Chemical Reactions on Solid Surfaces - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 432

B þ s % B $s (9.89b)

A $s þ B $s % C $s þ D $s (9.89c)

C $s % C þ s (9.89d)

D $s % D þ s (9.89e)

þÞ ¼ overall A þ B C þ D (9.89f)

/

Let us assuming that the surface reaction (9.89c) is the rate-limiting step, whereas the
adsorption e desorption steps are fast equilibrium steps. The adsorption step (9.89a) is a
fast equilibrium step, that is

0 ¼ r addes;A ¼ k A n s qC A k A n s q A (9.90)

which yields after rearrangement

q A ¼ K A qC A (9.91)

where

k A
k A

K A ¼ (9.92)

is the Langmuir isotherm constant for species A on the catalyst surface. Similarly, one can
write for B, C, and D:

q B ¼ K B qC B (9.93)

q C ¼ K C qC C (9.94)

q D ¼ K D qC D (9.95)

There are four species adsorbed on the same type of active sites. Site balance leads to the
summation of the fraction of vacant (i.e. available sites) and the fractions of the sites occupied
by A, B, C, and D make unity:

1 (9.96)
Substituting Eqns (9.91) and (9.93) through (9.95) into Eqn (9.96) , we obtain after
rearrangement

q þ q A þ q B þ q C þ q D ¼

1
K A C A þ K B C B þ K C C C þ K D C D

q ¼ (9.97)

Since the surface reaction (9.89c) is the rate-limiting step, the overall reaction rate for (9.89f) is
identical to that for (9.89c) as the stoichiometric coefficients are the same for A, B, C and D.
That is,

r ¼ r s ¼ k s n 2

s q A q B

(9.98)