Page 175 - Introduction to chemical reaction engineering and kinetics

7.1 Simple Homogeneous Reactions 157

Thus, the mechanism provides a first-order rate law with

klk2
k o b s = kkl + 2k2 m

(c) Note that, although a simple reaction order arises from this mechanism, the observed
rate constant is a combination of elementary rate constants for steps (1) and (2) and can ex-
hibit non-Arrhenius temperature dependence. The effective activation energy varies from
one extreme, (i), in which step (2) is relatively fast (large k2), to the other, (ii), in which
step (2) is so slow (small k2) as to be the rate-determining step (rds).
(i) In the first case, k2 Z+ k-,, and equation (G) becomes

rO* = W2h,o, (k2 law3 (J)
with the result that the experimental activation energy is the same as that for forward
step (1); that is, applying the Arrhenius equation, 3.1-6, to k,, = k,/2, we obtain

E Asobs = EA1 (k2 large) (K)

(ii) In the other extreme, k2 < k-i, and equation (G) becomes

r02 = (k, k2/k-1h205 (k2 small) CL)

This implies that step (1) is so rapid as to be in virtual equilibrium. Then, from
equation 5.3-11 (with n = l),

k,lk-, = Keql

where Z&i is the equilibrium constant for step (1). From the Arrhenius equation,
3.1-6, applied to kobs = k,k2/k-, = k2Keq1, we obtain

E A,obs = EA2+EA1 - EA,-l E EAT + AHi (N)

where EA,-1 and EA2 are the activation energies for reverse step (1) and step (2),
respectively, and AH, is the enthalpy of reaction for step (1); the second part of
equation (N) comes from the van? Hoff equation 3.1-5, dlnK,,,IdT = AH,IRT2.

Many mechanisms involve reversible steps which are rapid (and therefore in virtual
equilibrium) followed by the critical rds. In these cases, the equilibrium constant for
each of the rapid steps appears as a multiplicative factor in the rate law. The effective
activation energy is the sum of the enthalpies of the equilibrium steps and the activation
energy of the rds.

7.1.3 Closed-Sequence Mechanisms; Chain Reactions
In some reactions involving gases, the rate of reaction estimated by the simple collision
theory in terms of the usually inferred species is much lower than observed. Examples of
these reactions are the oxidation of H, and of hydrocarbons, and the formation of HCl
and of HBr. These are examples of chain reactions in which very reactive species (chain
carriers) are initially produced, either thermally (i.e., by collision) or photochemically
(by absorption of incident radiation), and regenerated by subsequent steps, so that re-
action can occur in chain-fashion relatively rapidly. In extreme cases these become “ex-
plosions,” but not all chain reactions are so rapid as to be termed explosions. The chain