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A radioactive isotope 40/19 K undergoes decay either into stable 40/20 Ca with a decay constant of 4.5 × 10⁻¹⁰ per year or into stable 40/18 Ar with a decay constant of 0.5 × 10⁻¹⁰ per year. Assuming that all stable 40/20 Ca and 40/18 Ar nuclei originate from the decay of 40/19 K, determine the time t × 10⁹ years for which the ratio of the combined number of stable 40/20 Ca and 40/18 Ar nuclei to the remaining 40/19 K nuclei becomes 99. (Given ln 10 = 2.3).

1. 2.3
2. 9.2
3. 1.5
4. 4.6

Correct answer: 9.2

Solution

The decay constants for 40/19 K allow calculation of the total decay rate. Using the ratio of stable nuclei to remaining nuclei and the decay equation, the time required for the ratio to reach 99 is found to be 9.2 × 10⁹ years.

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