How to Calculate Half-Life

The half-life of a radioactive isotope is the time taken for half the radioactive nuclei in a sample to decay, or equivalently, the time for the activity (count rate) to halve.

Key facts

• Half-life is constant for a given isotope. It does not change with temperature, pressure or the amount of substance remaining.
• Radioactive decay is random and spontaneous. You cannot predict when a specific nucleus will decay, but you can predict how many will decay on average.

Calculating remaining activity or mass

After $n$ half-lives, the fraction remaining is:

$\text{fraction remaining} = \left(\frac{1}{2}\right)^n$

Half-lives elapsed	Fraction remaining	Percentage remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
4	1/16	6.25%

Worked example

A radioactive source has an initial activity of 800 counts per minute. Its half-life is 3.0 hours. Find the activity after 9.0 hours.

Number of half-lives: $9.0 / 3.0 = 3$

Activity after 3 half-lives: $800 \times (1/2)^3 = 800 / 8 = 100$ counts per minute.

Reading half-life from a graph

On a count rate vs time graph (decay curve):

1. Choose an initial count rate (e.g. 400).
2. Find the time when the count rate is half that value (200).
3. The half-life is the time interval between those two readings.
4. Check by repeating from another starting point to verify consistency.

Remember to subtract background radiation from all readings first.

Worked example: from a graph

A decay curve shows the activity drops from 600 to 300 cpm between $t = 0$ and $t = 4$ minutes, and from 300 to 150 cpm between $t = 4$ and $t = 8$ minutes. The half-life is 4 minutes.

Common errors and how to correct them

Dividing by the number of half-lives instead of halving repeatedly. After 3 half-lives, the activity is not $800/3$; it is $800 \times (1/2)^3 = 100$.

Not subtracting background count rate before reading the graph. If background is 20 cpm and the reading is 420, the corrected count rate is 400.

How to apply this in an exam

State the definition of half-life. Show each halving step clearly. For graph questions, draw horizontal lines to the curve and vertical lines to the time axis, and check consistency over two intervals.