# The half-life of a radioactive isotope 'X' is 50 years. It decays to another element 'Y' which is stable. The two elements 'X' and 'Y' were found to be in the ratio of 1:15 in a sample of a given rock. The age of the rock was estimated to be Option 1) 150 years Option 2) 200 years Option 3) 250 years Option 4) 100 years

Answers (1)

As we learnt in

Number of nuclei in terms of half life -

$N=\frac{N_{0}}{2^{t/t_{1/2}}}$

- wherein

Very useful to determine number of nuclei in terms of half life

At any time Ratio of X and Y is 1:15

if number of nuclei of X is N

then number of nuclei of Y is 15 N

Total number of nuclei initially and finally must be equal

$\\ \Rightarrow N_{o}=16 N \\ \\ \Rightarrow N=\frac{N_{o}}{16} \: = \frac{N_{o}}{2^{\tfrac{t}{t_\frac{1}{2}}}} \: or \: \: \frac{t}{t_\frac{1}{2}}=4$

$t=4 \: t_\frac{1}{2} \: = \: 200 years$

Option 1)

150 years

Incorrect

Option 2)

200 years

Correct

Option 3)

250 years

Incorrect

Option 4)

100 years

Incorrect

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