UPDATED 02/10/03 - AW

7A. A BRIEF SUMMARY OF ELEMENTARY GAS-PHASE REACTIONS

We will classify these as:

- bimolecular

- trimolecular

- unimolecular

7A-1. Bimolecular Reactions

Bimolecular gas phase reactions can occur:

- between two molecules

- between a free radical and a molecule

- between an ion and a molecule

- between two free radicals

For bimolecular reactions among molecular species the comparison between experimentally observed pre-

exponential (frequency) factors of the rate constant and those predicted by collision and transition state

theory is given in Table 1. Clearly, simple kinetic theory leads to A = 1010 to 1011 (L/mol s) which is

significantly too high. CTST values are reasonable.

TABLE 1: Kinetic parameters for some biomolecular reactions

3 1 1

10log A dm mol s

Calculated

Reaction Observed 1

aE kJ mol

Observeda CTST Simple collision

theory

3 2 2

3 3

2 2 2

2 2

2 2

2 2

2

2

2 2 2

2 2

2 2

2 2

2

NO O NO O

NO O NO O

NO F NO F F

NO CO NO CO

NO NO O

NO NO Cl NOCl NO

NOCl NO Cl

NO Cl NOCl Cl

F ClO FClO F

ClO Cl O

10.5

29.3

43.5

132

111

28.9

103

84.9

35.6

0

8.9

9.8

9.2

10.1

9.3

8.9

10.0

9.6

7.5

7.8

8.6

8.1

8.1

9.8

9.7

8.9

8.6

9.1

7.9

7.0

10.7

10.8

10.8

10.6

10.6

10.9

10.8

11.0

10.7

10.4

a The observed and calculated values are from D.R. Herschbach, H.S. Johnston, K.S. Pitzer, and R.E. Powell, J. Chem. Phys.,

25, 736 (1956).

Kinetic results are also available for bimolecular reactions involving atoms and free radicals. Many of

these are abstraction (e.g. metathetical) reactions such as 3 2 6 4 2 5CH C H CH C H

Table 2 lists a comparison between experimentally determined pre-exponential factors for abstraction

reactions (the rates of which are often determined by isotopic labeling) and the values predicted by CTST.

Collision theory estimates lead to values about 1011 (L/mol s).

2

TABLE 2: Observed and calculated kinetic parameters for some abstraction reactions

3 1 1

10log A dm mol s

Reaction Observed 1

aE kJ mol

Observeda Calculated using

CTST

2 2

4 2 3

2 6 2 2 5

3 2 4

3 2 6 4 2 5

3 4 10 4 4 9

2

3 3 3 4 2 3

H H H H

H CH H CH

H C H H C H

CH H CH H

CH C H CH C H

CH i C H CH C H

Br H HBr H

CH CH COCH CH CH COCH

31.8

49.8

38.1

51.0

50.6

28.0

82.4

40.2

10.6

11.1

10.9

9.5

9.3

8.5

11.4

8.5

10.7 , 10.8 , 10.7

10.3

10.1

9.0 , 9.4 , 9.0

8.0

6.8

11.1

8.0

b c d

c

b

b c d

b

b

d

e

a The observed parameters were obtained from sources listed in the bibliography at the end of this chapter.

b S. Bywater and R. Roberts, Can. J. Chem., 30, 773 (1952).

c J.C. Polanyi, J. Chem. Phys., 23, 1505 (1955); 24, 493 (1956).

d D.J. Wilson and H.S. Johnston, J. Am. Chem. Soc., 79, 29 (1957).

e T.L. Hill, J. Chem. Phys., 17, 503 (1949).

Often abstraction reactions are important over a wide range of temperatures (e.g. 1000ºC) in which case

often a temperature dependence of activation energy is observed. This is accounted for by temperature

dependence of the pre-exponential factor which can be predicted by theory as shown in Table 3.

One should note that a popular method for estimation of the pre-exponential factor is the method of

Benson:

*2 /oS RkTA e e

h (1)

where *oS is the entropy of the activated complex estimated by group contribution theory from

configurations of similar molecular structures. Good command of chemistry is needed. Details can be

found in S. Benson's book.

For all bimolecular reactions the rate is second order and given by:

/E RT

A Br Ae C C (2a)

or

/' PE RT

A Br A e p p (2b)

The pre-exponential factor can be estimated from collision or transition state theory.

