Introduction 291.4 A Nobel Prize Winning Application
Generally a mathematical model is constructed to approximate a physical
problem. Often this model includes a differential equation. Unless the model
was poorly constructed, the solution of the differential equation, if one exists,
will usually approximate the solution of the physical problem.
Radioactive Decay Physical experimentation has shown that radioac-
tive substances decompose at a rate which is proportional to the quantity of
radioactive substance present. If we let Q(t) represent the quantity of radioac-
tive substance present at time t, then the statement above may be expressed
mathematically by the differential equation
(1) dQ = kQ
dtwhere k is the constant of proportionali ty. Multiplying equation (1) by dt and
dividing by Q, we obtain
(2)dQ
Q = kdt.The variables Q and t are "separated" in this equation in the sense that Q
and its differential dQ appear on the left-hand side of the equation, while the
differential oft, dt, appears on the right-hand side of the equation. Integrating
equation (2), we find
or lnlQI = kt+c
where C is an arbitrary constant of integration. Exponentiating the right-
hand equation, we see
IQI = ekt+c = ec ekt.
Since Q and e^0 are positive constants, we may rewrite this last equation as
(3) Q(t) = Pektwhere P is a positive constant. To determine the two constants k and P
in equation (3), we need to specify two physical conditions to be satisfied.
Suppose at time t 0 the amount of radioactive substance present is Qo and
suppose at some later time t 1 the amount of substance present is Q1. Then
stated mathematically, the two conditions to b e satisfied a re
(4) Q(to) = Qo