Radon, a radioactive noble gas, is an environmental problem...

Radon, a radioactive noble gas, is an environmental problem in some areas, where it can seep out of the ground and into homes. Exposure to radon-222, an $\alpha$ emitter with a half-life of 3.823 days, can increase the risk of lung cancer. At an exposure level of $4 \mathrm{pCi}$ per liter (the level at which the EPA recommends action), the lifetime risk of death from lung cancer due to radon exposure is estimated to be 62 out of 1000 for current smokers, compared with 73 out of 10,000 for nonsmokers. If the air in a home was analyzed and found to have an activity of $4.1 \mathrm{pCi} \mathrm{L}^{-1}$, how many atoms of ${ }^{222} \mathrm{Rn}$ are there per liter of air?

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Key Concepts

Activity and its Conversion to Atom Count

Activity measures the rate of radioactive decay and is expressed in units such as picocuries (pCi). The conversion from activity to the number of atoms relies on the formula A = ?N, where A is the activity, ? is the decay constant, and N is the number of radioactive atoms. This equation allows one to calculate the quantity of radioactive atoms present in a given sample from its measured activity.

Radioactivity

Radioactivity refers to the process by which unstable atomic nuclei lose energy by emitting radiation such as alpha particles, beta particles, or gamma rays. In this context, radon-222 is radioactive and decays by emitting alpha particles, which is a key concern when assessing environmental health risks.

Half-Life and Decay Constant

The half-life of a radioactive isotope is the time required for half of the atoms in a sample to decay. It is fundamentally related to the decay constant (?), which quantifies the probability per unit time that a given atom will decay. The relationship is given by ? = ln(2)/half-life, providing a bridge between the measurable half-life and the intrinsic decay properties of the isotope.

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