Half-Life 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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