Potassium-40 decays to argon-40 with a half-life of 1.27 * 109 yr. What is the age of a rock in which the mass ratio of 40Ar to 40K is 4.2?

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Identify the decay process: Potassium-40 (\(^{40}\text{K}\)) decays to Argon-40 (\(^{40}\text{Ar}\)) through a radioactive decay process.

Use the given half-life of \(^{40}\text{K}\), which is 1.27 \(\times\) 10^9\) years, to determine the decay constant (\(\lambda\)) using the formula \(\lambda = \frac{\ln(2)}{\text{half-life}}\).

Apply the formula for radioactive decay: \(N_t = N_0 e^{-\lambda t}\), where \(N_t\) is the number of \(^{40}\text{K}\) atoms remaining, \(N_0\) is the initial number of \(^{40}\text{K}\) atoms, and \(t\) is the time elapsed.

Relate the mass ratio of \(^{40}\text{Ar}\) to \(^{40}\text{K}\) to the number of atoms: \(\frac{N_{\text{Ar}}}{N_{\text{K}}} = 4.2\), where \(N_{\text{Ar}}\) is the number of \(^{40}\text{Ar}\) atoms and \(N_{\text{K}}\) is the number of \(^{40}\text{K}\) atoms.

Solve for the age of the rock (\(t\)) using the relationship between \(N_{\text{Ar}}\), \(N_{\text{K}}\), and the decay constant \(\lambda\).

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주요 개념

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Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive substance to transform into a different element or isotope.

Half-Life

Half-life is a specific time period in which half of a given quantity of a radioactive isotope decays into its daughter product. For potassium-40, the half-life is 1.27 billion years, meaning that after this time, half of the original potassium-40 will have decayed into argon-40.

Mass Ratio and Age Calculation

The mass ratio of daughter to parent isotopes can be used to determine the age of a rock through the equation derived from the decay law. By knowing the mass ratio of argon-40 to potassium-40 and the half-life, one can calculate the time elapsed since the rock formed, providing an estimate of its age.