Agriculture Reference
In-Depth Information
Equilibrium
A
100
k 1 = 0.01
k 2 = 0.1
80
B
k 1 = 0.001
k 2 = 0.01
60
40
k 1 = 0.0001 day -1
k 2 = 0.001
C
20
Time = 2 days
0 0
20
40
60
80
100
120
140
160
c, mgL -1
FIGURE 6.1
Effect of rate coefficients on sorption isotherms using the second-order kinetic model. (From
H. M. Selim and M. C. Amacher. 1997. Reactivity and Transport of Heavy Metals in Soils . Boca
Raton, FL: CRC Press. With permission.)
( C i = S i = 0 at t = 0) and a soil-to-solution ratio (ρ/Θ) of 1:10, which is com-
monly used in batch experiments. Since the amount sorbed was assumed to
be initially zero, larger values for k 2 than k 1 were selected in our simulations
to induce reverse (desorption) reactions.
As shown in Figure 6.1, after 2 days of reaction, isotherm A, where k 1 and
k 2 were 0.01 and 0.1 day -1 , respectively, appears closer to the equilibrium iso-
therm than other cases shown. The equilibrium case was calculated using
Equation 6.9 and represents an isotherm at t → 4 for a soil having values of
k 1 and k 2 that are extremely large. Isotherms B and C represent cases where
both k 1 and k 2 values were reduced in comparison to those for isotherm A, by
one and two orders of magnitude, respectively. For both cases, the isotherms
deviate significantly from the equilibrium case. It is apparent from curve C
that 2 days of reaction is insufficient to attain equilibrium and a sorption
maximum is not apparent from the shape of the isotherm. Moreover, it is per-
haps possible to consider a linear-type isotherm for the concentration range
shown. However, as much as 100 days or more of reaction time is necessary
to achieve apparent equilibrium conditions. This is illustrated in Figure 6.2,
where the influence of time of reaction using the second-order model is
shown.
The influence of the sorption maxima ( S max ) on the retention isotherms
is shown in Figure  6.3. The parameters selected were similar to those of
Figure 6.2 except that a contact time of 10 days was chosen. As expected, the
isotherms reached their respective maxima at lower C values with decreasing
S max . The results of Figure 6.3 also indicate a steep gradient of the retention
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