# SS: Half Life

Show/Hide Sub-topics (Radioactivity and the nuclear atom | O Level)

The number of radioactive atoms in a substance decreases exponentially over time. Therefore every radioactive element has a characteristic time called half-life. After that time, the number of radioactive nuclei in the sample would have decreased to half the number originally present.

The half-life of a sample of a radioactive isotope is defined as the time taken for half the original unstable radioactive nuclei to decay.

• Different radioactive elements decay at different rates and hence different half-life. Uranium-238 has a half-life of over 400 million years. Strontium-90 has a half-life of 28 years and the half-life of oxygen-18 is less than 0.01 s.
• The radioactive decay of Na-24 can be studied using a GM-tube and a counter to find the number of counts per second at various time intervals.The half-life of Na-24 is 15 hours. Initially, there are 20 million undecayed Na-24 nuclei. After 15 hours (1 half-life), there will be 10 million undecayed Na-24 nuclei left.
• The half-life of a radioactive element gives an indication of its stability. Radioactive elements with longer half-life are more stable.

Formula:

$$\frac{N_{\text{final}}}{N_{\text{initial}}} = \left( \frac{1}{2} \right)^{n}$$

, where

$N_{\text{final}}$ is the number of remaining radioactive element

$N_{\text{initial}}$ is the number of initial radioactive element

n is the number of half-life

Note: You can apply logarithm to the above equation if you need to find the number of half-life (when you are given both final and initial number of elements)

$$n \text{log} \left( \frac{1}{2} \right) = \frac{N_{\text{final}}}{N_{\text{initial}}}$$