# How is exponential decay related to a half-life?

May 5, 2018

Half-life is a form of exponential decay over time.
See below.

#### Explanation:

Assume you have an initial quantity (${Q}_{0}$) of a substance with a half-life period of $h$, then after $t$ time periods the quantity remaining (${Q}_{t}$) will be given by:

${Q}_{t} = {Q}_{0} \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

This means that the quantity will halve in value after $h$ time periods.

An example:

The half-life of Plutonium-241 is 14.4 years. If I start with 1000 gm after 5 years and 50 years I will have:

${Q}_{5} = 1000 \cdot {\left(\frac{1}{2}\right)}^{\frac{5}{14.4}} \approx 786.1$ gm

${Q}_{50} = 1000 \cdot {\left(\frac{1}{2}\right)}^{\frac{50}{14.4}} \approx 90.1$ gm

This is a form of exponential decay.