Chemical Reactions on Solid Surfaces - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 414

the first layer, and q 2 is the fraction of active centers (sites) that are covered by the adsorbate
molecules on both the first layer and the second layer. On the bottom (or first) layer,

0 ¼ r ad;1 ¼ k 1 ð 1 q 1 Þp A k 1 ðq 1 q 2 Þ (9.47)

Let,

k 2

k 2 ¼ a

(9.48)

be the adsorption isotherm constant for the upper layer. The adsorption isotherm constant for
the base (or lower) layer is

k 1
k 1 ¼ ac

(9.49)

Eqns (9.46) and (9.47) give

aðq 1 q 2 Þp A ¼ q 2 (9.50)

acð 1 q 1 Þp A ¼ q 1 q 2 (9.51)

which can be solved to yield

acp A ð 1 þ ap A Þ

q 1 ¼ (9.52)

1 þ acp A ð 1 þ ap A Þ

2 cp 2 A

a

q 2 ¼ (9.53)

1 þ acp A ð 1 þ ap A Þ

The number of moles of adsorbate on the adsorbent surface is thus the sum of the molecules
on the first layer and the second layer. That is,

n max
2

n 2 acp A ð 1 þ 2ap A Þ

n As ¼ ðq 1 þ q 2 Þ¼ (9.54)

1 þ acp A ð 1 þ ap A Þ

where n s is the total number of active sites on the adsorbent surface, which is equivalent to the
maximum number of adsorbate molecules that could be adsorbed or number of saturation
adsorbate molecules. In terms of concentration, the “bi-layer adsorption” can be written as

n 2 cK A C A ð 1 þ 2K A C A Þ

n As ¼

þ K A C A Þ
Fig. 9.10 shows a schematic of the isotherm as described by Eqn (9.55) . One can observe
that the feature differs qualitatively from the Langmuir isotherm ( Fig. 9.4 ) at low coverage
ratios where the adsorbate concentration in the adsorbent increases more slowly.
At this point, one may ask why would the adsorbate e adsorbent surface interaction be
limited to just two pseudo-layers? In fact, as the adsorbate molecules “pack” tighter and tighter
on the adsorbent surface, the interactions among adsorbate molecules become stronger.
Fig. 9.11 shows a schematic of monolayer uniform hard spheres packed on a flat surface.
One can infer from Fig. 9.11 that for closely distributed active centers, one “hard sphere”
can have up to six neighboring “hard spheres” of the same size closely packed on a perfectly

1 þ cK A C A ð 1