Page 87 - Introduction to chemical reaction engineering and kinetics

4.3 Dependence of Rate on Concentration 69
equation 4.2-5, on taking logarithms and differentiating with respect to T, we have

dlnk, _ IZ d In k$
dT
dT
--T+-
and using equation 3.1-6, we convert this to

EA = EAp -I nRT (4.2-9)

For the relation between the corresponding pre-exponential factors A and AL, we use
equations 3.1-8, and 4.2-5 and -9 to obtain

A = Ab(RTe)n (4.2-10)

where e = 2.71828, the base of natural logarithms.
If A and EA in the original form of the Arrhenius equation are postulated to be inde-
pendent of T, then their analogues AL and E,& are not independent of T, except for a
zero-order reaction.

4.2.3.2 Rate Defined by - dpildt
Applying the treatment used in the previous section to relate EA and EAp, corresponding
to kip, and A and A,, corresponding to kip, with equation 4.2-5 replaced by equation 4.2-
8, we obtain

EA = EAp + (n - l)RT (4.2-11)

and

A = A,(RTe)“-1 (4.2-12)

These results are similar to those in the previous section, with n - 1 replacing IZ, and
similar conclusions about temperature dependence can be drawn, except that for a first-
order reaction, EA = EAp and A = A,. The relationships of these differing Arrhenius
parameters for a third-order reaction are explored in problem 4-12.

4.3 DEPENDENCE OF RATE ON CONCENTRATION

Assessing the dependence of rate on concentration from the point of view of the rate
law involves determining values, from experimental data, of the concentration param-
eters in equation 4.1-3: the order of reaction with respect to each reactant and the rate
constant at a particular temperature. Some experimental methods have been described
in Chapter 3, along with some consequences for various orders. In this section, we con-
sider these determinations further, treating different orders in turn to obtain numerical
values, as illustrated by examples.

4.3.1 First-Order Reactions

Some characteristics and applications of first-order reactions (for A -+ products,
(-T*) = k*c*) are noted in Chapters 2 and 3, and in Section 4.2.3. These are summa-
rized as follows:
(1) The time required to achieve a specified value of fA is independent of CA0 (Ex-
ample 2-1; see also equation 3.4-16).