

0

[ ] [ ] [ ][ ] [ ]E E ESS S

= +

≈

0

0

[ ][ ]11

[ ]

bb

a b

a

k Ev k ESk k

k S

= =⎛ ⎞′ +

+ ⎜ ⎟⎝ ⎠

vmax

KM – Michaelis constantmax

0

1[ ]

M

vv KS

=+

Enzymes• Experimental enalysis of enzyme catalysed reaction

max max 0

1 1 1[ ]

MKv v v S

⎛ ⎞= + ⎜ ⎟

⎝ ⎠

Lineweaver-Burk plot

max

0

1[ ]

M

vv KS

=+

Eadie-Hofstee plot

max0[ ]M

vv K vS

= − +

Enzymes• Catalytic efficiency of enzymes

Turnover number (catalytic constant) of an enzyme kcat is the number of catalytic cycles performed by the active site in a unit of time.

max

0[ ]cat bvk kE

= =

Catalytic efficiency of an enzyme is defined as kcat/KM .

b a

cat b a

M b bk k

a

k k kK k k

k

ε

ε

= =′ +

⎯⎯⎯→

,a a

b

E S ES k k

ES P k

′+ ←⎯→

⎯⎯→

Enzymes• Mechanism of enzyme inhibition

,

[ ][ ][ ]

[ ][ ][ ]

a a

b

I

I

E S ES k k

ES P kE IEI E I KEI

ES IESI ES I KESI

′+ ←⎯→

⎯⎯→

←⎯→ + =

′←⎯→ + =

Inhibition occurs to the enzyme or to ES complex, or to both:

max

0

max max 0

1 [ ] , 1 [ ] ,

[ ]

1 1[ ]

I IM

M

vv I K I KKS

Kv v v S

α ααα

αα

′ ′= = + = +′ +

⎛ ⎞′= + ⎜ ⎟

⎝ ⎠

Inhibition modes:- competitive inhibition – binding to the active site of an enzyme – (a)- uncompetitive inhibition – binding to an other site of the enzyme

only if the substrate present – (b);- non-competitive (mixed) inhibition – binding to other site that reduces

enzyme ability to bind the substrate

Photochemistry

• Photochemical processes are initiated by absorption of radiation by at least one component of reaction mixture

• Primary process – products are formed directly from the excited state of reactant

• Secondary process – products are formed from intermediates that are formed from excited state of a reactant

number of eventsnumber of photons absorbed abs

ii

ii

vI

vv

φ

φ

= =

=∑

Photochemistry

Photochemistry

• Of importance of timing

Electronic transition caused by adsorption: 10-16 – 10-15 sFluorescence: 10-12 – 10-6 sIntersystem crossing: 10-12 – 10-4 sPhosphorescence 10-6 – 10-1 s

Chemical reaction involving a photo excited state requires a comparable reaction constant to run

Example: consider a unimolecular chemical reaction with rate constant k=1.7x104 s-1 involving a reactant with fluorescence lifetime of 1ns and phosphorescence life time 1ms.Reaction relaxation time t=1/k=59us. Therefore only triplet state can be a suitable precursor for the reaction.

Photochemistry

• Primary quantum yield:a number of photophysical or photochemical events that lead to a primary product per number of photons absorbed

number of eventsnumber of photons absorbed

1 and

abs

ii i abs

i iabs

ii

ii

vI

v v II

vv

φ

φ

φ

= =

= = =

=

∑ ∑ ∑

∑

Photochemistry• Mechanism of decay of excited singled state

*

* *

* *

* * *

Absorption:

Fluorescence: [ ]

Internal conversion: [ ]

Intersystem crossing: [ ]

i abs abs

f f f

IC IC

ISC ISC

S hv S v I

S S hv v k S

S S v k S

S T v k S

+ ⎯⎯→ =

⎯⎯→ + =

⎯⎯→ =

⎯⎯→ =

If the absorbance of sample is not too high [S*] can be assumed small and constant:

** * *

*

*

*

[ ] [ ] [ ] [ ] 0

( )[ ]

