Page 56 - The engineering of chemical reactions

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

and integrated from CA = CAM at t = 0 to CA at t = 1
CA t
de.4
- = -k
f - I dt
J C’A J
CAo t=o
to give

or

kt
CA = C.&e-
I I
The above two simple equations are those you will use most often in this course.
The solution of CA(t) for the first-order irreversible reaction is plotted in Figure
2-5. For a first-order irreversible reaction CA decreases from CA0 at t = 0 to CAO/e at
t = l/k and to CA0/e2 at t = 2/k. For these kinetics doubling the reaction time increases
the conversion by a factor of 10.

Example 2-2 The reaction A -+ B has k = 0.01 set-‘. For CA,, = 2.0 moles/liter, what
time is required for 90% conversion in a constant-volume batch reactor? For 99%? For
99.9%?

Simple application of the preceding equation for 90% conversion (CA = 0.2)
yields

t = -f In $ = +k In CAo = & In & = 100 ln 10
CA .
= 100 x 2.303 = 230 set
For 99% conversion (CA = 0.02), we obtain

t = & In 100 = 100 x 2 x 2.303 = 460 set
For 99.9% conversion (CA = 0.002), we obtain
1
t = o.ol In 1000 = 100 x 3 x 2.303 = 690 set
These problems are easy! Note that the reactor residence time (proportional to
reactor size) increases markedly as the required conversion increases. Note also
that for this example (first-order kinetics) we did not need even to specify CA,,
because the equation involves only the ratio CAo/CA.

We can use this solution for any first-order irreversible reaction

A + products, r = kCA