Page 136 - Materials Chemistry, Second Edition

Mass-Balance Concept and Reactor Design 119

concentration is mainly attributed to natural biodegradation and volatiliza-
tion. Assume that both removal mechanisms are first-order reactions and
that the reaction-rate constants for both mechanisms are independent of TPH
concentration and are constant. Estimate how long it will take for the concen-
tration to drop below 100 mg/kg due to these natural attenuation processes.

Strategy:
Only the initial concentration and the concentration at day 5 are given.
We need to take a two-step approach to solve the problem: First
determine the rate constant, and then use the rate constant to deter-
mine the time needed to reach a final concentration of 100 mg/kg.
Two removal mechanisms (i.e., biodegradation and volatilization) are
occurring simultaneously, and both of them are first-order. These
two mechanisms are additive, and they can be represented by one
single equation with a combined rate constant.

dC =− kC kC− 2 = − k ( 1 + k C) =− kC
dt 1 2 (4.12)

Solution:
(a) Insert the initial concentration and the concentration at day 5
into Equation (4.11) to obtain k:

2,750
ln =− k(20)
3,000

So, k = 0.00435/day.
(b) For the concentration to drop below 100 mg/kg, it will take (from
Equation 4.11):
100
t
ln =− 0.00435()
3000
t = 782 days

Example 4.3: Estimate the Rate Constant from Two
Known Concentration Values (2)

The subsurface soil at a site was impacted by an accidental spill of gasoline.
A soil sample, taken 10 days after removal of the polluting source, showed
a TPH concentration of 1,200 mg/kg. The second sample taken at 25 days
showed a drop of concentration to 1,100 mg/kg. Assume that a combination
of all the removal mechanisms, including volatilization, biodegradation, and
oxidation, shows first-order kinetics. Estimate how long it will take for the