How To Calculate Using Half Life

The atoms of radioactive substances have unstable nuclei that emit alpha, beta and gamma radiation to achieve a more stable configuration. When an atom undergoes radioactive decay, it can transform into a different element or into a different isotope of the same element. For any given sample, the decay doesn't occur all at once, but over a period of time characteristic of the substance in question. Scientists measure the rate of decay in terms of half life, which is the time it takes for half of the sample to decay.

Half lives can be extremely short, extremely long or anything in between. For example, the half life of carbon-16 is just 740 milliseconds, while that of uranium-238 is 4.5 billion years. Most are somewhere in between these almost immeasurable time intervals.

Half-life calculations are useful in a variety of contexts. For example, scientists are able to date organic matter by measuring the ratio of radioactive carbon-14 to stable carbon-12. To do this, they make use of the half life equation, which is easy to derive.

After the half life of a sample of radioactive material has elapsed, exactly one half of the original material is left. The remainder has decayed into another isotope or element. The mass of the remaining radioactive material (​m​_R) is 1/2 ​m​_O, where ​m​_O is the original mass. After a second half life has elapsed, ​m​_R = 1/4 ​m​_O, and after a third half life, ​m​_R = 1/8 ​m​_O. In general, after ​n​ half lives have elapsed:

\(m_R=\bigg(\frac{1}{2}\bigg)^n \; m_O\)

Americium-241 is a radioactive element used in the manufacture of ionizing smoke detectors. It emits alpha particles and decays into neptunium-237 and is itself produced from the beta decay of plutonium-241. The half life of the decay of Am-241 to Np-237 is 432.2 years.

If you throw away a smoke detector containing 0.25 grams of Am-241, how much will remain in the landfill after 1,000 years?

​Answer​: To use the half life equation, it's necessary to calculate ​n​, the number of half lives that elapse in 1,000 years.

\(n = \frac{1,000}{432.2} = 2.314\)

The equation then becomes:

\(m_R=\bigg(\frac{1}{2}\bigg)^{2.314} \; m_O\)

Since ​m​_O = 0.25 grams, the remaining mass is:

\(\begin{aligned}
m_R&=\bigg(\frac{1}{2}\bigg)^{2.314} \; ×0.25 \; \text{grams} \
m_R&=\frac{1}{4.972} \; ×0.25 \; \text{grams} \
m_R&=0.050 \;\text{grams}
\end{aligned}\)

The ratio of radioactive carbon-14 to stable carbon-12 is the same in all living things, but when an organism dies, the ratio starts changing as the carbon-14 decays. The half life for this decay is 5,730 years.

If the ratio of C-14 to C-12 in a bones unearthed in a dig is 1/16 of what it is in a living organism, how old are the bones?

​Answer​: In this case, the ratio of C-14 to C-12 tells you that the the current mass of C-14 is 1/16 what it is in a living organism, so:

\(m_R=\frac{1}{16}\;m_O\)

Equating the right hand side with the general formula of half life, this becomes:

\(\frac{1}{16}\;m_O = \bigg(\frac{1}{2}\bigg)^n\;m_O\)

Eliminating ​m​_O from the equation and solving for ​n​ gives:

\(\begin{aligned}
\bigg(\frac{1}{2}\bigg)^n &=\frac{1}{16} \
n&=4
\end{aligned}\)

Four half lives have elapsed, so the bones are 4 × 5,730 = 22,920 years old.