The activity of a radioactive source is the rate at which hat source decays and it is defined using:

(1)

where

is the activity

is the decay constant

is the number of undecayed nuclei

Since the activity is the rate at which undecayed nuclei decay, we can also write;

(2)

where

is the change in the number of undecayed nuclei

is the change in time (in which decay over)

Equating equations (1) and (2) gives us;

(the negative sign arises because the number if undecayed nuclei is decreasing over time.)

Rearranging this so as to integrate;

This becomes:

Since we are integrating, limits are required;

- We are integrating from an initial time to some point in time later.
- The number of undecided nuclei will change from an initial amount of to a new amount some time () later, .

This becomes;

Using log rules whereby , this becomes;

Taking the exponential of both sides gives;

Rearranging for gives:

where

is the amount of undecayed nuclei remaining after time

is the initial amount of undecayed nuclei at

is the decay constant (which is dependent on the type of nuclei material)

is the chosen time frame in which the nuclei are decaying over

Since

Rearranging for gives;

The maximum activity will occur when there is the maximum number of undecided nuclei (because there are more unstable nuclei and therefore a possibility for more to decay per unit time), so we can write;

Substituting this in to gives;

Multiplying both sides by gives;