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

Appendix C 270
C.3 Application to parallel and consecutive first-order reactions
Consider the reaction system
A 1
−→ P
A 1
−→ B 2
−→ P
(C.39)
(C.40)
which is described by the following rate equations
[A]
[B]
[P]
= − ( 1 + 1 )[A]=− 1 [A] (C.41)
=+ 1 [A] − 2 [B]
(C.42)
=+ 1 [A] + 2 [B]
(C.43)
with 1 =( 1 + 1 ). 55
The solution for Eq. C.41 is
[A] = [A] 0
−( 1+ 1 ) =[A] 0
− 1
(C.44)
Thus we have to find the solution of the inhomogeneous DE for [B] (Eq. C.42)
[B]
+ 2 [B] = + 1 [A]
0
− 1
using the above standard procedure:
(C.45)
• Solution of the homogeneous DE by separation of variables:
y
[B]
= − 2 [B] (C.46)
[B]
= × − 2
(C.47)
• Determination of a particular solution by variation of constant:
y
[B]
= ()
[B]
= () × − 2
= ()
× − 2 − () × 2 − 2
(C.48)
(C.49)
(C.50)
55 An example is the radiationless deactivation of a ∗ electronically excited molecule A directly to
the ground state P or via an intermediatate optically dark ∗ state B, which decays to the ground
statemoreslowly.