12.5 Chain Reactions 559
The Rate Law of the Photochemical Reaction
The rate of the initiation step of Eq. (12.5-9) is proportional to the rate at which photons
are absorbed. We letJbe the average rate of absorption of light, measured ineinsteins
per unit volume per second (one einstein is 1 mol of photons). The rate of photochemical
production of Br atoms is equal to 2J, so that the steady-state equation for Br atoms is
now
d[Br]
dt 2 J−k 2 [Br][H 2 ]+k 3 [H][Br 2 ]+k 2 ′[HBr][H]− 2 k 1 ′[Br]^2 ≈0 (12.5-10)We neglect the collisional (thermal) production of Br atoms, which is much slower than
the photochemical production. The steady-state equation for [H] is still Eq. (12.5-4c):
When this equation is added to Eq. (12.5-10), we obtainJ−k′ 1 [Br]^2 0 (12.5-11)Equation (12.5-11) is combined with Eq. (12.5-5) and substituted into Eq. (12.5-7) to
obtain the rate law−
d[H 2 ]
dtk 2(
1 /k′ 1) 1 / 2
[H 2 ]J^1 /^2
1 +
k′ 2 [HBr]
k 3 [Br 2 ](12.5-12)
which agrees with experiment.Exercise 12.20
Verify Eq. (12.5-12).Thequantum yieldΦof a photochemical reaction is defined as the number of
molecules of product produced per photon absorbed. It is also equal to the number of
moles of product per einstein of photons absorbed. The quantum yields of photochem-
ical reactions can range from nearly zero to about 10^6. Quantum yields greater than
unity ordinarily indicate a chain reaction.Reaction of Hydrogen with the Other Halogens
Hydrogen reacts in the gas phase with chlorine or iodine as well as with bromine,
but there are differences between the three reactions. The reaction of hydrogen with
chlorine is inhibited by oxygen. An approximate rate expression for the thermally
initiated reaction is
d[HCl]
dtklJ[H 2 ][Cl 2 ]^2
km[Cl 2 ]+[O 2 ]([H 2 ]+kn[Cl 2 ])(12.5-13a)wherekl,km, andknare temperature-dependent parameters. An approximate empirical
rate law for the photochemical reaction is^21
d[HCl]
dtkaJ[H 2 ][Cl 2 ]
kb[C1 2 ]+[O 2 ]([H 2 ]+kc[Cl 2 ])(12.5-13b)(^21) K. L. Laidler,op. cit., p. 295ff (note 3).