A radioactive parent isotope P decays into a stable daughter isotope D. At the time a mineral crystallised, no daughter isotope was present. At present the measured number ratio is N(D):N(P) = 15:1. If the half-life of P is 500 million years, find the age of the mineral in million years.
Physics · Nuclear physics · radioactivity and law of decay
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Question

A radioactive parent isotope P decays into a stable daughter isotope D. At the time a mineral crystallised, no daughter isotope was present. At present the measured number ratio is N(D):N(P) = 15:1. If the half-life of P is 500 million years, find the age of the mineral in million years.

Physics · Nuclear physics · radioactivity and law of decay

G1 Physics · Accepted Answer

Let the present number of parent atoms be x. Then present daughter atoms = 15x. Since there was initially no daughter and every daughter atom came from one parent decay: Initial parent atoms = x+15x = 16x. Therefore the surviving fraction is x ÷ 16x = 1 ÷ 16 = 1 ÷ 2⁴. Thus four half-lives have elapsed. Age = 4×500 million years = 2000 million years. Hence N = 2000.