[ ]( )[ ]

abs f IC ICS

abs f IC ICS

f f f

abs f IC ICS f IC ICS

d S I k S k S k Sdt

I k k k S

v k S kI k k k S k k k

φ

= − − − =

= + +

= = =+ + + +

0/* *0

0

[ ] [ ]1

tt

f IC ICS

S S e

k k k

τ

τ

−=

=+ +

Fluorescence life time

Photochemistry• Quenching – shortening of the lifetime of the excited state• Can be described by opening an additional channel for

deactivation of S*:* *Quenching: [ ][ ]Q QS Q S Q v k S Q+ ⎯⎯→ + =

** * * *[ ] [ ] [ ] [ ] [ ][ ] 0

[ ]

abs f IC ICS Q

f

f IC ICS Q

d S I k S k S k S k S Qdt

kk k k k Q

φ

= − − − − =

=+ + +

Stern-Volmer plot:01 [ ]f

Qk Qφ

τφ

= +

Photochemistry• Quenching mechanisms

*S Q S Q+ → +Collision deactivation* *S Q S Q+ → +Resonance energy transfer*S Q S Q or S Q− + + −+ → + +Electron transfer

Photochemistry

• Förster theory energy transfer (proposed by T.Förster in 1959)– Energy donor and acceptor are separated by a short distance– Photons emitted by an excited state of the donor can be absorbed by the

acceptor

60

6 60

TRE

R R=

+Transfer efficiency

* *S Q S Q+ → +Resonance energy transfer

,0

1 fT

f

Eφφ

= −Transfer efficiency

Complex photochemical reactions

• Complex reactions involving secondary processes might have quantum yield higher than 1

2*

2

HI hv H IH HI H I

I I M I M

+ → ++ → +

+ + → +

i ii ii i

Example: photolysis of HI. One photon breaks 2 HI molecules

Oscillating reactions• Belousov-Zhabotinsky reaction

In 1950 Boris Belousov found periodic oscillations in a mixture of bromate, citric acid and ceric ion.

Oscillating reactions• Concentration of reactants, intermediates and products of certain chemical

reactions can vary periodically in space or in time as a result of feedback mechanism in the reaction. Can be sustained indefinitely if condition are kept far from equilibrium

Belousov-Zhabotinski reaction: spatial patterns

Autocatalysis• Autocatalysis – catalysis of a reaction by the products.

[ ][ ]A Pv k A P⎯⎯→=

Belousov-Zhabotinsky reaction (occurs in a mixture of potassium bromate, malonic acid and cerium(III) at acidic condition):

3 2 3 2 2

2 3 2 2

2 2

2 2 ( ) 2 2 2 ( ) 2

BrO HBrO H O BrO H O

BrO Ce III H O HBrO Ce IV H O

− +

+

+ + ⎯⎯→ +

+ + ⎯⎯→ + +

Autocatalytic mechanism of oscillating reactions• The Lotka-Volterra mechanism

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ]

a

b

c

d AA X X X k A Xdt

d XX Y Y Y k X Ydt

d BY B k Ydt

+ ⎯⎯→ + = −

+ ⎯⎯→ + = −

⎯⎯→ =

Concentration of A is held constant, product B is not participating in the reaction but is normally removed as well.

Autocatalytic mechanism of oscillating reactions• Brusselator

2

[ ] [ ]

[ ] [ ][ ]

[ ] [ ] [ ]

[ ] [ ]

a

b

c

d

d XA X k Adt

d YB X Y C k B Xdt

d YX X Y X X X k X Ydt

d XY D k Xdt

⎯⎯→ =

+ ⎯⎯→ + =

+ + ⎯⎯→ + + =

⎯⎯→ = −

Concentrations of A and B are held constant

Limit cycle,attractor

Bistability• A number of chemical reactions carried out in flow reactors

can have multiple steady states with different chemical composition

• If we add a feedback we can start oscillations e.g. using species Z that react with X and Y in such way that

Y Z XX Z Y+ ⎯⎯→

+ ⎯⎯→

Chemical Chaos

• For some reactions the solution can be infinitely sensitive to the initial condition: deterministic chaos

• Certain systems can go to chaos through period doubling

Problems• 23.1a Derive the rate law for the decomposition of ozone in the reaction

2 O3(g) → 3 O2(g) on the basis of the following proposed mechanism:

(1) O3 ↔ O2 + O k1, k1′(2) O + O3 → O2 + O2 k2

• 23.6a The enzyme-catalysed conversion of a substrate at 25°C has a Michaelis constant of 0.035 mol dm-3. The rate of the reaction is 1.15 × 10-3

mol dm-3 s-1 when the substrate concentration is 0.110 mol dm-3. What is the maximum velocity of this enzymolysis?

• 23.8a In an experiment to measure the quantum efficiency of a photochemical reaction, the absorbing substance was exposed to 490 nm light from a 100 W source for 45 min. The intensity of the transmitted light was 40 per cent of the intensity of the incident light. As a result of irradiation, 0.344 mol of the absorbing substance decomposed. Determine the quantum efficiency.