### Half-life

- The half-life ( ${T}_{\frac{1}{2}}$ ) of a quantity subject to exponential decay is the time required for the quantity (mass, number of atom or activity) to decay to half of its initial value.
- The decay curve of a radioactive source can be found experimentally - provided the activity of the source decreases over a practicable time period (minutes, hours or days for example).
- The corrected count-rate is then plotted against time to give the decay curve. The half-life of the source can be found from the curve as indicated above.

(Decay Curve) |

**Example 1**

A radioisotope has half-life of 8 hours. Initially, there were 3.6 x 10

^{18}radioisotope atoms in a sample. How much time is taken for the number of atoms of the radioisotope to fall to 4.5 x 10

^{17}?

**Answer**:

The sample take 3 half-life to decay from 3.6 x 10

^{18}radioisotope atoms to 4.5 x 10

^{17}. Therefore, the time taken

$$\begin{array}{l}t=3{T}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ t=3(8)=24h\end{array}$$

**Example**2

The diagram shows the graph of the activity of a radioisotope, X, against time. What is the half-life of the radioactive substance?

**Answer**:

The half-life is the time taken for the activity to decrease to become half of the initial value.

From the graph we can see that the radioisotope take 6 days for the activity to become half. Therefore

Half-life = 6 days