The half-life of a radioactive isotope $X$ is 20 years. It decays to another element $Y$ which is stable. The

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:

Accepted Answer

1. **Understand the Decay Process:** The radioactive isotope $X$ decays into the stable element $Y$. The amount of $Y$ present represents the amount of $X$ that has decayed (assuming no $Y$ was present initially).
2. **Determine Initial Amount ($N_0$):** 
   - Ratio of remaining parent ($X$) to daughter ($Y$) nuclei = $1:7$.
   - Let the number of nuclei of $X$ be $x$ and $Y$ be $7x$.
   - Total initial nuclei $N_0 = N_X + N_Y = x + 7x = 8x$.
3. **Calculate Fraction Remaining:**
   - Fraction of $X$ remaining = $\frac{N_X}{N_0} = \frac{x}{8x} = \frac{1}{8}$.
4. **Relate to Half-Lives:**
   - The fraction remaining is given by $(\frac{1}{2})^n$, where $n$ is the number of half-lives passed.
   - $\frac{1}{8} = (\frac{1}{2})^3$, so $n = 3$ half-lives.
5. **Calculate Age:**
   - Age = $n 	imes T_{1/2} = 3 	imes 20 	ext{ years} = 60 	ext{ years}$.

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