Chemical Reactions on Solid Surfaces - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 431

9.2. LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING
STEPS

At the beginning of this chapter, it was stated that reaction sequences involving surface
steps (for example, Eqn 9.1 ) could be visualized as a type of chain reaction. This is indeed
so and it naturally leads to the utility of pseudo-steady state hypothesis (PSSH) derivations
for kinetic rate simplifications. However, also associated with the development of the theory
of surface reaction kinetics has been the concept of the rate-limiting or rate-controlling step.
This presents a rather different view of sequential steps than does pure chain reaction theory,
since if a single step controls the rate of reaction then all other steps must be at (rapid) equi-
librium. This is a result that is not a consequence of the general PSSH. As we have learned
that both PSSH and rapid equilibrium steps lead to similar asymptotic rate expressions, there
are no clear advantages gained by using one or the other for a complicated reaction network.
Langmuir isotherm is generally applied in surface reaction analyses, which was formalized
by Hinshelwood, Hougen, and Watson.
In fact, pursuing the example in Eqn (9.1) a bit further in this regard, if the surface reaction
is rate limiting, we can express the net rate of reaction directly in terms of the surface species
concentrations, n s q A and n s q B :

r ¼ k s n s q A k s n s q B (9.85)

where k S and k

S are rate constants for the forward and reverse surface reaction steps. This
represents a considerable simplification from the normal chain reaction analysis, because the
first and third steps of (9.1) are at equilibrium and the surface species concentrations are
entirely determined by their adsorption/desorption equilibrium on the surface.
Substitution of Eqn (9.15) into Eqn (9.85) gives the rate of reaction in terms of the bulk fluid
phase concentrations of the reacting species.

K A ð C A C B = K C Þ
1

r ¼ k s n s (9.86)

þ K A C A þ K B C B

where

K A k s
K B k s

K C ¼ (9.87)

is the equilibrium constant. In fact, the value of n may be a somewhat elusive quantity, so

that in practice this is often absorbed into the rate constant as,

s

C A C B =K C

r ¼ k (9.88)

1 þ K A C A þ K B C B

The rates of catalytic reactions are usually expressed in terms of unit mass or unit total
surface (not external surface) of the catalyst. This is understood because of the active sites
are proportional to the total surface area.
As a second example of surface reaction rate control, consider the slightly more compli-
cated bimolecular reaction on one single type of active sites:

þ s %

A $s

(9.89a)

A