Half-life and calculating radioactive decay

In a nutshell

Radioactive decay is completely random. The half-life is the time it takes for the number of nuclei in a sample to decrease to half its initial value. The activity of a sample is how many nuclei decay per second.

Decay and half-life

Radioactive decay is completely random. At any given time, a nucleus in a radioactive sample has a chance that it will decay. The nuclei will either decay by alpha, beta or gamma decay.

The half-life is the time it takes for the number of nuclei in a sample to decrease by half. After two half-lives, the number of nuclei will half again and decrease to a quarter of the initial value. This process continues as the number of nuclei remaining continues to decrease.

If an isotope has a short half-life, the nuclei will decay very quickly, meaning the sample emits a lot of radiation in a short amount of time. The sample is dangerous initially, but will quickly become safe. Whereas a sample with a long half-life will release radiation over a long period of time which could be dangerous is kept in the same place for millions of years.

Measuring radiation

The count rate is the number of nuclear decays detected per second. The count rate of a source can be measured with a Geiger-Muller tube. 

Rate of radioactive decay

Definition

Activity is the rate of radioactive decay and is measured in Becquerels ($$Bq$$). For a source of atomic nuclei, an activity of $$1\,Bq$$​ means one nucleus decays every second.

The activity of a source decreases over time. This is due to the source trying to become more stable. 

The half-life of a source also corresponds to the time it takes the activity to halve.

Example

Carbon-$$14$$​ has a half-life of $$5700 \,years$$. This means that after $$5700 \, years$$​ there is $$50\,\%$$​ of the original amount of carbon-$$14$$​ remaining. After two half-lives ($$11700\,years$$​) there is $$25\,\%$$​ of the carbon-$$14$$​ remaining (half of half of the original sample).

Calculating half life

Calculating the half life of a sample is useful as it can be used to determine when a sample is safe to handle. 

The graph below shows the activity over time of a sample of a radioactive isotope. The half life, $$t_0$$, can be determined by finding the point on the $$y$$-axis where activity is half the initial value, $$\frac{A_0}{2}$$​. A line can be drawn down to the $$x$$-axis to determine the half life. After two half-lives, $$2t_{1/2}$$, the activity of the source falls to a quarter of its initial value, $$\frac{A_0}{4}$$.​

The half-life is constant for a particular isotope. 

A Activity B Time 1. Activity is equal to half the initial activity. 2. Activity is equal to a quarter the initial activity. 3. Half life (time taken for activity to decrease to half the initial activity). 4. Two half lives.

The ratio of remaining radioactive nuclei after a period of time can be worked out using the formula $$\left( \dfrac{1}{2} \right) ^{n}$$​​,  where $$n$$ is the number of half-lives in the period of time.

This can be converted into a percentage by multiplying by $$100\,\%$$ if the question specifies to give your answer as a percentage.

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Example

A radioactive sample has a half-life of $$4 \,years$$​, what is the ratio of decayed to original nuclei after $$24\, years$$​?

First, write down the values give in the question and check they are in the correct form:

$$t_{1/2} = 4\ years\\ \ \\ T = 24\ years$$

The number of half-lives elapsed is $$n = \frac{24}{4}=6$$.

Next, write down the equation needed:

$$\left( \dfrac{1}{2} \right) ^{n}$$​​

Then, substitute the values into the equation:

 $$\left( \dfrac{1}{2} \right) ^{6} = \dfrac{1}{64}$$

The ratio of decayed to original nuclei is $$\underline{\dfrac{1}{64}}$$.

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