Environmental Engineering Reference
In-Depth Information
where Q
t
and Q
∞
are amounts adsorbed after time t and after infinite
time respectively
6
)
(11.18)
2
2
F
1
(/
1
n
)exp(
n B
t
2
1
where F is the fractional attainment of equilibrium at time 't' and is
obtained by using Equation 11.18 and 'n' is the Freundlich constant of the
adsorbate.
2
D
(11.19)
B
i
Time Constant
t
2
()
r
o
where B
t
= time constant, Di
i
= effective diffusion coefficient of ion in the
adsorbent phase (cm
2
.s
-1
) and r
o
= radius of the adsorbent particle assumed
to be spherical.
Although the equation is a mathematical infinite series, only a few terms
can give a good approximation coefficient Di
i
for an adsorption system.
For every observed value of F, a corresponding value of B
t
is derived from
Reichenberg's table [88]. The plot of B
t
versus time distinguishes between
film diffusion and particle diffusion controlled rates of adsorption. The
plot of 'B
t
' versus time (t) giving a straight line passing through the origin,
shows a particle diffusion mechanism [87].
It is to be noted that the treatment of diagnosing the particle diffusion
process as a rate-controlling process is the same as that for the isotopic
exchange. The only difference in the two processes is that in the devel-
oped mathematical equation the self-diffusion coefficient is replaced by the
effective diffusion coefficient of exchange ions [88-90].
For the system undergoing particle diffusion mechanism, the plot of log
D
i
versus 1/T is linear and permits the use of the Arrhenius equation [91].
Activation energy E
a
and D
o
values for such processes can be calculated
from the slope and intercept of this plot.
E
RT
a
DD x
p
(11.20)
i
o
where D
i
is the diffusion constant, Di
o
is the maximum diffusion con-
stant, E
a
is the activation energy (KJ.mol
-1
), R is universal gas constant
(J.mol
-1
K
-1
) and T is temperature (K).
By increasing the temperature the increase in value of Di
i
ascertains
increased mobility of the diffusing ions due to decreased retarding force.
The plot between 1/Temperature and log Di
i
defines the diffusibility of
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