a. Assuming a fast reaction between colloidal particles and impurities upon contact, the reaction rate can be expressed using the rate equation for a second-order reaction:
Rate
=
�
×
[
Colloidal particles
]
×
[
Impurities
]
Rate=k×[Colloidal particles]×[Impurities]
where
�
k is the rate constant,
[
Colloidal particles
]
[Colloidal particles] is the concentration of colloidal particles, and
[
Impurities
]
[Impurities] is the concentration of impurities. This equation assumes that the reaction only occurs when colloidal particles and impurities collide, and the rate of collision is proportional to both concentrations.
b. If colloidal particles can react with any number of impurity molecules without being consumed, the reaction follows pseudo-first-order kinetics. The rate-limiting step is the concentration of impurities since colloidal particles are in excess. The equation for the decay of impurities over time is given by:
[
Impurities
]
�
=
[
Impurities
]
0
×
�
−
�
′
×
[
Colloidal particles
]
×
�
[Impurities]
t
=[Impurities]
0
×e
−k
′
×[Colloidal particles]×t
where
[
Impurities
]
�
[Impurities]
t
is the concentration of impurities at time
�
t,
[
Impurities
]
0
[Impurities]
0
is the initial concentration of impurities,
[
Colloidal particles
]
[Colloidal particles] is the concentration of colloidal particles,
�
′
k
′
is the effective rate constant incorporating
�
k and reaction stoichiometry, and
�
t is time.
To remove 99% of impurity molecules (
99
%
=
1
%
99%=1% remaining),
1
%
1% of the initial concentration should be left:
[
Impurities
]
�
=
0.01
×
[
Impurities
]
0
=
[
Impurities
]
0
×
�
−
�
′
×
[
Colloidal particles
]
×
�
[Impurities]
t
=0.01×[Impurities]
0
=[Impurities]
0
×e
−k
′
×[Colloidal particles]×τ
Solving for
[
Colloidal particles
]
[Colloidal particles]:
[
Colloidal particles
]
=
−
ln
(
0.01
)
�
′
×
�
[Colloidal particles]=−
k
′
×τ
ln(0.01)
Assumptions: Fast reaction kinetics, excess colloidal particles, well-mixed system, neglecting potential surface effects, and no particle aggregation.