Page 60 - The engineering of chemical reactions

44 Reaction Rates, the Batch Reactor, and the Real World

However, CA = 0 at t = CA,/k and the concentration becomes negative for longer times.
Therefore, this expression must be modified to become
CA = CA0 - kt, t 5 C&k
C* =o, t 2 CAo/k
Similar expressions must be used for any orders zero or fewer because reactant concentra-
tions go to zero at finite time and can never be negative.
In fact, any kinetics of this type must be an approximation of a more complicated rate
expression. We will show later that’catalytic reactions frequently obey expressions such as
kKCA
A + products, r =
l+KCA
where k and K are temperature-dependent coefficients. In fact, K is an adsorption-
desorption equilibrium constant, as we will consider in Chapter 7. Note that whenever
KCA >> 1, this expression becomes r x k, to give zeroth-order kinetics. Howeve;, as
CA + 0, the rate becomes approximately
r M kKCA
and the reaction approaches first-order kinetics so that the solution for CA(~) in a batch
reactor varies smoothly for all times.
Similarly the reaction
kKCA
A -+ products,
’ = (1 + KcA)2
obeys negative-order kinetics,
k
r=-
KCA
if KCA >> 1, but again approaches first-order kinetics,
r = kKCA
if KCA << 1
We will encounter similar rate expressions of this type when we consider surface
or enzyme-catalyzed reactions in Chapter 7. These rate expressions are called Langmuir-
Hinshelwood and Michaelis-Menten kinetics, respectively.
These rates versus time would be plotted as shown in Figure 2-7.

Bimolecular reactions

Consider next a bimolecular reaction
A + B + 3C, Y = kCACB
The mass balance on A is
dCA
- = -kCACe
dt
which cannot be solved without eliminating Cg , We showed previously that the number of
moles of all species in a batch reactor are related by the relation