Page 83 - Introduction to chemical reaction engineering and kinetics

4.1 The Rate Law 65

where r and k are the species-independent rate and rate constant, respectively, and ri
and ki refer to species i . Since ki is positive for all species, the absolute value of vi is used
in the last part of 4.1-3. In this equation, n indicates a continued product (c;“lcp . . .),
and (Y~ is the order of reaction with respect to species i . In many cases, only reactants
appear in the rate law, but equation 4.1-3 allows for the more general case involving
products as well.
We also assume that the various rate constants depend on T in accordance with the
Arrhenius equation. Thus, from equations 3.1-8 and 4.1-3,

I I
k = A exp(-E,lRT) = ?!- = A exp(-E,lRT) (4.1-4)
l’il lvil

Note that, included in equations 4.1-3 and -4, and corresponding to equation 1.4-8 (r =
lilvi), are the relations

k = killvil; i = 1,2,...,N (4.1-3a)

A = Aillvil; i = 1,2,...,N (4.1-4a)

As a consequence of these various defined quantities, care must be taken in assigning
values of rate constants and corresponding pre-exponential factors in the analysis and
modeling of experimental data. This also applies to the interpretation of values given
in the literature. On the other hand, the function n csi and the activation energy EA are
characteristics only of the reaction, and are not specific to any one species.
The values of (Y~, A, and EA must be determined from experimental data to establish
the form of the rate law for a particular reaction. As far as possible, it is conventional
to assign small, integral values to al, (Ye, etc., giving rise to expressions like first-order,
second-order, etc. reactions. However, it may be necessary to assign zero, fractional and
even negative values. For a zero-order reaction with respect to a particular substance,
the rate is independent of the concentration of that substance. A negative order for a
particular substance signifies that the rate decreases (is inhibited) as the concentration
of that substance increases.
The rate constant ki in equation 4.1-3 is sometimes more fully referred to as the spe-
cific reaction rate constant, since lril = ki when ci = 1 (i = 1,2, . . . , N). The units of ki
(and of A) depend,on the overall order of reaction, IZ, rewritten from equation 3.1-3 as

N
n=C(Yi (4.1-5)
i=l
From equations 4.1-3 and -5, these units are (concentration)i-” (time)-‘.

4.1.2 Empirical versus Fundamental Rate Laws
Any mathematical function that adequately represents experimental rate data can be
used in the rate law. Such a rate law is called an empirical orphenomenological rate law.
In a broader sense, a rate law may be constructed based, in addition, on concepts of
reaction mechanism, that is, on how reaction is inferred to take place at the molecular
level (Chapter 7). Such a rate law is called a fundamental rate law. It may be more
correct in functional form, and hence more useful for achieving process improvements.