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-life 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.

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