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Question:

In a radioactive sample, 1940K nuclei either decay into stable 2040Ca nuclei with decay constant 4.5 × 10⁻¹⁰ per year or into stable 1840Ar nuclei with decay constant 0.5 × 10⁻¹⁰ per year. Given that in this sample all the stable 2040Ca and 1840Ar nuclei are produced by the 1940K nuclei only. In time t × 10⁹ years, if the ratio of the sum of stable 2040Ca and 1840Ar nuclei to the radioactive 1940K nuclei is 99, the value of t will be?

9.2 × 10⁹

1.5 × 10⁶

none

both

Solution:

Correct option is A. 9.2 × 10⁹
t = 0
dN/dt = λ₁ + λ₂
loge(N/N₀) = −(λ₁ + λ₂)t
2.3 × log₁₀(N₀/N₀/100) = 5 × 10⁻¹⁰t
2.303 × 2 = 5 × 10⁻¹⁰t
2.303 × 0.4 × 10¹⁰ = t
t = 9.2 × 10⁹ year.