Chapter X

Kinetics of Complex Reactions

Levine: p.559 17.9

§10.1 Typical complex reactions

Complex reactions:

reaction contains more than one elementary reaction

Typical complex reactions

1) Opposing Reaction :

2) Parallel Reaction:

3) Consecutive Reaction:

+

-

A B G Hk

ka b g h

1 2A B Ck k

1.1 Opposing Reaction / reversible reaction

majority of the reactions are reversible, i.e., the forward and the backward / reverse reaction take place simultaneously.

(1) kinetic equilibrium constant

for opposing reaction consisting of elementary reactions:

[A] [B]a br k [G] [H]g hr k

As reaction proceeds, r+ increases while r decreases. When r+

becomes equal to r, equilibrium is reached.

+

-

A B G Hk

ka b g h

[A] [B] [G] [H]a b g hk k

[G] [H]

[A] [B]

g h

ca b

kK

k

k

kKc

In this way we arrive at a very important connection between the

equilibrium constant and the rate coefficients of simple reactions. This

relation, named as kinetic equilibrium constant, is correct only for

elementary reactions.

therefore

For first-first order opposing reaction:

(2) rate equation

t = 0 a 0

t = t a-x x

t = te a-xe xe

The total rate is xkxakdt

dx )(

A Bk

k

e e( )k a x k x e

e

( )k a xk

x

Under equilibrium conditions

e

e

( )( )

a xdxk a x k x

dt x

e

e

( )x xdxk a

dt x

e e

e

ln( )

x xk

at x x

e e

e

ln( )

x xk

at x x

e e

e

ln( )

a x xk

at x x

e e

e

ln( )

a x xk

at x x

which suggests that k+ and k can be determined by measuring x at t and at equilibrium concentration. Relaxation method.

e

e

1ln

( )

xk k

t x x e

e

1ln

( )

xk k

t x x

e

e

ln ( )( )

xk k t kt

x x e

e

ln ( )( )

xk k t kt

x x

Similar to the rate equation of first-order reaction

1-2 opposing reaction

2-2 opposing reaction

A B Ck

k

A + B C + Dk

k

Principle of relaxation method for studying fast reaction

1.2 Parallel reaction / Competing reaction

1

[B][A]

dk

dt

2

[C][A]

dk

dt

))(()()( 2121 xakkxakxakdt

dx

21 kk When )(1 xakdt

dx

When 21 kk )(2 xakdt

dx

The rate of parallel reaction is determined mainly by the faster one.

))(( 21 xakkdt

dx

Integration of the equation yields:

tkkxa

a)(ln 21

])(exp[)( 21 tkkaxa

A B C

a 0 0

a-x y z

x = y + z

)(1 xakdt

dy

)(2 xakdt

dz

a

t

For production of B and C:

)(1 xakdt

dy ])(exp[)( 21 tkkaxa ])(exp[)( 21 tkkaxa

])(exp[ 211 tkkakdt

dy ]})(exp[1{ 21

21

1 tkkkk

aky

]})(exp[1{ 21

21

1 tkkkk

aky

2

1

k

k

z

y

2

1

k

k

z

y

The composition of the final products is fixed.

selectivity of the reaction.

]})(exp[1{ 2121

2 tkkkk

akz

]})(exp[1{ 21

21

2 tkkkk

akz

A

B C

t

c

Optimum temperature for better selectivity

Example

A B A1 Ea, 1

A C A2 Ea, 2

logA2

1/T

logA1

log k

B

C

logA2

1/T

logA1

logk

B

C

The selectivity of the parallel reaction can be improved by adoption of appropriate catalyst.

Using catalyst to better selectivity

Main reaction and Side reaction:

reaction with higher k is taken as the main reaction, while others side reactions.

Reaction that produces the demanded product is the main reaction.

product

consumed

nS

nSelectivity:

1.3 Consecutive reaction

Some reactions proceed through the formation of intermediate.

