Geoscience Reference
In-Depth Information
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
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 1 þa 2 Þ
a 2
C
¼
C 0
(18)
a 2
1
Search WWH ::




Custom Search