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

Appendix C 268
C.2 Application to two consecutive first-order reactions
Consider the reaction system
A 1
−→ B
B 2
−→ C
(C.11)
(C.12)
which is described by the following rate equations
[A]
[B]
[C]
= − 1 [A] (C.13)
=+ 1 [A] − 2 [B]
(C.14)
=+ 2 [B]
(C.15)
The solution for Eq. C.13 is
[A] = [A] 0
− 1
(C.16)
Thus we have to find the solution of the inhomogeneous DE for [B] (Eq. C.14)
[B]
+ 2 [B] = 1 [A] 0
− 1
(C.17)
• Solution of the homogeneous DE by separation of variables:
y
[B]
= − 2 [B] (C.18)
[B]
= × − 2
(C.19)
• Determination of a particular solution by variation of constant:
y
[B]
= ()
[B]
= () × − 2
= ()
× − 2 − () × 2 − 2
(C.20)
(C.21)
(C.22)
Insertion of [B]
and [B]
into Eq. C.17 gives
()
which simplifies to
× − 2 − () × 2 − 2 + () × 2 − 2 = 1 [A] 0
− 1
()
= 1 [A] 0
( 2− 1 )
This DE is easily integrated to obtain ():
(C.23)
(C.24)