Page 135 - Materials Chemistry, Second Edition

118 Practical Design Calculations for Groundwater and Soil Remediation

following general mathematical expression describes the rate that the con-
centration of species A (C ) changes with time:
A
γ= dC A =− kC A n (4.9)
A
dt
where n is the order of the reaction, k is the reaction-rate constant, and γ is
A
the rate of conversion of species A. If n is equal to 1, it is called a first-order
reaction. It implies that the reaction rate is proportional to the concentration
of the species. In other words, the higher the compound concentration, the
faster is the reaction rate. The first-order kinetics is applicable in many envi-
ronmental engineering applications. Consequently, discussion in this book
will be focused on the first-order reactions and their applications. A first-
order reaction can then be written as:

γ= dC A =− kC A (4.10)
A
dt
The rate constant itself provides lots of valuable information with regard
to the reaction. A larger k value implies a faster reaction rate, which, in turn,
demands a smaller reactor volume to achieve a specific conversion. The
value of k varies with temperature. In general, the higher the temperature,
the larger the k value will be for a reaction.
What would be the units of a reaction-rate constant for a first-order reac-
tion? Let us take a close look at Equation (4.10). In that equation, the units of
dC /dt is concentration/time and that of C is concentration; therefore, the
A
units of k should be 1/time. Consequently, if a reaction-rate constant has a
value of 0.25 day , the reaction should be a first-order reaction. The units of
−1
k for zeroth-order reactions and second-order reactions are [(concentration)/
time] and [(concentration)(time)] , respectively.
−1
Equation (4.10) tells us that the concentration of compound A is changing
with time. This equation can be integrated between t = 0 and time t:

ln C A =− kt or C A = e − kt (4.11)
C A0 C A0

where C is the concentration of compound A at t = 0, and C is the concen-
A
A0
tration at time t.
Example 4.2: Estimate the Rate Constant from Two
Known Concentration Values (1)
An accidental gasoline spill occurred at a site 20 days ago. The total petro-
leum hydrocarbon (TPH) concentration at a specific location in soil dropped
from an initial 3,000 mg/kg to the current 2,750 mg/kg. The decrease in