3

TABLE 3: Temperature dependencies of partition functions and of pre-exponential factors

according to CTST

Temperature dependence of partition function

Type of species Translation Rotation Totala

Monatomic (A)

T1.5

--

T1.5

Linear (L) T1.5

T T2.5

Nonlinear (N) T1.5

T1.5

T3

bTemperature dependence of according to CTSTA

Type of reaction Linear activated complex Nonlinear activated complex

A A

A L

A N

L L

L N

N N

A A A

A A L

A L L

L L L

A A N

A L N

L L N

L N N

N N N

Bimolecular 0.5

0.5

1

1.5

2

2.5

T

T

T

T

T

T

Trimolecular 1

2

3

4

2.5

3.5

4.5

5

5.5

T

T

T

T

T

T

T

T

T

0

0.5

1

1.5

2

T

T

T

T

T

0.5

1.5

2.5

3.5

2

3

4

4.5

5

T

T

T

T

T

T

T

T

T

a Neglecting the temperature dependence of the vibrational partition function and assuming no free rotation. Each degree of

free rotation contributes 0.5T .

a Neglecting the temperature dependence of the vibrational partition function and assuming no change in free rotation when the

activated complex is formed. An additional 0.5T appears for each degree of freedom lost in forming the activated complex.

4

7A-2. Trimolecular Reactions

The first gas phase reaction of this type that was observed was

NOCCNO 22 2

Later it was found that indeed all reactions of the type

XNOXNO 22 2

with BrOCX ,, , etc. are trimolecular and occur in one step. Hence, the rate is third order, second order

in N O and first order in X 2. At constant temperature the rate is given by

2

2 XNOkr (3)

where 2 and XNO indicates the concentration of N O and X 2 respectively. Of course the rate can

also be represented in terms of partial pressures but overall reaction order is always 3 (three).

Collision theory postulates formation of a complex of two molecules which reacts with the third and leads

to a factor of 105 in overestimating the pre-exponential factor. CTST however accounts for the decrease

in enthropy due to the formation of a complex with 3 molecules (this decreases the pre-exponential factor)

and leads to reasonable estimates for

RTEa

XNO

B eqq

q

h

Tkk /

2

2

(4)

where q’s are the molecular partition functions. Although the activated complex contains 6 atoms and

hence 3 x 6 - 7 = 11 vibrational states but these vary only slightly with temperature so that

k T e Ea RT3 5. / (5)

The pre-exponential factor is predicted by Table 3 for a reaction of the type L + L + L.

For the reaction between N O and O 2, Ea = 0, so that k T 3 5. and the rate constant decays with

temperature. Accounting for one free rotation (dependence T 0.5) yields

RTEaeTk /3 (5a)

which means that

TfTntconsnkn 3tan (5b)

when f ( T ) is the contribution from vibrational partition functions. For oxidation of N O the comparison

between the above theory and data is shown in Figure 1 and Table 4.

5

FIGURE 1: Plot of ln k + 3 ln T – f(T) [see Eq. (5.11)] versus 1/T for the reaction between nitric

oxide and oxygen

10

3 K/T

TABLE 4: Calculated and observed rate constants for the nitric oxide-oxygen reaction

3 6 2 110k dm mol s

T/K Calculated Observed

80

143

228

300

413

564

613

662

86.0

16.2

5.3

3.3

2.2

2.0

2.1

2.0

41.8

20.2

10.1

7.1

4.0

2.8

2.8

2.9

7A-3. Unimolecular Reactions

Originally simple decompositions were thought to be elementary unimolecular reactions such as

2 5 2 4 22 2N O N O O

but this was later proven to be untrue. However, examples of unimolecular (hence first order reactions)

do exist such as isomerization of cyclopropane to propylene, dissociation of molecular bromine and

decomposition of sulfuryl chloride (i.e 2 2 2 2S O C S O C ).

Unfortunately the original simple hypothesis that unimolecular reactions do not depend on molecular

collisions and hence occur only by absorbing energy and forming an activated complex was later

disproven by showing that believed unimolecular reactions do not remain first order at very low pressures.

6

Hence, although a unimolecular reaction is supposed by definition to involve one molecule of the reactant

species only, and proceed in one step, the definition was broadened to allow mechanisms for explanation

of unimolecular reactions. A mechanism is a postulated sequence of elementary steps that leads from

reactants to products.

