First slide

Radio activity

Question

The half life of a radioactive isotope 'X' is 20 years. It decays to another element 'Y' which is stable. The two elements ‘X’ and 'Y' were found to be in the ratio 1 : 7 in a sample of a given rock. The age of the rock is estimated to be

Moderate

A

80 years
B

100 years
C

40 years
D

60 years

Solution

$X \to Y$
Initial number of atoms, $N_{0} 0$
Number of atoms after time t $N N_{0} - N$
As per question $\frac{N}{N_{0} - N} = \frac{1}{7}$
$7 N = N_{0} - N 8 N = N_{0} \frac{N}{N_{0}} = \frac{1}{8}$
As $\frac{N}{N_{0}} = {(\frac{1}{2})}^{n}$ where n is the no. of half lives
$∴ \frac{1}{8} = {(\frac{1}{2})}^{n} {(\frac{1}{2})}^{3} = {(\frac{1}{2})}^{n} ∴ n = 3 n = \frac{t}{T_{1 / 2}} o r t = n T_{1 / 2} = 3 \times 20 y e a r s = 60 y e a r s$
Hence, the age of rock is 60 years.

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