Repeated halving
- After each half-life, the number of undecayed nuclei, the activity and the count rate all fall to half of their previous value.
- After 111 half-life, 12\frac{1}{2}21 of the original nuclei remain.
- After 222 half-lives, 12×12=14\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}21×21=41 of the original nuclei remain.
- After 333 half-lives, 18\frac{1}{8}81 remain, after 444 half-lives 116\frac{1}{16}161 remain, and after 555 half-lives 132\frac{1}{32}321 remain.
- After nnn half-lives, the fraction remaining is (12)n=12n\left(\frac{1}{2}\right)^n = \frac{1}{2^n}(21)n=2n1.
- The number of half-lives that have passed is found by dividing the time elapsed by the half-life, so n=time elapsedhalf-lifen = \frac{\text{time elapsed}}{\text{half-life}}n=half-lifetime elapsed.
Net decline as a ratio
Definition
Net decline
The decrease in the number of undecayed nuclei, or in activity, between the start and the end of a period of radioactive decay.
- The net decline is the amount by which the number of undecayed nuclei, or the activity, has fallen.
- Net decline is found from net decline=initial value−value remaining\text{net decline} = \text{initial value} - \text{value remaining}net decline=initial value−value remaining.
- After nnn half-lives, the fraction that has decayed is 1−(12)n1 - \left(\frac{1}{2}\right)^n1−(21)n.
- The decline can be written as a ratio of the net decline to the initial value.
- After 111 half-life, half has decayed, so the ratio of net decline to initial value is 1:21:21:2.
- After 222 half-lives, 34\frac{3}{4}43 has decayed, so the ratio is 3:43:43:4.
- After 333 half-lives, 78\frac{7}{8}87 has decayed, so the ratio is 7:87:87:8.
- The decline can also be written as the ratio of the value remaining to the initial value, which is 1:2n1:2^n1:2n, so after 333 half-lives it is 1:81:81:8.
- Ratios have no units, because they compare two values of the same quantity.
Example
Net decline of a sample
- A sample contains 480048004800 undecayed nuclei, and 444 half-lives pass.
- The ratio of the number remaining to the initial number is 300:4800300 : 4800300:4800, which simplifies to 1:161 : 161:16.
Example
Activity after a given time
- A source has an activity of 960 Bq960\ \text{Bq}960 Bq and a half-life of 555 days, and its activity is measured again after 202020 days.
- The ratio of the final activity to the initial activity is 60:960=1:1660 : 960 = 1 : 1660:960=1:16, and the ratio of the net decline to the initial activity is 900:960=15:16900 : 960 = 15 : 16900:960=15:16.
Half-life graphs
Definition
Decay curve
A graph of the number of undecayed nuclei, activity or count rate of a radioactive sample plotted against time.
- A decay curve is a graph of the activity, count rate or number of undecayed nuclei plotted against time.
- The curve falls steeply at first and then more and more gently, because the amount that decays each second gets smaller as fewer nuclei remain.
- The curve approaches zero but never quite reaches it.
- To find the half-life from a decay curve, follow these steps.
- Read the starting value from the vertical axis.
- Halve this value and draw a horizontal line across to the curve.
- Draw a vertical line down from the curve to the time axis and read off the time, which is the half-life.
- Repeat from a different starting point, such as the value at the first half-life, and calculate the mean of your answers.
- Count-rate data should have the background count subtracted before the curve is drawn, otherwise the curve levels off at the background value instead of approaching zero.
- Real data points scatter slightly about the curve because decay is random, so a smooth curve of best fit is drawn through them.
Example
Reading a half-life from a decay curve
- A decay curve starts at a corrected count rate of 600600600 counts per minute.
- Half of this value is 300300300 counts per minute, and the curve passes through 300300300 counts per minute at 121212 minutes.
- Half of 300300300 is 150150150 counts per minute, and the curve passes through 150150150 counts per minute at 242424 minutes.
- Both readings give a half-life of 121212 minutes, so after 363636 minutes, which is 333 half-lives, the count rate will have fallen to 600×18=75600 \times \frac{1}{8} = 75600×81=75 counts per minute.
Common Mistake
- Do not divide by 2n2n2n to find the fraction remaining, because after 333 half-lives the fraction is 18\frac{1}{8}81, not 16\frac{1}{6}61.
- Do not use the total time in place of the number of half-lives, because you must first divide the time by the half-life.
- Do not confuse the amount remaining with the net decline, because the decline is the amount that has decayed.
Exam technique
Setting out decay calculations
- Show the number of half-lives first, for example n=205=4n = \frac{20}{5} = 4n=520=4, because this step is often credited on its own.
- Write the halving chain, such as 960→480→240→120→60960 \rightarrow 480 \rightarrow 240 \rightarrow 120 \rightarrow 60960→480→240→120→60, when you are unsure, because it is easy to check.
- Give a ratio in the order the question asks for and in its simplest whole-number form, such as 15:1615 : 1615:16.
- When reading a half-life from a graph, draw your construction lines on the graph so that the examiner can follow your method.
Self review
- What fraction of a radioactive sample remains undecayed after 333 half-lives?
- How do you find the number of half-lives that have passed in a given time?
- What is the ratio of net decline to initial value after 222 half-lives?
- Why does a decay curve never quite reach zero?
- Why should background radiation be subtracted before drawing a decay curve?
