Elements such as ${\displaystyle {}_{92}^{238}U}$ 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:

$R=I\left ( \frac{1}{2} \right )^{n}$

where I represents the initial mass of the element and R represents the mass remaining.

Example $$\PageIndex{1}$$:

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:

$R=(1.00\; gram)\left ( \frac{1}{2} \right )^{6.00}=(1.00\; gram)(0.0156)=0.0156\; gram$

Exercise $$\PageIndex{1}$$

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:

$_{0}^{1}n+_{7}^{14}N \rightarrow _{1}^{1}\rho +_{6}^{14}C$

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:

$_{6}^{14}C \rightarrow _{-1}^{0}\beta +_{7}^{14}N$

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.

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