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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
A
100 years
B
150 years
C
200 years
D
250 years
Detailed Solution
X : Y = 1 : 15. If 1 part of X remains, 15 parts have decayed into Y, so the original amount of X was 1 + 15 = 16 parts.
Fraction of X remaining: $\frac{N}{N_0} = \frac{1}{16}$
$\frac{N}{N_0} = \left(\frac{1}{2}\right)^n$, where n is the number of half-lives.
$\frac{1}{16} = \left(\frac{1}{2}\right)^4$, so n = 4
Age of the rock $t = n\times T_{1/2} = 4\times50$
t = 200 years
Fraction of X remaining: $\frac{N}{N_0} = \frac{1}{16}$
$\frac{N}{N_0} = \left(\frac{1}{2}\right)^n$, where n is the number of half-lives.
$\frac{1}{16} = \left(\frac{1}{2}\right)^4$, so n = 4
Age of the rock $t = n\times T_{1/2} = 4\times50$
t = 200 years