Physical Chemistry Third Edition

12.1 Reaction Mechanisms and Elementary Processes in Gases 525

that lead to chemical reaction in a gas-phase bimolecular elementary process depends

only on the temperature.We will discuss the validity of this assumption in Section 12.3.

Consider first a gaseous bimolecular elementary process involving two molecules

of substance number 1:

F 1 +F 1 −→products (12.1-3)

where we abbreviate its formula byF 1. Eq. (9.8-22) gives the rate of two-body colli-

sions of a single substance, which is proportional to the square of the number density

of the substance. If we assume that at a fixed temperature a fixed fractionfof the

collisions leads to reaction, the reaction is second order:

rate−

1

2

d[F 1 ]

dt



fZ 11

NAv



1

NAv

fπd 12

(

4 kBT

πm 1

) 1 / 2

N 12

NAvfπd 12

(

4 kBT

πm 1

) 1 / 2

[F 1 ]^2 (12.1-4)

wherem 1 is the mass of a molecule of the reacting substance,d 1 is its effective hard-

sphere diameter,N 1 is its number density (equal toN 1 /V), and [F 1 ] is its molar concen-

tration. The quantityπd 12 is thecollision cross sectionthat we defined in Section 9.8.

The quantityfπd^21 is an effective cross-sectional area, or the effective area of the

“target” that actually leads to reaction. It is called thereaction cross section. The

divisorNAvin the first line of Eq. (12.1-4) is included to express the rate in moles

rather than molecules.

By comparison with Eq. (11.2-8) we can write an expression for the forward rate

constant:

kfπd^21

(

4 kBT

πm 1

) 1 / 2

NAv

(bimolecular gas-phase

reaction, one reactant)

(12.1-5)

The units of this second-order rate constant are m^3 mol−^1 s−^1. If concentrations and

rates are to be measured in mol L−^1 , an additional conversion factor of 1000 L m−^3 is

needed, givingkthe units L mol−^1 s−^1.

Exercise 12.1

Show that Eq. (12.1-5) is correct. Remember that a factor of 1/2 occurs in the definition of

the rate as in Eq. (11.1-6), that each reactive collision uses up two molecules of substance

number 1, and that a factor of 1/2 was introduced into Eq. (9.8-22) to avoid overcounting of

collisions.

In the case of a bimolecular elementary process involving one molecule of substance

1 and one molecule of substance 2, the elementary bimolecular process is first order in

each substance and second order overall. The rate constant is given by

kfπd 122

(

8 kBT

πμ 12

) 1 / 2

NAv

(bimolecular gas-phase

reaction, two reactants)

(12.1-6)