Lindemann-Christansen (L-C) Hypothesis asserts the following mechanism for a unimolecular reaction

A P R

1

1

2

*

*

k

k

k

A A A A

A P R

(6a-b)

Since [A * ] << [A] its rate of change is very small so that

2

* 1 1 2

2

1

1 2

* *

*

AR k A k A A k A O

k AA

k A k

Now from stoichiometry R p = R A - so that

2

1 2 1

2

1 2

*p

k k AR k A k A

k A k (7)

When pressure is not too low k - 1 [A ] >> k 2 and

11 2

1

p

k kR A k A

k (8)

If pressure is sufficiently low k - 1 [A ] << k 2

2

1pR k A (9)

and second order is observed. This analysis indicates that the observed rate constant 1k the unimolecular

reaction would be pressure, i.e concentration of A, dependent since

12

1 2 11

21 2

1

1

kk

k k A kk

kk A k

k A

(10)

The plot of 1

1 1vs

k A should be a straight line

1

1

1 2 1

1 1k

k k k k A (10a)

which unfortunately is not quantitatively confirmed as evident from Figure 2. However the dependence

of the "first order" rate constant k 1 on pressure is evident. This means that while Lindemann's theory

qualitatively predicts correctly the change in reaction order with pressure it does not explain it

7

quantitatively.

FIGURE 2: Schematic plots of 1/k1 versus 1/[A]

1/[A]

Hinshelwood's treatment dealt with the temperature dependence of the rate constant and postulated

1

**

/*

1

1

1 !

s

o Bk To

B

k Z es k T

(11)

where s is the number of degrees of freedom. The postulate is that activation energy is distributed among

many degrees of freedom in the energized molecule A* and after a significant number of vibrations finds

its way into the appropriate degrees of freedom so that A * can pass on to products. The theoretical *

o is

related to the experimental activation energy per molecule

bya

* 3

2o a Bs k T (12)

Hinshelwood's treatment did not resolve the inadequacy of L - C theory in explaining Figure 2 .

Rice-Ramsperger-Kassel (RRK) Treatment generalizes the L - C mechanism to

*A M A M (13a)

*A A P R (13b)

where M is any molecule (could be A ). Hinshelwood's treatment of k 1 was now extended to

k 2 and k 1/k -1 so that

*

* * *

21

*

2 11 /o

k f dk

k k M (14)

where *

1 1/f k k

8

RRK theory was arrived at by both classical statistical mechanics and quantum treatment. The RRK

theory regards a molecule as a system of loosely coupled oscillators all of which have the same frequency

of vibration. In the energized molecule A * the amount of energy * is distributed among the normal

modes of vibration and energy flows freely among them. After a sufficient number of vibrations the

critical amount of energy *

o may be in a particular normal mode and reaction can occur. According to

RRK, energized A * molecules have random life times so that the transition *A A is strictly

statistical.

The events that produce A * are strong collisions ( energy >> k B T ). Ultimately,

* / 11

1

11 ! 1 /

o Bk T s x

s

o

k e x e dxk

s k k M x b (15)

where k is the rate constant corresponding to the free passage of the system through the dividing

surface * *, /o BA A b k T and [M ] concentration of the inert (or A ) with s taken as one-half

the total number of normal modes in the molecule. The above equation for k 1 is in good agreement with

experiments.

The only drawback of the theory is that s cannot be 'a priori' predicted.

Rice-Ramsperger-Kassel-Marcus (RRKM) Theory remedies this. It considers the individual vibrational

frequencies of the energized species and activated complexes explicitly, takes account of the way the

various normal-mode vibrations and rotations contribute to reaction and allowance is made for the zero

point energies. (See W. Forst, Theory of Unimolecular Reactions). Schematic representation is provided

in Figure 3.

9

FIGURE 3: Energy scheme for the RRKM mechanism, in which (1) f( *)d * is calculated using

quantum-statistical mechanics, and (2) k2( *) is expressed using CTST. The energy in

A* and A‡ is either inactive (zero-point, overall translational, overall rotational) or

active (all vibrations and free rotations).

In the limit of high pressure RRKM yields the transition state theory result. Agreement with experiments

is very good. In summary, the rate of unimolecular reactions

r = k 1 [A ] (16)

is not always first order since the rate constant is composition dependent as indicated by the simple L - C

theory, eq. (10), or by the modern RKKM theory, eq (15). The relationship between various theories for

unimolecular reactions is illustrated in Figure 4.

10

FIGURE 4: Diagram showing the relationship between some of the theories of unimolecular gas

reactions. The symbols are defined in the text.

7A-5. Combination and Disproportionation Reactions

Examples of these are

- reactions between atoms

2H H H

- reactions between free radicals

* *

3 3 2 6C H C H C H

These have essentially zero activation energy and occur due to every collision.

