Page 208 - Introduction to chemical reaction engineering and kinetics

190 Chapter 8: Catalysis and Catalytic Reactions

(a) Using the integral method of experimental investigation (Section 3.4.1.1.2) obtain
a linear form of the CA-t relation from which kA may be determined.
(b) What is the value of t,,,, the time at which the rate is (- rJmax, in terms of the
parameters cAo, c&, and kA?
(c) How is fA related to t? Sketch the relation to show the essential features.

SOLUTION
(a) Integration (e.g., by partial fractions) of the material-balance equation for A with the
rate law included,

-dc,/dt = kACA(lbfo - c/,) (8.3-9)

results in

h(cA/cB) = ln(cA,/cB,) - bfokAt (8.3-10)

Thus, ln(cA/cn) is a linear function oft; from the slope, kA may be determined. (Compare
equation 8.3-10 with equation 3.4-13 for a second-order reaction with VA = vg = - 1.)
Note the implication of the comment following equation 8.3-8 for the application of equa-
tion 8.3-10.
(b) Rearrangement of equation 8.3-10 to solve for t and substitution of CA,+ from equa-
tion 8.3-7, together with 8.3-3, results in CA = cn at t,,,, and thus,

(8.3-11)
t man = (1/j%‘td ln(cAo/%o)
This result is valid only for CA0 > cnO, which is usually the case; if CA0 < cnO, this result
suggests that there is no maximum in (-rA) for reaction in a constant-volume BR. This
is examined further through fA in part (c) below. A second conclusion is that the result
of equation 8.3-11 is also of practical significance only for cnO # 0. The first of these
conclusions can also be shown to be valid for reaction in a CSTR, but the second is not
(see problem 8-4).
(c) Since fA = 1 - (cAIcA~)for constant density, equation 8.3-10 can be rearranged to

eliminate cA and cn so as to result in

1 - eXp(-M,k,t)
fA = (8.3-12)
1 + c,eXp(-M,k,t)
where

co = cAofcEo (8.3-12a)
Some features of the fA-t relation can be deduced from equation 8.3-12 and the first and
second derivatives of fA. Thus, as t + 0, fA + 0; df,/dt(SlOpe) + kAcBo > 0 (but = 0,
if cnO = 0); d2fA/dt2 + k&&A0 - cnO) > 0, Usually, with CA0 > cnO, but < 0 other-
wise. As t -+ ~0, fA + 1; df,/dt + 0; d2 fA/dt2 + 0(-). The usual shape, that is, with
cAO > cBO, as in part (b), is sigmoidal, with an inflection at t = t,,, given by equation
8.3-11. This can be confirmed by setting d2 fA/dt2 = 0.
The usual case, CA0 > cn,, is illustrated in Figure 8.7 as curve A. Curve C, with no
inflection point, illustrates the unusual case Of CA0 < c&,, and Curve B, with CAM = CB~,
is the boundary between these two types of behavior (it has an incipient inflection point at
t = 0). In each case, kA = 0.6 L mol-’ mm-’ and M, = 10/6 mol L-l; C, = 9, 1, and
1/9 in curves A, B, and C, respectively.