Page 205 - Essentials of physical chemistry

More Kinetics and Some Mechanisms 167

Now divide the numerator and denominator by (k 3 [Br 2 ]), and we obtain Bodenstein–Lind formula:

1
2
k 1 1
2k 2 [H 2 ] [Br 2 ] 2
d[HBr] k 5
:
¼
dt k 4 [HBr]
1 þ
k 3 [Br 2 ]
While this is an amazing explanation of the Bodenstein–Lind rate equation, it is not an isolated
incident. There are many other complex reactions, which can be solved with this method.

STEADY-STATE EXAMPLE NO.2:THERMAL CRACKING OF ACETALDEHYDE [10]
This next example is not quite perfect because it gives a solution with a leftover radical unaccounted
for. However, it is shown here as an example of what to expect in research. Suppose we want to
understand the thermal decomposition of acetaldehyde. Rice and Herzfeld [10] studied the thermal
‘‘cracking’’ of hydrocarbons as part of a very important study related to petroleum processing. Here,
we present the thermal cracking of acetaldehyde. Consider the following scheme for the thermal
decomposition reaction [11]:

D
CH 3 CHO ƒƒƒƒ! CH 4 þ CO þ CH 3 CH 3 :

k 1
.
(I) CH 3 CHO ƒƒƒƒ! CH 3 þ . CHO E* ¼ 76:0 kcal=mol
k 2
.
(II) CH 3 þ CH 3 CHO ƒƒƒƒ! CH 4 þ . CH 2 CHO E* ¼ 10:0 kcal=mol
k 3
(III) . CH 2 CHO ƒƒƒƒ! CO þ . CH 3 E* ¼ 18:0 kcal=mol
k 4
(IV) . CH 3 þ . CH 3 ƒƒƒƒ! CH 3 CH 3 E* ¼ 0 kcal=mol
.
d[CH 3 ] 2
¼ k 1 [CH 3 CHO] k 2 [ . CH 3 ] [CH 3 CHO] þ k 3 [ . CH 2 CHO] k 4 [ . CH 3 ]
0 ffi
dt
d[ . CH 2 CHO]
¼ k 2 [ . CH 3 ] [CH 3 CHO] k 3 [ . CH 2 CHO]
dt
0 ffi
.
d[ CHO]
¼ k 1 [CH 3 CHO]
0 ffi
dt
The third steady-state species ( . CHO) is apparently constant and only depends on the amount of
acetaldehyde, so that is a dead end as far as the steady-state substitution process goes. There may be
some other steps leading to H 2 CO but here . CHO does not contribute to the production of CH 4 .
However, from the second equation, we find that

k 2
[ . CH 3 ] [CH 3 CHO]:
[ . CH 2 CHO] ss ¼
k 3
Then from the first steady-state equation, we have by substitution of the [ . CH 2 CHO] ss expression:

k 3 k 2 2
[ . CH 3 ] [CH 3 CHO] k 4 [ . CH 3 ] ffi 0,
k 1 [CH 3 CHO] k 2 [ . CH 3 ] [CH 3 CHO] þ
k 3