- free radical-molecule reaction * *

2 4 2 5H C H C H

- disproportionation

11

* *

2 5 2 5 2 4 2 6C H C H C H C H

These are not simple bimolecular reactions since they are the reverse of unimolecular reactions and hence

have special features. If the rate of the reverse reaction is known or calculable by RRKM theory, then the

equilibrium constant can be used to derive the dependence of the forward reaction.

Consider

* *

3 3 2 6

2 6

2*

3

with c

C H C H C H

C HK

C H

The reverse decomposition reaction

* *

2 6 3 3C H C H C H

at sufficiently high pressure is 1-st order

r - 1 = k - 1 [C 2 H 6 ]

From the equilibrium expression

62131

11

HCkCHk

rr

Since

2*

3

62

1

1

CH

HCK

k

kc

2*

1 1 3

2 and

r k C H (16)

At very low pressure the decomposition is 2nd order

2

1 1 2 6r k C H

which implies that

2*

1 1 3 2 6r k C H C H (17)

The forward rate is now third order.

Mechanisms of Atom and Radical Combinations

In the dissociation of a molecule R 2 into two radicals R * the initial step is the energization process

*

2 2R M R M (18a)

the second step is the dissociation

* *

2 2R R (18b)

12

The initial step (initation) was brought about by a molecule M which may be R 2 or an added substance

(third body).

Termination of the free radical R * follows then the reverse of the above process

1

1

* *

22

k

k

R R

2*

2 2

kR M R M

For combination (termination) reactions the third body M is known as the chaperon. The energy of *

2R

is transferred to M, which provides the energy transfer mechanism.

Using the fact that the net rate of *

2R formation is zero, 30*2R

R as shown below, the concentration of

*

2R can be found

2

2

* * *

1 1 2 2 2

*

1*

2

1 2

0k R k R k R M

k RR

k k M

The rate of *R termination is twice the rate of *

2R formation since the overall stoichiometry is 2

*2 RR :

-

2*

1 2*

* 2 2

1 2

22R

k k R MR k R M

k k M (19)

Hence, at sufficiently high pressures

2

1 2

*

* 12R

k k M

R k R (20)

second order termination rate is observed.

At sufficiently low pressures

1 2

2*

* 1 22R

k k M

R K k R M (21)

and third order termination is observed. Here 1 1 2/K k k is the equilibrium constant for the first step.

Combination and termination of free radicals most likely follows the above mechanism.

The combination of atoms represents an extreme situation. If energy transfer applies two atoms will come

together and separate within the period of first vibration ~ 10-13 s (Figure 5) unless an effective chaperon

molecule arrives and collides with the complex within such short a time. Only at gas pressures of 104 to

105 atm are collision frequencies high enough for them to be likely. The combination rates are low and

3rd order.

However, an alternative mechanism often plays a role - atom molecule complex mechanism.

13

MRRRM

MRMMRM

RMMR

k

k

k

k

k

2

*

*

**

2

2

2

1

1

The predicted rate is

MRKKkR R 2*

213* 2 (22)

where 111 kkK and 222 kkK .

The kinetics remains third order at all pressures. This mechanism is favored if k1 is large,

R M * is a strong complex.

FIGURE 5: Potential-energy curve for a diatomic molecule A2, showing that when the two A

atoms come together, they separate in the first vibration.

14

Appendix to Chapter 7A

Glossary of commonly used terms:

A * = energized molecule acquired which has all the energy needed to become an activated

molecule but to do so must undergo vibrations i.e redistribute the energy

A = activated molecule which has passed the dividing barrier (col) on the energy surface

Z = zero point energy for the reactant. The uncertainty principle (i.e position and momentum of

electron cannot both be known) precludes that the energy levels of a harmonic oscillator

could be given by vhEv which for 0 would yield 0vE of the classical ground

state. Instead 1

2vE v h so that zero point energy is

1

2ZE h where h is the

Planck constant

* = activation energy i.e energy needed to be acquired by A to become A * zero point energy

* = * *

inactive active

*

inactive = energy of A * that remains in the same quantum state during reaction and cannot contribute

to breaking of bonds. This includes zero point energy, energy of overall translation and

rotation.

*

active = energy of A * that can contribute to reaction consisting of vibrational energy and energy of

internal rotations.

Z = zero point energy for the activated complex A

= total energy above zero point in activated complex A

= *

inactive active

inactive inactive

o = difference in zero point energies of andA A