Biomedical Engineering Reference
In-Depth Information
TABLE E9-1.3
Least Square Parameter Estimation for the One-Parameter Nonideal Adsorption Isotherm.
T
he Final Solution Shown Are for C
As
N
¼
185.0 mg-phenol/g-Activated Carbon and
0.07557 (mg/L)
1
, Which Were Obtained by Minimizing the Variance between th
e
Nonideal Adsorption Model and the Experimental Data, s
L
, While Changing C
As
N
and K
K
A
¼
A
Error
2
(
Aqueous solution
C
A
, mg/L
On activated carbon
C
0
As
, mg/g
Eqn
(9.30)
C
As
, mg/g
C
As
LC
0
As
)
2
0
0
0
0
1.5
53.0
47.91687
25.83822
2.0
62.4
56.77735
31.61424
5.1
82.8
91.23779
71.19638
6.6
105.0
101.5365
11.99601
22.9
137.0
146.0282
81.50787
51.6
172.2
164.7061
56.15872
80.2
170.9
171.2536
0.125006
s
P
i
¼
1
ð
C
As
;
i
C
0
As
;
i
Þ
2
s
L
¼
¼
6
:
306872
8
1
To have the theory useful, we must extend the derivations to multicomponent adsorptions.
To do this, we note that the interaction energy between the adsorbate molecule and the
adsorption center (or active site) canbe divided into twoparts: “surface energy” and“adsorbate
bonding energy.” The “adsorbate bonding energy” is dependent on adsorbate molecule and
the adsorbent pair but not dependent on the location of the surface where the adsorption
occurs. At this point, we can assume that the “adsorbate bonding energy” is equivalent to
the minimum heat of adsorption or adsorption heat under ideal conditions. The “surface
energy” is a property of the adsorbent surface only, unrelated to the adsorbate molecules,
although it has an effect only when an adsorbate molecule approaches or is bonded on.
Thus, the “surface energy” is the nonideal component of the adsorbate
e
adsorbent interaction
energy. Based on this assumption, we canwrite the adsorption heat in the Langmuir adsorption
isotherm as
DH
ad;j
¼ DH
ad;j0
E
s
(9.32)
where
H
ad,
j
0
is the
“bonding energy” of the species
j
to the adsorbent surface, which can be thought of as the
adsorption energy of species
j
on the adsorbent surface under ideal adsorption conditions;
E
s
is the “surface energy” and different for different adsorption site.
For each group (
i
) of sites that has identical “surface energy,” the site coverage according to
the Langmuir adsorption isotherm is
Eqn (9.17)
,or
D
H
ad,
j
is the adsorption heat for species
j
adsorbing onto the adsorbent;
D
K
ji
C
j
q
ji
¼
(9.17)
þ
P
N
s
m¼
1
1
K
mi
C
m
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