CH4 + Cl2 CH3Cl CH2Cl2 CHCl3 CCl4

A B C

t = 0 a 0 0

t = t x y z

a = x + y + z

General reaction 1 2A B Ck k

xkdt

dx1

tkx

a1ln )exp( 1tkax

ykxkdt

dy21

)exp()exp( 2112

1 tktkkk

aky

)exp()exp( 21

12

1 tktkkk

aky

ykdt

dz2

C

tmax t

A

C

B

)exp(

)exp(1

212

1

112

2

tkkk

k

tkkk

k

az

)exp(

)exp(1

212

1

112

2

tkkk

k

tkkk

k

az

shows that the intermediate’s concentration rises from zero to a maximum and then drops back to zero as A is depleted and C dominates in the mixture. ��

)exp()exp( 2112

1 tktkkk

aky

)exp()exp( 21

12

1 tktkkk

aky

If C is the demanded product, the reaction time should be prolonged. If B is the demanded product, the reaction should be interrupted at optimum time, i.e., tmax.

0dt

dy

21

21max

)/ln(

kk

kkt

At tmax, the concentration of B = ?

C

tmax t

A

C

B

21

21max

)/ln(

kk

kkt

)exp()exp( 21

12

1 tktkkk

aky

)/ln

exp()/ln

exp(21

212

21

211

12

1max kk

kkk

kk

kkk

kk

aky

21

2

21

1

)/exp(ln)/ln(exp 212112

1max

kk

k

kk

k

kkkkkk

aky

21

2

21

1

)()(2

1

2

1

12

1max

kk

k

kk

k

k

k

k

k

kk

aky

1)()( 1

2

1

2

1

12

1max

21

2

k

k

k

k

kk

aky kk

k 2

2 11

2

k

k kmax

ky a( )

k

2

2 11

2

k

k kmax

ky a( )

k

k2/k11/5 5 10 100 103 108

tmax2.01 0.40 0.25 0.047 710-3 10-7

ymax/a 0.67 0.13 0.08 7 10-3 10-3 0

Ea,1Ea,2-0.4 4.0 5.7 11.5 17.2 46.1

When k2 >> k1, ymax would be

very small, and the tmax would

be very short.

2 1

2 1max

ln( k / k )t

k k

2 1

2 1max

ln( k / k )t

k k

t

y

0

k1/k2

0dt

dy 0dt

dy

2

2 11

2

k

k kmax

ky a( )

k

2

2 11

2

k

k kmax

ky a( )

k

Physical meaning of k2 >> k1

B is a active intermediate (Such as active atom: Cl, H, etc., radicals:

CH3•, H2C:, C+, C-, etc., activated molecules: A*), it is difficult to form

but easy to decompose to product.

For consecutive reaction with large k2/k1 ratio, once the reaction take

place, the active intermediate (B) rapidly attains its maximum

concentration and its concentration keeps nearly unchanged during the

whole reaction.

0dt

dy 0dt

dy Steady-state approximation

1 2A B Ck k

)exp()exp(1 2

12

11

12

2 tkkk

ktk

kk

kaz

1 21 2

2 1

Zdc ak kexp( k t ) exp( k t )

dt k k

1 2

1 22 1

Zdc ak kexp( k t ) exp( k t )

dt k k

When k2 >> k1 1 1Zdc

ak exp( k t )dt

1 1Zdc

ak exp( k t )dt

The total rate is determined mainly by k1

When k2 << k12 2

Zdcak exp( k t )

dt 2 2

Zdcak exp( k t )

dt

The total rate is determined mainly by k2

The rate of the overall consecutive reaction depends only on the smaller rate constant (rate-determining step).

rate-determining step (r. d. s.): the step with the slowest rate.

?? !! It’s a r.d.s

patient !

The rate of the elementary step with the lowest rate constant, i.e., r.d.s., can be used to express the actual rate of the overall reaction.

Its activation energy should be 10 kJmol-1 more than that of other steps.

What is a eligible r. d. s.?

Rate-determining step approximation