Page 117 - Introduction to chemical reaction engineering and kinetics

5.3 Reversible Reactions 99

5.3.4 Optimal T for Exothermic Reversible Reaction
An important characteristic of an exothermic reversible reaction is that the rate has an
optimal value (a maximum) with respect to Tat a given composition (e.g., as measured
by fA). This can be shown from equation 5.3-14 (with n = 1 and Keq = KC,eq). Since gf
and g, are independent of T, and Y = r,,lvo (in equation 5.3-l),

(5.3-21)

(5.3-22)

since Keq = greqlgf,eq, and dKeqK.s = d InK,,. Since dk/dT is virtually always posi-

tive, and (gfkkgf,eqkreq) > 1 kf > gf,e4 and g, < gr,J, the first term on the right
in equation 5.3-22 is positive. The second term, however, may be positive (endothermic
reaction) or negative (exothermic reaction), from equation 3.1-5.
Thus, for an endothermic reversible reaction, the rate increases with increase in tem-
perature at constant conversion; that is,

(drDldT), > 0 (endothermic) (5.3-23)

For an exothermic reversible reaction, since AH” is negative, (drDldT), is positive or
negative depending on the relative magnitudes of the two terms on the right in equation
5.3-22. This suggests the possibility of a maximum in r,, and, to explore this further, it
is convenient to return to equation 5.3-3. That is, for a maximum in rb,

dr,idT = 0,and (5.3-24)
dk, dk,
gfz = ET,* (5.3-25)

Using equation 3.1-8, k = A exp(-E,lRT) for kf and k, in turn, we can solve for the
temperature at which this occurs:

T opt = EAr iE”f ln(;;kf)l’ (5.3-26)

(a) For the reversible exothermic first-order reaction A * D, obtain Topr in terms of
fA, and, conversely, the “locus of maximum rates” expressing fA (at ro,,,,) as a
function of T. Assume constant density and no D present initially.
(b) Show that the rate (rn) decreases monotonically as fA increases at constant T,
whether the reaction is exothermic or endothermic.

SOLUTION
(a) For this case, equation 5.3-3 (with r = rD) becomes

rD = kfCA - k,.c, (5.3-15)