Chemical Reactions on Solid Surfaces - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 436

K 1

S K B C B

q A ¼ K 1

q B ¼ (9.109)

S

þ K B C B þ K 1

1

S K B C B

Substituting Eqns (9.107) and (9.109) into (9.100) , we obtain

k A C A k A K 1 C A k A k A K 1

S K B C B S K B C B

r ¼ C s

S K B C B ¼ k A C s

(9.110)

þ K B C B þ K 1 þ K B C B þ K 1

1 1

S K B C B

At equilibrium, r ¼ 0, we have

C B;eq
C A;eq ¼

1
k A k A K 1

K eq ¼ (9.111)

S K B

also noting that

k A
k A

K A ¼ (9.112)

we have

K S K A
K B

K eq ¼ (9.113)

Eqn (9.110) can be rewritten as

C A K 1

eq C B

r ¼ k A C s (9.114)

þ K B C B þ K 1

1

eq K A C B

Eqn (9.114) can also be written as

C A K 1

eq C B

r ¼ k A C s (9.115)

þ K B C B þ K A C A 1

By noting that

C A ¼ K 1

eq C B

(9.116)

as the equilibrium constant is not dependent on the nature of the surfaces. C A is the virtual or
equilibrium concentration of A in the bulk fluid phase with the surface overage. This is deriv-
able because the concentration of A on the surface is in equilibrium with other components
(or species) in the system as the rest of the steps that govern the overall reaction are fast equi-
librium steps. One can observe from Table 9.3 that it holds true for all other cases. Comparing
Eqn (9.115) with Eqn (9.110) , we note that

1

q ¼ (9.117)

þ K B C B þ K A C A 1

and

K A C A

q A ¼ (9.118)

þ K B C B þ K A C A 1