A radio-isotope has a half-life of `5` yeard. The fraction of the atoms of this material that would decay in `15` years will be

Options

A. `1//8`
B. `2//3`
C. `7//8`
D. `5//8`

Answer

To solve the problem of determining the fraction of a radioisotope that decays over a period of 15 years, given that its half-life is 5 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Half-Life**: The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as 5 years.

2. **Determine the Number of Half-Lives in 15 Years**:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{15 \text{ years}}{5 \text{ years}} = 3 \]
This means that 15 years corresponds to 3 half-lives.

3. **Calculate the Remaining Fraction After 3 Half-Lives**:
After each half-life, the remaining amount of the substance is halved. Therefore:
- After 1 half-life (5 years):
\[ N = \frac{N_0}{2} \]
- After 2 half-lives (10 years):
\[ N = \frac{N_0}{4} \]
- After 3 half-lives (15 years):
\[ N = \frac{N_0}{8} \]
This means that after 15 years, 1/8 of the original amount remains.

4. **Calculate the Decayed Fraction**:
The fraction of the original sample that has decayed can be calculated as:
\[ \text{Decayed Fraction} = 1 - \text{Remaining Fraction} \]
Substituting the remaining fraction:
\[ \text{Decayed Fraction} = 1 - \frac{1}{8} = \frac{7}{8} \]

5. **Conclusion**:
The fraction of the atoms of the radioisotope that would decay in 15 years is:
\[ \frac{7}{8} \]

### Final Answer:
The fraction of the atoms that decay in 15 years is \(\frac{7}{8}\).