Page 173 - Modeling of Chemical Kinetics and Reactor Design

Reaction Rate Expression 143

To obtain the maximum concentration of B, differentiate Equation
3-143 with respect to time t, which gives

dC B = kC AO − { ke − kt 1 + k e }
−
1
k t 2
dt k − k 1 2 (3-144)
2 1
The values of k and k govern the location and maximum con-
1 2
centration of B, and this occurs at dC /dt = 0, t = t . Equation
B max
3-144 becomes:
2
2
1
0=− kC AO e −kt + kk C AO e −kt
2
1
1
k − k k − k
2 1 2 1
2
kC e − kt 1 kk C e − kt 2
1 AO = 12 AO
k − k k − k (3-145)
2 1 2 1
( k t k 2
e k 2 − ) max = (3-146)
1
k
1
t ( k − )= ln k 2
k
max
2
1
k
1
The maximum concentration of B occurs at
 k 
2
ln   k  
t max = 1 (3-147)
k − k 1
2
At t
max

ke t − max k 2 = k e t − max k 1 (3-148)
2
1
Substituting for e − kt 1 in Equation 3-147 gives

C 1
B max = ke − 2 − k e − 2 } = e − 2 (3-149)
kt max
kt max
kt max
k
C k − ) { 2 1
AO ( 2 1