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solution of the equation describing the nonlinear sorption kinetics according to (
11
)
was adopted, assuming exemplary numerical value of the constant Freundlich
exponent (
N
2
) in relation to the Freundlich nonlinear isotherm (
4
).
Taking into consideration the above assumptions, (
11
) can be written as:
¼
@
C
@
C
2
a
1
ð
t
¼
k
1
þ
k
1
C
0
C
Þ
(14)
After separation of the variables
C
and
t
and double-sided integration of (
14
), the
final form of the analytical solution was defined as an exponential function; using
appropriate transformations (Bronsztejn and Siemiendiajew
1990
), we get:
e
k
1
t B
a
1
ð
e
k
1
t B
B
ð
A
þ
1
Þ
A
1
Þ
C
¼
(15)
2
ð
A
e
k
1
t B
1
Þ
with the auxiliary expressions:
p
D
p
4
¼
k
1
C
0
k
1
a
1
2
A
p
D
and
B
¼
a
1
C
0
þ
a
1
(16)
a
1
þ
2
k
1
C
0
k
1
It should also be noticed that the value of the discriminant (
D
) of the expression
C
2
a
1
ð
[
k
1
þ
k
1
C
0
C
Þ
]in(
14
) is less than zero (
D
<
0) and amounts to
2
1
.
Having determined the rate constant of adsorption (
k
1
) from (
15
) and (
16
), the
rate constant of desorption (
k
2
) can easily be determined, based on the parameter of
the Freundlich nonlinear isotherm (
K
2
) occurring in (
4
) as the relationship
K
2
¼
k
1
a
1
k
1
a
D
¼
4
C
0
k
1
=ð
k
2
a
0
Þ
, in the following form:
k
1
a
0
k
1
m
K
2
r
k
2
¼
K
2
¼
(17)
However, for other empirical numerical values of nonlinear exponents (
N
) in the
Freundlich nonlinear isotherms, the solution of (
14
) can be obtained using various
more complex nonanalytical methods, for example, numerical ones, as presented in
Aniszewski (
2009
).
In further analysis, the analytical solution of the equation describing the linear
sorption kinetics according to (
13
) was accepted, assuming numerical value of the
constant Freundlich exponent (
N
1) in relation to the Henry linear isotherm (
5
).
After separation of the variables
C
and
t
and double-sided integration of (
13
),
the final form of analytical solution was defined as the exponential function; using
the appropriate transformations (Bronsztejn and Siemiendiajew
1990
), we get:
¼
e
k
1
tð
1
þa
2
Þ
a
2
C
¼
C
0
(18)
a
2
1