Question

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. 150 years
• B. 200 years
• C. 250 years
• D. 100 years

Answer 1

The Correct Option is B

 The problem involves the decay of a radioactive isotope, and we need to calculate the age of a rock based on the given ratio of isotopes $X$ and $Y$. Here’s how to solve it:

1. The radioactive isotope $X$ decays into a stable element $Y$. The half-life of $X$ is given as 50 years.
2. In the problem, the ratio of the remaining $X$ to the produced element $Y$ is 1:15. This implies that for every 1 part of $X$ that remains, there are 15 parts of $Y$ that have been produced.
3. This ratio also implies that the original quantity of $X$ would have been 16 parts in total (1 part remaining + 15 parts decayed).
4. The amount of $X$ remaining is 1/16 of the original. This situation after $n$ half-lives can be expressed using the formula for radioactive decay:
   • Where \(N\) is the final amount, \(N_0\) is the initial amount, and \(n\) is the number of half-lives.
5. Here, \(\frac{1}{16} = \left( \frac{1}{2} \right)^n\).
6. Solving for \(n\) requires considering that \(\left( \frac{1}{2} \right)^4 = \frac{1}{16}\):
   • This implies \(n = 4\) half-lives.
7. The half-life of the isotope $X$ is 50 years, so the total age of the rock is:
8. Thus, the age of the rock is estimated to be 200 years.

Therefore, the correct answer is 200 years.