Chemical Reactions on Solid Surfaces - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 419

where n s 1 is the maximum number of adsorption sites available on each layer. Solving the
discrete Eqn (9.60) , we obtain,

i

q i ¼ A þ Bb (9.65)

i , similar to one would do when solving a homogeneous
ordinary differential equation. We have two free parameters in Eqn (9.65) . Eqn (9.65) needs to
be satisfied also by the first layer, Eqn (9.62) and the topmost layer, Eqn (9.57) . Substituting
Eqn (9.65) into Eqn (9.57) , we obtain

which is obtained by assuming q i ¼ g

Nþ 1

A þ Bb ¼ 0 (9.66)

Substituting Eqn (9.65) into Eqn (9.62) , we obtain

cð 1 A BbÞ¼ B ð 1 bÞ (9.67)

Solving Eqns (9.66) and (9.67) , we obtain

c

B ¼ (9.68a)

Nþ 1

1 ð 1 cÞb cb

Nþ 1

cb

A ¼ (9.68b)

Nþ 1

1 ð 1 cÞb b

Substituting Eqn (9.68) into Eqn (9.65) , we obtain the fractional coverage of layer i

i Nþ 1

cðb b Þ

q i ¼ (9.69)

Nþ 1

1 ð 1 cÞb b

Therefore, the total number of molecules adsorbed is

i Nþ 1

n As;N ¼ n s1 P

N

1 q i ¼ n s1 c P

N

b b

Nþ 1

1 ð 1 cÞb cb

i¼ i¼ 1

(9.70)

N

1 b

Nþ 1

b

b Nb

N

1 ½ 1 þ Nð 1 bÞb

1

¼ n s1 c

1 ¼ n s1 cb Nþ Nþ 1

1 ð 1 cÞb cb ð 1 bÞ½ 1 ð 1 cÞb cb

Since

k i
k i

b ¼ p A ¼ ap A (9.71)

Eqn (9.70) is reduced to

N

1 ½ 1 þ Nð 1 ap A Þðap A Þ

n As;N ¼ n s1 cap A (9.72)

Nþ 1

ð 1 ap A Þ½ 1 ð 1 cÞap A cðap A Þ