Chemical Reactions
Problem:
"Two chemicals A and B react to form a new chemical C.
Assuming that the concentrations of both A and B
decrease by the amount of C formed, find the differential
equation governing the concentration x(t) of the
chemical C if the rate at which the chemical reaction takes
place is proportional to the product of the remaining
concentrations of A and B."
Solution:
The statement of the problem is a little muddled.
The concentration of a chemical in a solution is the ratio
of the amount of the chemical to the amount of the solution.
This is often, though not necessarily, expressed as the molarity
of the solution, i.e., the number of moles (6.0249x10 23
molecules) of the chemical per liter of the solution. To say
that the concentration of one chemical is decreased by the
amount of another chemical is to use the terms inconsistently.
Therefore, the problem might better have read:
"Two chemicals A and B react to form a new chemical C.
Assuming that the amounts of both A and B
decrease by the amount of C formed, find the differential
equation governing the amount x(t) of the chemical C
if the rate at which the chemical reaction takes place is
proportional to the product of the remaining amounts
of A and B."
Alternatively, the word concentration could have been used
throughout the statement of the problem. To use
concentration in some places and amount in others, however,
is to combine apples and oranges.
With the statement reworded for consistency, there's still a
little ambiguity in the problem. Exactly how do the amounts
of A and B decrease by the amount of C formed? We'll
make the easiest and most natural assumption, that for each
unit (either moles or molecules) of C formed, one unit of A
is lost and one unit of B is lost. If a represents the number
of units of chemical A originally present, and b represents
the number of units of chemical B originally present, then
the amount of chemical A remaining at any time t and the
amount of chemical B remaining at any time t are given by
a  x(t) and b  x(t), respectively.
If we denote the constant of proportionality by k , then the
differential equation governing the amount x(t) of C
formed at time t is
dx t 
 ka  x t b  x t 
dt
If the units in which the amounts of the chemicals are given
are units of mass (or weight) rather than numbers of moles or
molecules, then the problem is slightly different. Suppose, for
instance, that the problem stated that for each gram of
chemical A, one gram of chemical B was used to form
chemical C. That would mean that 1 gram of A and 1 gram
of B react to form 2 grams of C. Then,
the amount of chemical A remaining at any time t and the
amount of chemical B remaining at any time t are given by
a
1
x t  and
2
b
1
x t ,
2
respective ly .
The differential equation governing the amount x(t) of C
formed at time t is then
dx t 
1
1



 k a  x t  b  x t 
dt
2
2



More generally, if the problem stated that for each M grams
of chemical A, N grams of chemical B were used to form
chemical C, that would mean that M grams of A and N
grams of B react to form M  N grams of C. Then,
the amount of chemical A remaining at any time t and the
amount of chemical B remaining at any time t are given by
a
M
x t 
MN
and
b
N
x t  , respectively.
MN
The differential equation governing the amount x(t) of C
formed at time t is then
dx t 
M
N



 k a 
x t  b 
x t 
dt
M

N
M

N


