HalfLife
An unstable nucleus can wait a relatively long time before it gets around to
decaying
.
After it does so, it typically becomes a different element that may or may not be
radioactive itself. Since the amount of time that passes before decay occurs is essentially
random, the radioactivity of a large number of atoms is best treated statistically.
Suppose that
N
measures the number of radioactive atoms in a sample that have not yet
decayed. In this case, the change in N with respect to time must be proportional to the
number N itself. If there are no nuclei left to decay, after all, we can’t very well expect
the number to keep decreasing. The differential equation is therefore
dN/dt =
λ
N
, where
lambda is a proportionality constant and the negative implies a
decrease
in the number of
atoms. The solution to this equation is an exponential:
N = N
₀
e

λ
t
, where lambda is
now called the
decay constant
and
N
is the starting number of atoms. This can be
difficult to conceptualize, however, so a second measurement called the
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 Spring '11
 Tibbets
 Radioactive Decay, HalfLife, Isotope, onequarter, millionyear, 1019 years

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