6. MODELING WITH FIRST-ORDER EQUATIONS
43
Radioactive Decay
Suppose ∆t is a sufficiently small time interval (i.e., as small as we need) and
P (∆t) is the probability that a given nucleus will decay over any time period
of length ∆t.
We assume P (∆t) ∝ ∆t =⇒ P (∆t) ≈ λ∆t, λ > 0 and that P (∆t) << 1
approx
(P (∆t) is much smaller than 1).
P (∆t)
Also assume that lim
= λ.
∆t→0 ∆t
Let N (t) = # of nuclei at time t.
Then the number of atoms decaying over a time period of ∆t beginning at time
t is
N (t)P (∆t) ≈ N (t) · λ · ∆t =⇒
N (t + ∆t) = N (t) − N (t)P (∆t) ≈ N (t) − N (t) · λ · ∆t =⇒
dN
N (t + ∆t) − N (t)
−N (t) · λ · ∆t
= lim
= lim
=
∆t→0
∆t→0
dt
∆t
∆t £
§
lim − N (t) · λ = −λN(t)
∆t→0
Thus the model is
dN
= −λN(t), N (0) = N0.
dt
However, this model is flawed due to the discreteness of N(t), whose values are
integers, resulting in a discontinuous function.