Biomedical Engineering Reference
In-Depth Information
An important step in the consideration of surface reactions is the equilibrium level of
adsorption on a surface. With
Eqn (9.1a)
, the rate is shown to be reversible and equilibrated
at a long enough timescale, so the rate of adsorption equals the rate of desorption. It is conve-
nient to think of this as a process of dynamic equilibrium, where the net rate of change is zero.
Assuming A is the only species adsorbed on the surface or at least on the active site of
interest, from
Eqns (9.2) and (9.4)
, we have
q
A
k
ad
k
ad0
k
des0
k
ad0
k
des0
E
ad
E
des
RT
e
D
H
ad
e
qC
A
¼
k
des
¼
¼
(9.5)
RT
This expression is extraordinarily useful, since it permits us to obtain some information con-
cerning the surface coverage factors,
q
and
q
A
, about which we have been rather vague. The
point is that these factors cannot be measured conveniently, while macroscopic quantities
such as
C
A
and the heat of absorption can. Eqn
(9.5)
provides a link between the two. Solving
for the ratio
q
A
/
q
it can be seen that the surface coverage at equilibrium (or at least some func-
tion of it) is determined by the temperature of the system and the partial pressure of adsor-
bate. Such an equation for fixed temperature and varying partial pressure expresses the
adsorption isotherm
for the adsorbate, or, for fixed partial pressure and varying temperature,
the
adsorption isobar.
The heat of adsorption,
H
ad
, appears in
Eqn (9.5)
since in solving for the
ratio of surface coverage functions the difference (
E
des
e E
ad
) appears in the exponential;
from
Fig. 9.3
we see that this is equal in magnitude to the heat of adsorption.
Eqn
(9.5)
is termed the Langmuir isotherm, which we may write in more general
notation as
D
q
A
q
¼ K
A
C
A
¼ K
A
e
D
H
ad
RT
C
A
(9.6)
where
k
ad
k
des
K
A
¼
(9.7)
The total number of active sites or centers is fixed for a given amount of surfaces. Therefore,
an active site (just the same as the active sites in enzymes) balance leads to
1 (9.8)
if A is the only adsorbate on the particular type of active centers. The site balance is not
dependent on temperature or concentration. Combining
Eqns (9.6) and (9.8)
, we obtain
q
A
þ q ¼
K
A
C
A
q
A
¼
(9.9)
1
þ K
A
C
A
which is the Langmuir isotherm equation for the nondissociative adsorption of a single
species on a surface (i.e. single molecule on a single-type active site).
In many cases of practical importance, the adsorbate molecule will dissociate on adsorp-
tion (e.g. H
2
on many metals) or occupy two adjacent active sites by bonding at two points in
the molecule (e.g. ethylene on nickel). In such cases,
A
2
þ
2s
%
2
A
$s
(9.10)
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