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

2.6 Kinetics of simple composite reactions 34
2.6.5 Competing second-order reactions*
A+A−→ 1
A 2 (2.143)
A+B−→ 2
P 1 (2.144)
B −→ 3
P 2 (2.145)
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Solution for the case [A] À [B]:
(1)
[A]
= −2 1 [A] 2 − 2 [A] [B] (2.146)
≈−2 1 [A] 2 (2.147)
y
or
1
[A] − 1
[A] 0
=2 1 (2.148)
[A] = ¡ [A] 0 −1 +2 1 ¢ −1
(2.149)
=
[A] 0
1+2 1 [A] 0
(2.150)
(2)
[B]
= − 2 [A] [B] − 3 [B] (2.151)
This is another inhomogeneous first-order DE, for which the solution can be found
as outlined in Appendix C, giving
[B] = [B] 0
× (1 + 2 1 [A] 0
) − 22 1
× exp (− 3 ) (2.152)
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Example:
1
CH 3 +CH 3 −→ C 2 H 6 (2.153)
CH 3 +OH−→ 2
CH 2 +H 2 O (2.154)
OH −→ 3
P 2 (2.155)
In the experimental study of reaction 2.154 (Deters 1998), the rate coefficient for the
reaction ( 2 according to Eq. 2.152) was fitted to measured OH concentration-time
profiles in the presence of high concentrations of CH 3 using the Marquardt-Levenberg
nonlinear least squares fitting algorithm (Press 1992). The rate coefficients 1 and
3 were determined by independent measurements. The absolute CH 3 concentrations,
which are needed for the evaluation of 2 according to Eq. 2.152, were obtained using
a quantitative titration reaction.