Page 203 - Essentials of physical chemistry

More Kinetics and Some Mechanisms 165

CHAIN REACTIONS AND THE STEADY STATE

So far here and in Chapter 7, we have considered straightforward analysis of reaction rates measured
by ‘‘the extent of the reaction’’ based on mole turnover rate, but there are many reactions which
follow more complicated rate equations. The classic problem of this more complicated type was first
treated by Bodenstein and Lind [6] and later by Christiansen [7], Herzfeld [8], and Polanyi [9]. The
first experimental data for the reaction of hydrogen and bromine were fitted accurately to an
expression, which is far from what might be expected:

1
d[HBr] k[H 2 ] [Br 2 ] 2
H 2 þ Br 2 ! 2HBr; ¼ :
dt k [HBr]
0
1 þ
[Br 2 ]
The value of k was 0.10, and the overall rate constant was fitted to an Arrhenius plot with a value
0
for E* of 175 kJ=mol. This mystery was successfully explained later by Herzfeld and by Polanyi as
due to a series of intermediate reactions. While this overall reaction is largely due to the ease with
which H 2 and Br 2 can be broken apart to free radical atomic species, it is now known that there are
many similar reactions which can be treated by the same mathematical approximation called ‘‘the
steady-state approximation’’:

1. Specifically, write down the postulated intermediate reactions with their rate constants
and assign an identifying label to each reaction.
2. Identify transient, short-lived species, typically free radicals.
3. Write the equations for creation and annihilation of the transient species and set the time
derivative to zero to approximate the idea that the concentration of these transient species
reaches some steady value, but it is constant so the time derivative is zero.
4. Use the steady-state equations to solve for the expressions of the transient species.
5. Perform ‘‘clever algebra’’ to consolidate and simplify the steady-state equations in terms of
the concentrations of the reactants and products. In some cases, one or more of the steady-
state equations may lead nowhere, which probably means that you postulated a nonexistent
reaction or wrote an incomplete description of the creation and annihilation of that
transient.
6. Substitute the steady-state concentrations into the basic rate reaction either for disappear-
ance of reactants or for appearance of product(s).

Before we show examples of this procedure, it may be worth mentioning that this approach is a very
good ‘‘pencil-and-paper’’ way to test postulated mechanisms. Even today, with sophisticated spec-
troscopic equipment and the latest quantum chemistry computer programs, one should try to test a
mechanism for a research problem with pencil, paper, and the steady-state method; it is that useful.

STEADY-STATE EXAMPLE NO.1:H 2 þ Br 2 ! 2HBr
Now we are ready to treat the Bodenstein–Lind rate equation with the steady-state method. We write
the individual steps with roman numbers for better manipulation:
k 1
(I) Br 2 ƒƒƒƒ! 2Br .
k 2
(II) Br . þ H 2 ƒƒƒƒ! HBr þ H .
k 3
(III) H . þ Br 2 ƒƒƒƒ! HBr þ Br .
k 4
(IV) H . þ HBr ƒƒƒƒ! H 2 þ Br .
k 5
(V) Br . þ Br . ƒƒƒƒ! Br 2