The radioisotope potassium-40 decays to argon-40 by positron emission with a half-life of 1.3 * 10⁹ yr. A sample of moon rock was found to contain 78 argon-40 atoms for every 22 potassium-40 atoms. The age of the rock is

Question

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The ratio of argon-40 to potassium-40 (78:22) suggests that 78/100 of the original potassium-40 has decayed to form argon-40. This means that 22/100 of the original potassium-40 remains.

Using the half-life equation, t = (ln 2) / λ, where λ is the decay constant (λ = ln 2 / half-life), we can find the age of the rock:

t = (ln 2) / λ = (ln 2) / (0.693 / 1.3x10^10) = 2.8x10^9 years

Therefore, the age of the rock is 2.8x10^9 years, which is equivalent to 2.8x10^9 / 1.3x10^10 = 0.215 or 21.5% of the half-life. Multiplying this fraction by the half-life gives:

Age of rock = 1.3x10^10 years x 0.215 = 2.8x10^9 years

Thus, the best choice is C, which corresponds to an age of 2.8x10^9 years.

The Radioisotope Potassium-40 Decays to Argon-40 by Positron Emission with a Half-Life