Biomedical Engineering Reference
In-Depth Information
9.1.3. Common Empirical Isotherms
While our discussions in ideal (Langmuir) and nonideal adsorption isotherms have led to
more general adsorption isotherms that could be applicable to a variety of surfaces, we shall
make a detour and mention a couple of empirical isotherms. These isotherms were thought of
as accounting for nonideality of the adsorbent surfaces. They are approximations to a variety
of ideal and nonideal isotherms. As such, their utilities are more restrictive than the above
isotherms. The reason we introduce these isotherms is that there are a few classic applications
that have utilized these empirical isotherms.
In the low coverage region well before the coverage levels off as the bulk concentration is
increased, the Langmuir isotherm (9.9) as well as the generalized logarithm coverage
Eqn
(9.27)
can be approximated by
q
A
¼ cC
A
(9.41)
In addition, one can note that the dissociative adsorption
Eqn (9.14)
can be approximated by
Eqn (9.37)
as well in the low coverage region. This versatile approximation,
Eqn (9.41)
,is
called the
Freundlich isotherm
.
In the intermediate coverage region,
<< K
A
C
A
e
E
max
1
and
1
>> K
A
C
A
RT
Eqn
(9.40)
may be approximated by
RT
E
max
ln
ðK
A
C
A
e
Emax
q
A
z
Þ
(9.42)
RT
which is the Temkin isotherm. Eqn
(9.42)
correlates experimental data better in the midrange
of surface coverage and is best known for the adsorption of Hydrogen and Nitrogen in
ammonia synthesis reactions.
To show the closeness of the Freundlich isotherm and Temkin isotherm to the UniLan
isotherm,
Fig. 9.7
shows the UniLan isothermwith
E
max
¼
5
RT
together with ExLan isotherm
and Temkin isotherm with identical parameters. One can observe that there is significant
difference between ExLan and UniLan isotherms since ExLan has a lower apparent adsorp-
tion heat than UniLan when the same parameters are taken. Temkin approximation,
Eqn
(9.42)
, is reasonably close to the original UniLan isotherm if
K
A
C
A
˛
(0.01, 1), which is less
0.9128(
K
A
C
A
)
0.2636
. However, when the Temkin
isotherm is regarded
as
an empirical model,
accurate than the Freundlich isotherm,
q
A
¼
its correlation in
K
A
C
A
˛
(0.1, 10),
804
1
ln
14212K
A
C
A
e
10:804
q
A
¼
, showed reasonable approximation to the UniLan
isotherm in this region. Therefore, both Freundlich and Temkin isotherms can be applied
to correlate adsorption isotherms.
10
:
ð
0
:
Þ
Example 9-2
Temkin and Freundlich Isotherms.
Recorrelating the adsorption data in Example 9-1,
Table E9-1.1
, with Temkin and Freund-
lich isotherm models. Compare the quality of the fits with that of the Langmuir isotherm fit.
Solution.
We first look at the power-law model (Freundlich isotherm). Substituting
Eqn
(9.28)
into
Eqn (9.41)
, we obtain
C
As
¼ c
0
C
A
(9.43)
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