Page 149 - Modeling of Chemical Kinetics and Reactor Design

Reaction Rate Expression 119

Rearranging and integrating Equation 3-28 between the limits gives

C Af dC t 2
∫
− ∫ A = kdt (3-29)
C AO C A t 1


 C Af

−ln C A  = ( k t 2 − ) (3-30)
t
1
 C AO  
− [C −C ] = ( k t − ) (3-31)
ln
t
Af AO 2 1
C
t
−ln Af = ( kt 2 − ) (3-32)
1
C
AO
At t = 0 and t = t. Therefore,
1 2
C
− ln Af = kt (3-33)
C AO
The fractional conversion X for a given reactant A is defined as
A
the fraction of the reactant converted into product or

X = N AO − N A
A (3-34)
N AO
For a constant density system, the volume V remains constant

C = N A = N (1 − X ) = C (1 − X ) (3-35)
A
AO
A AO A
V V
Differentiating Equation 3-35 gives
dC = –C AO dX A (3-36)
A
Substituting Equation 3-36 into Equation 3-28 gives

dX A = ( X )
k −1
dt A (3-37)