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 by F 1 . Eq. (9.8-22) gives the rate of two-body collisions 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 fraction f of the
collisions leads to reaction, the reaction is second order:
rate −
1
2
d[F 1 ]
dt
f Z 11
N Av
1
N Av
f πd
2
1
4k B T
πm 1
1/2
N
2
1
N Av f πd
2
1
4k B T
πm 1
1/2
[F 1 ]
2
(12.1-4)
where m 1 is the mass of a molecule of the reacting substance, d 1 is its effective hardsphere diameter, N 1 is its number density (equal to N 1 /V ), and [F 1 ] is its molar concentration. The quantity πd 2
1 is the collision cross section that we defined in Section 9.8.
The quantity f πd 2
1 is an effective cross-sectional area, or the effective area of the
“target” that actually leads to reaction. It is called the reaction cross section. The
divisor N Av in 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:
k f πd 2
1
4k B T
πm 1
1/2
N Av
(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, giving k the 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
k f πd 2
12
8k B T
πµ 12
1/2
N Av
(bimolecular gas-phase
reaction, two reactants)
(12.1-6)
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 by F 1 . Eq. (9.8-22) gives the rate of two-body collisions 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 fraction f of the
collisions leads to reaction, the reaction is second order:
rate −
1
2
d[F 1 ]
dt
f Z 11
N Av
1
N Av
f πd
2
1
4k B T
πm 1
1/2
N
2
1
N Av f πd
2
1
4k B T
πm 1
1/2
[F 1 ]
2
(12.1-4)
where m 1 is the mass of a molecule of the reacting substance, d 1 is its effective hardsphere diameter, N 1 is its number density (equal to N 1 /V ), and [F 1 ] is its molar concentration. The quantity πd 2
1 is the collision cross section that we defined in Section 9.8.
The quantity f πd 2
1 is an effective cross-sectional area, or the effective area of the
“target” that actually leads to reaction. It is called the reaction cross section. The
divisor N Av in 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:
k f πd 2
1
4k B T
πm 1
1/2
N Av
(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, giving k the 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
k f πd 2
12
8k B T
πµ 12
1/2
N Av
(bimolecular gas-phase
reaction, two reactants)
(12.1-6)