Page 256 - Soil and water contamination, 2nd edition

Chemical transformation 243

We may write the rate law for mass transfer from solution to the adsorbed phase in

groundwater as (see also Box 13.I) as:
M C C
r s s b s (13.5)
V t n t
-1
-3
where r = the sorption rate [M L T ], ρ = the dry bulk density of the bed sediment , and
b
n = the water-filled porosity of the sediment [-]. If we assume local equilibrium and a linear

Freundlich isotherm and differentiate it with respect to time, we obtain:
C C
s K w (13.6)
t d t
Combining Equations (13.5) and (13.6) gives:
C w
b
r K d (13.7)
n t
When we enter this reaction rate into the general transport equation (Equation 13.1), where
C = C , we obtain:
w
C C 2 C b C b C C 2 C
u x D x K d 1 K d u x D x (13.8)
t x x 2 n t n t x x 2

The bracketed term on the left-hand side is a constant called the retardation factor R :
f
R f 1 b K d (13.9)
n
So, the transport equation with sorption reaction becomes:

C u x C D x 2 C (13.10)

t R x R x 2
f f
The retardation factor as defined above is a number equal to 1 in absence of adsorption
(K = 0) or greater than 1 if there is adsorption. It has the effect of slowing down the
d
chemical transport (retardation). The retardation factor can also be computed as the ratio
of the mean velocity of the water to the mean velocity of the chemical. For example, if a
column experiment is carried out using a chemical that is readily sorbed by the mineral
fraction in the column (e.g. potassium ), the breakthrough curve shows a delayed response
(Figure 13.3). The first curve in Figure 13.3 is the breakthrough curve of a conservative
substance that does not sorb. The slight S-shape is due to dispersion . The time taken for the
half of the input concentration to pass is equal to the mean residence time of the solution
in the column. The second curve shows the breakthrough curve of a substance subject
to sorption . So, the travel time of the substance is increased by a factor that equals the
retardation factor R relative to the travel time of the conservative substance. It is also more
f
S-shaped because more time has elapsed for dispersion.
Note that the simple retardation factor should only be applied when a single distribution
coefficient K is adequate to describe sorption . It is conceivable that this will not always be
d
the case, because the sorption isotherm may not be linear or the sorption kinetics may be
slow. In the case of a convex isotherm, the distribution coefficient and the retardation increase
with decreasing concentration. This means that low concentrations are transported at a slower
rate than high concentrations. Consequently, the front of the breakthrough curve remains
sharp (see Figure 13.3). On the other hand, it also results in a delayed release of the residual
substance after the bulk of the chemical substance has passed. In such cases, the breakthrough

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