Derivation of a Half-Life Equation

To begin, we declare the general equation for exponential decay: where N is the number of half-lives that have elapsed at time t, m_o is the initial mass (or number of moles/atoms) of radioisotope (i.e. when t = 0), and m_t is the mass (or number of moles/atoms) of radioisotope remaining at time t.  If t_1/2 is the half-life of the radioisotope in the same unit as t, then:

, which implies .
Taking the (natural) logarithm of both sides of the equation yields .
By properties of logarithms, then:

We thus conclude that .

This equation lets us do easily what was hard with the basic equation for exponential decay: relate the fraction of a sample of a radioisotope remaining at a given time with the isotope's half-life. Note that in many textbooks you will see ln(2) written as its numerical approximation, .693.  Also, note that nothing in this equation is specific to nuclear decay; anything that undergoes exponential decay, from the death phase in bacteria to the decomposition of many molecules can be modeled with similar or identical mathematics. We will use these ideas again in our study of chemical kinetics.

Author: C. Shultz