Variation of Radioactivity Over Time
The rate at which a given radionuclide sample decays as being equal to the product of the number of atoms and the decay constant. From this basic relationship it is possible to use calculus to derive an expression which can be used to calculate how the number of atoms present will change over time. The derivation is beyond the scope of this text, but Equation (1-1) is the useful result.

     (1-1)

where:
N = number of atoms present at time t
N_o = number of atoms initially present
λ = decay constant (time^-1)
t = time

Since the activity and the number of atoms are always proportional, they may be used interchangeably to describe any given radionuclide population. Therefore, the following is true.

    (1-2)

where:
A = activity present at time t
A_o = activity initially present
λ = decay constant (time^-1)
t = time