Changes

After 2 seconds, the fraction of material left is P².  After 10 seconds, it is P¹⁰, and so on.  So the fraction remaining is an exponential function of time.  This is the reason for the characteristic exponential decay of radioactive substances.  The type of decay doesn't matter; the decay is always exponential.  The usual way this is described is the ''half-life'', which is the time for one half of the sample to disintegrate.  The half-life can be seen to be log(1/2) / log(P).  Half-lives can vary from quantities like 10^-21 seconds for Beryllium-13, to 10¹⁸ years (far more than the age of the universe) for Tungsten-182.

After 2 seconds, the fraction of material left is P².  After 10 seconds, it is P¹⁰, and so on.  So the fraction remaining is an exponential function of time.  This is the reason for the characteristic exponential decay of radioactive substances.  The type of decay doesn't matter; the decay is always exponential.  The usual way this is described is the ''half-life'', which is the time for one half of the sample to disintegrate.  The half-life can be seen to be log(1/2) / log(P).  Half-lives can vary from quantities like 10^-21 seconds for Beryllium-13, to 10¹⁸ years (far more than the age of the universe) for Tungsten-182.

Since the rate at which radiation is admitted is proportional to the amount of the substance present, and the amount of substance remaining undergoes exponential decay, the rate of particle emission (e.g. Geiger counter clicks per second) also decays exponentially.

Since the rate at which radiation is emitted is proportional to the amount of the substance present, and the amount of substance remaining undergoes exponential decay, the rate of particle emission (e.g. Geiger counter clicks per second) also decays exponentially.

Specifically, the formula for the fraction of substance remaining after time t is:

Specifically, the formula for the fraction of substance remaining after time t is: