Page 162 - Modeling of Chemical Kinetics and Reactor Design

132 Modeling of Chemical Kinetics and Reactor Design

or

−
t = 1 C { 1 A − n − C } (3-87)
1
n
(
AO
kn − ) 1
The half-life is defined as the time required for the concentration
to drop to half of its initial value, that is, at t = t , C = 1/2C .
1/2 A AO
From Equation 3-86,
2 ( − 1 C AO ) 1−n −C 1−n = kt 12 (n − 1) (3-88)
AO
2 ( − 1 • 2 − 1) C 1−n = kt 12 (n − 1) (3-89)
n
AO
2 ( n 1 − 1) C 1− n
−
t 12 = kn 1) AO (3-90)
(
−
Similarly, the time required for the concentration to fall to 1/p of
its initial value gives

p ( n 1 − 1) C 1− n
−
t 1 p = kn 1) AO (3-91)
(
−
Taking the natural logarithm of both sides of Equation 3-90 gives

1 (
ln n 1)
−
ln t 12 = ln 2 ( n 1 − 1) +− n) lnC AO − ln k − ( − (3-92)
−
 2 n 1 − 1
ln t 12 = ln  kn 1)  + ( 1− n ) lnC AO (3-93)
 ( − 

Plots of ln t versus ln C AO from a series of half-life experiments
1/2
are shown in Figure 3-9. Table 3-3 gives some expressions for reaction
half-lives.
The reaction rate constant k is

(2 n−1 −1 C ) 1 − n
k = AO (3-94)
( n − ) 1 t 12