Applications of First Order Differential Equations
1. A Chemical Reaction.
Two chemicals X and Y react in such a way that one gram of X combines with 4
grams of Y to produce a compound Z which is written by the chemists as
X + 4Y → Z.
The law of mass action in chemical kinetics states that, at a given time t, the rate
at which X and Y react (i.e. the rate at which Z is produced) is proportional to the
product of the amounts of X and Y that remain untransformed at that time (so this is
a second order reaction). Assume that there are initially 50 grams of X and 32 grams
of Y , and it is known that 30 grams of Z are formed in 10 minutes.
(a) Let z(t) grams of Z be present at time t. Derive a differential equation initial
value problem for z(t) and solve it by hand calculation.
(b) Check your solution in (a) using MAPLE.
(c) How much of the compound Z is present after 15 minutes?
(d) How much Z is present after a very long time (t → ∞)?
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