Page 138 - Materials Chemistry, Second Edition

Mass-Balance Concept and Reactor Design 121

4.3.2 Half-Life
The half-life can be defined as the time needed to have one-half of the COC
degraded. In other words, it is the time required for the concentration to
drop to half of the initial value. For first-order reactions, the half-life (often
shown as t ) can be found from Equation (4.11) by substituting C by one-
1/2
A
half of C (i.e., C = 0.5 C ):
A0
A
A0
ln 2 0.693
t 1/2 = = (4.13)
k k
As shown in Equation (4.13), the half-life and the rate constant are inversely
proportional for the first-order reactions. If a value of half-life is given, we
can find the rate constant readily from Equation (4.13), and vice versa.

Example 4.4: Half-Life Calculation (1)
The half-life of 1,1,1-trichloroethane (1,1,1-TCA) in subsurface was deter-
mined to be 180 days. Assume that all the removal mechanisms are first-
order. Determine (1) the rate constant and (2) the time needed to drop the
concentration down to 10% of the initial concentration.

Solution:
(a) The rate constant can be easily determined from Equation (4.13)
as:
0.693
t 1/2 = 180 =
k

Thus, k = 0.00385/day.
(b) Use Equation (4.11) to determine the time needed to drop the
concentration down to 10% of the initial value (i.e., C = 0.1C ):
0
C 1
= = e − (0.00385)( t)
C 0 10

Therefore, t = 598 days.

Example 4.5: Half-Life Calculation (2)

On some occasions, the decay rate is expressed as T instead of t . T is the
1/2
90
90
time required for 90% of the compound to be converted (or the concentration
to drop to 10% of the initial value). Derive an equation to relate T with the
90
first-order reaction-rate constant.