A radioactive parent isotope P decays into a stable daughter D. At the time the mineral formed, the number of daughter atoms was equal to the initial number of parent atoms. At present the measured ratio is N(D):N(P) = 7:1. If the parent half-life is 1000 years, find the age of the sample in years.
Physics · Nuclear physics · radioactivity and law of decay
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Question

A radioactive parent isotope P decays into a stable daughter D. At the time the mineral formed, the number of daughter atoms was equal to the initial number of parent atoms. At present the measured ratio is N(D):N(P) = 7:1. If the parent half-life is 1000 years, find the age of the sample in years.

Physics · Nuclear physics · radioactivity and law of decay

G1 Physics · Accepted Answer

Let the initial number of parent atoms be N₀. The initial daughter population is also N₀. Let the surviving parent fraction after time t be f, so N(P) = fN₀. The number of parent atoms that have decayed is N₀−fN₀ = N₀(1−f). Therefore present daughter population is N(D) = initial daughter + newly produced daughter = N₀ + N₀(1−f) = N₀(2−f). Given N(D):N(P) = 7:1, (2−f) ÷ f = 7. Thus, 2−f = 7f ⇒ 2 = 8f ⇒ f = 1 ÷ 4. A surviving fraction 1 ÷ 4 means two half-lives. Age = 2×1000 = 2000 years. Hence N…