Page 175 - Modeling of Chemical Kinetics and Reactor Design

Reaction Rate Expression 145

The concentrations of components A, B, and C vary with time. The
concentration profiles of A, B, and C in a batch system using the
differential Equations 3-127, 3-128, and 3-129 and velocity constants
k and k , and employing the Runge-Kutta fourth order numerical
1
2
method are reviewed in Chapter 5. Important features of consecutive
reactions occur in substitution processes. For example CH + Cl =
2
4
CH Cl + HCl and CH Cl + Cl = CH Cl + HCl, and so forth. They
2
2
2
3
3
also occur frequently in oxidation processes, where the desired product
may further oxidize to give an undesired product. An example is the
oxidation of methanol, where the desired formaldehyde is readily
degraded to carbon dioxide: CH OH → HCHO → CO .
3
2
The formation of resins, tarry matter by consecutive reaction, is
prevalent in organic reactions. Figure 3-13a shows the time variations
in the concentrations of A, B, and C as given by these equations. The
concentration of A falls exponentially, while B goes through a maxi-
mum. Since the formation rate of C is proportional to the concentration
of B, this rate is initially zero and is a maximum when B reaches its
maximum value.
Kinetic Equations 3-143 and 3-153 are obeyed by nucleides under-
going radioactive decay, where the rate constant k is large and k is
2
1
small. The reactant A is converted rapidly into the intermediate B,
which slowly forms C. Figure 3-13b shows plots of the exponentials
− kt 1 and − kt 2 and of their difference. Since k is small, the exponential
e e 2
− kt 2 shows a slow decay while e − kt 1 shows a rapid decline. The
e
difference of e − kt 2 − e − k t 1 is shown by the dashed line in Figure 3-
13b. The concentration of B is (Equation 3-143) equal to this dif-
ference multiplied by C AO (since k k k ). Therefore, the concen-
1
2
tration of B rapidly rises to the value of C AO and then slowly declines.
The rise in concentration C then approximately follows the simple
first-order law. Conversely, when k is small and k is large (k k
2
1
2
k ), the concentration of B is given by Equation 3-143:
1
C = kC AO e ( − kt 1 − e − k t 2 ) (3-154)
1
B
k
2
At t = 0, C = 0, but after a short time, relative to the duration of
B
the reaction, the difference e − kt 1 − e − k t 2 reaches the value of unity. The
concentration of B is then C k /k , which is much less than C .
AO 1 2 AO
After this short induction period, the concentration of B remains almost