Question

The half-life period of a radioactive element A is 62 years. It decays into another stable element B. An archaeologist found a sample in which A and B are in 1 : 15 ratio. The age of the sample is:

• A. 248 years
• B. 186 years
• C. 124 years
• D. 310 years

Hint:

- The decay of radioactive substances follows an exponential decay model based on half-life.
- The ratio of remaining substance to decayed substance can be used to find the age of the sample.

Answer 1

The Correct Option is A

We are given that the half-life of A is 62 years, and the ratio of A to B in the sample is 1:15. The relationship between the amount of radioactive substance left and its half-life is governed by the equation: \[ N = N_0 \left( \frac{1}{2} \right)^{t/T} \] Where: - \( N \) is the amount of the radioactive substance left after time \( t \),
- \( N_0 \) is the initial amount of the substance,
- \( T \) is the half-life of the substance, and
- \( t \) is the time elapsed.
Here, we know the ratio of the remaining substance A to the product B is 1 : 15, which implies: \[ \frac{N_A}{N_B} = 1 : 15 \] Using the decay formula and the fact that B is formed as A decays, we can calculate the age of the sample: \[ \frac{N_A}{N_0} = \left( \frac{1}{2} \right)^{t/62} \] Given the ratio of 1:15, we substitute into the equation: \[ \left( \frac{1}{2} \right)^{t/62} = \frac{1}{16} \] This gives \( t = 248 \) years. Thus, the age of the sample is 248 years.