Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-1

• Ch. 22 The Rate of Chemical Reactions

1. Experimental Techniques

2. The Rate of Reactions

3. Integrated Rate Laws

4. Reactions Approaching Equilibrium

5. The Temperature Dependence of Reaction Rates

6. Elementary Reactions

7. Consecutive Elementary Reactions

8. Unimolecular Reactions Lecture 12

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-2

• A number of gas-phase reactions follows first-order kinetics as

in this example,

C3H6

Cyclopropane

C3H6

Propene]HC[ 63 cyclok

• Presumably, a molecule acquires enough energy to react as a

result of its collisions with other molecules.

• However, collisions are simple bimolecular events. How can

they results in a first-order rate law ?

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-3

• The mechanism for the unimolecular reactions was provided by

Frederick Lindemann in 1921, and then elaborated by Cyril

Hinshelwood.

• In the Lindemann-Hinshelwood mechanism, it is supposed

that a reactant molecule (A) becomes energetically excited by

collision with another molecule (A):

2]A[dt

]d[A* A A A A ak*

The energized molecule (A*) might lose its excess energy by

collision with another molecule:

]A*][A[dt

]d[A* A A A A 'ak*

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-4

• Alternatively, the excited molecule might shake itself apart and

form products P (i.e. the unimolecular decay).

]A*[dt

]d[A* P A bk*

• If the unimolecular step is slow enough to be RDS, the overall

reaction will be first-order, as observed.

• The summarized reaction scheme is:

A + A A* + Aka

ka' kb

P

slow

fast

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-5

A + A A* + Aka

ka' kb

P

slow

fast

• The QSSA can be applied to the net rate of formation of A* as

0]A*[]A*][A[]A[dt

]d[A* '2 baa kkk

Solving for [A*], ]A[

]A[]A*[

'

2

ab

a

kk

k

• So the rate law for the formation of P is ]A[

]A[]A*[

dt

]d[P'

2

ab

bab

kk

kkk

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-6

A + A A* + Aka

ka' kb

P

slow

fast

]A[

]A[]A*[

dt

]d[P'

2

ab

bab

kk

kkk

• At this stage the rate law is not first-order.

• However, if the rate of deactivation by (A,A*) collision is much

greater than the rate of unimolecular decay,

baba kkkk ]A[or ]A*[ ]A*][A[''

Then, '

with ]A[dt

]d[P

a

ba

k

kkkk

• Now the rate law is first-order.

]A[

]A[]A*[

dt

]d[P'

2

ab

bab

kk

kkk

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-7

• The Lindemann-Hinshelwood mechanism can be tested

because it predicts that as [A] is reduced, the reaction should

switch to overall second-order kinetics.

A + A A* + Aka

ka' kb

P

slow

fast

]A[

]A[]A*[

dt

]d[P'

2

ab

bab

kk

kkk

,0]A[ As

2]A[dt

]d[PakThen

• The physical reason for the change of order is that at low [A] the

RDS is the bimolecular formation of A*.

ba kk ]A[' ]A*[ ]A*][A[' ba kk

]A[

]A[]A*[

dt

]d[P'

2

ab

bab

kk

kkk

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-8

A + A A* + Aka

ka' kb

P

slow

fast

]A[

]A[]A*[

dt

]d[P'

2

ab

bab

kk

kkk

• If we write the full rate law as: ]A[

]A[ with ]A[

dt

]d[P'

ab

ba

kk

kkkk

Then the expression for the effective rate constant (k) can be

rearranged to

]A[

11 '

aba

a

kkk

k

k

• Hence, a test of the Lindemann-Hinshelwood mechanism is

to plot 1/k against 1/[A], and to expect a straight line.

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-9

• Although the rate of each step of a complex mechanism might

increase with T and show the Arrhenius behavior,…

Is that true of a composite reaction?

• Consider the high [A] limit of the Lindemann-Hinshelwood

mechanism.

'with ]A[

dt

]d[P

a

ba

k

kkkk

If each of the rate constants has an Arrhenius-like T dependence,

RT

aEbEaE

a

ba

RT

aE

a

RT

bE

bRT

aE

a

a

baaaa

a

aa

e

e

ee

k

kkk

)()()(

')(

'

)()(

'

'

'A

AA

A

AA

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-10

• The composite rate constant (k) also has an Arrhenius-like form

with activation energy

RT

aEbEaE

a

ba

a

baaaa

ek

kkk

)()()(

''

'

A

AA

Ea

A

)()()( ' aEbEaEE aaaa

• If , the activation energy of the composite

reaction is positive.

• However, if , the activation energy is

negative and the rate will decrease with T.

)()()( ' aEbEaE aaa

)()()( ' aEbEaE aaa

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-11

• In our example, reveals the reverse

reaction is so sensitive to T that its rate increases sharply with T.

The depletion of the steady-state concentration of A* at high T.

So the rate of the formation of P decreases with T.

)()()( ' aEbEaE aaa

A + A A* + Aka

ka' kb

P

• Note that the Lindemann-Hinshelwood mechanism is an

unlikely candidate for this type of behavior because the

deactivation of A* has only a small activation energy.

• The negative Ea (the decrease of the rate with T) is an evidence

of a composite reaction (i.e., non-elementary).

slow

fast

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-12

• For a two-step reaction with a pre-equilibrium, there are three

activation energies.

• The relative magnitudes of the activation energies determine

whether the overall activation energy is positive or negative.

Positive Ea Negative Ea

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 12-13

• Next Reading:

8th Ed: p.830 ~ 834

9th Ed: deleted.

• Problem set:

8th Ed: Discussion 22.7; Exercise 22.8b, 10b, 13b, 14b, 16b

9th Ed: Discussion 21.7; Exercise 21.10b, 11b, 13b, 14b, 17b

• Midterm Exam: Nov. 1 (Fri) ; 19:00 ~ 12:00; 별232

Chapt 21 & 22