Calculating the isotope remaining
It should also be possible to state how much of a sample remains or what the activity or count should become after a given length of time.
This could be stated as a fraction, decimal or ratio.
For example the amount of a sample remaining after four half-lives could be expressed as:
- a fraction - a \(\frac{\text{1}}{\text{2}}\) of a \(\frac{\text{1}}{\text{2}}\) of a \(\frac{\text{1}}{\text{2}}\) of a \(\frac{\text{1}}{\text{2}}\) remains which is \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) = \(\frac{\text{1}}{\text{16}}\) of the original sample.
- a decimal - \(\frac{\text{1}}{\text{16}}\) = 0.0625 of the original sample
This could then be incorporated into other data. So, if the half-lifeThe half-life of a radioactive source is the time taken for its activity to fall to half of the original activity. Half-life is measured in seconds. is two days, four half-lives is 8 days.
Question
If a sample with a half-life of 2 days has a count rate of 3,200 Bq at the start, what is its count rate after 8 days?
If a sample has a count rate of 3,200 Bq at the start, what is its count rate after 8 days?
8 days = 4 half lives.
After 4 half lives the activity remaining would be \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) = \(\frac{\text{1}}{\text{16}}\)
\(\frac{\text{1}}{\text{16}}\) of 3,200 Bq = 200 Bq.
The count rate after 8 days is 200 Bq.
Question
The half-life of cobalt-60 is 5 years. If there are 100 g of cobalt-60 in a sample, how much will be left after 15 years?
15 years is three half-lives so the fraction remaining would be \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) x \(\frac{\text{1}}{\text{2}}\) = \(\frac{\text{1}}{\text{8}}\)
\(\frac{\text{1}}{\text{8}}\) of 100g = 12.5g.
After 15 years the amount remaining will be 12.5 g.