Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

9.1 Chain reactions and chain explosions 205
I
Rate laws:
[X]
[P]
= 1 [A] + ( − 1) 3 [X] [B] − 4 [X] (9.10)
= 3 [B] [X] (9.11)
I
Solutions for three limiting cases:
(1) =1or 4 3 ( − 1) [B]:
[X]
≈ 0 (9.12)
y
1 [A]
[X] =
(9.13)
4 − ( − 1) 3 [B]
y
[P]
= 3 [B] [X] = (9.14)
The overall reaction is stable, because the radical concentration is stable. The
product formation rate is final (constant, well behaved).
(2) 4 = 3 ( − 1) [B]: Atshorttimes,[A] [B] ≈ const y
[X]
= 1 [A] = const (9.15)
y
[X] →∞ (9.16)
The overall reaction turns unstable, because the radical concentration goes to
infinity and thus the product formation rate too, leading to chain explosion!
(3) 4 3 ( − 1) [B]: Atshorttimes,[A] [B] ≈ const y
[X]
+ 4 [X] − 3 ( − 1) [X] [B] = 1 [A]
(9.17)
[X]
+[X]( 4 − 3 ( − 1) [B]) = 1 [A]
(9.18)
y
1 [A]
³
´
[X] =
× exp [( − 1) 3 [B] ] − 1 (9.19)
( − 1) 3 [B] − 4
The overall reaction turns unstable, because the radical concentration increases
exponentially, leading to an exponentially growing product formation rate, and
therefore chain explosion!