11.5: Radioactive Half-Life
Elements such as that emit radioactive particles do so at rates that are constant and unique for each element. The rate at which an radioactive element decays is measured by its half-life ; the time it takes for one half of the radioactive atoms to decay, emitting a particle and forming a new element. Half-lives for elements vary widely, from billions of years to a few microseconds. On a simple, intuitive level, if you begin with 1.00 gram of a radioactive element, after one half-life there will be 0.500 grams remaining; after two half-lives, half of this has decayed, leaving 0.250 grams of the original element; after three half-lives, 0.125 grams would remain, etc. For those that prefer equations, the amount remaining after n half-lives can be calculated as follows:
\[R=I\left ( \frac{1}{2} \right )^{n} \nonumber \]
where I represents the initial mass of the element and R represents the mass remaining.
The half-life of Actinium-225 is 10.0 days. If you have a 1.00 gram sample of Actinium-225, how much is remaining after 60.0 days?
Solution
The number of half-lives is 6.00 (that is n ) and I = 1.00 gram. Substituting:
\[R=(1.00\; gram)\left ( \frac{1}{2} \right )^{6.00}=(1.00\; gram)(0.0156)=0.0156\; gram \nonumber \]
The half-life of Antimony-124 is 60.20 days. If you have a 5.00 gram sample of Actinium, how much is remaining after 5.0 half-lives?
One of the interesting uses for half-life calculations involves radiocarbon dating, where the content of carbon-14 in organic (formally living matter) is used to calculate the age of a sample. The process begins in the upper atmosphere, where nitrogen is bombarded constantly by high-energy neutrons from the sun. Occasionally, one of these neutrons collides with a nitrogen nucleus and the isotope that is formed undergoes the following nuclear equation:
\[_{0}^{1}n+_{7}^{14}N \rightarrow _{1}^{1}\rho +_{6}^{14}C \nonumber \]
Plants take up atmospheric carbon dioxide by photosynthesis, and are ingested by animals, so every living thing is constantly exchanging carbon-14 with its environment as long as it lives. Once it dies, however, this exchange stops, and the amount of carbon-14 gradually decreases through radioactive decay with a half-life of about 5,730 years, following the nuclear equation shown below:
\[_{6}^{14}C \rightarrow _{-1}^{0}\beta +_{7}^{14}N \nonumber \]
Thus, by measuring the carbon-14/carbon-12 ratio in a sample and comparing it to the ratio observed in living things, the number of half-lives that have passed since new carbon-14 was absorbed by the object can be calculated.