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Half life, decay rate and the number of atoms

The problem is #12 on the attached document.

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Suppose that the half life is t_half. The number of atoms in the sample as a function of time is then given by:

N(t) = N(0) 2^(-t/t_half)

The rate at which atoms are decaying is thus given by minus the derivative (decay rate equals the rate at which N is decreasing):

-dN/dt = ...

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