If `N_(0)` is the original mass of the substance of half - life period `t_(1//2) = 5 year` then the amount of substance left after `15` year is

Options

A. (a)`(N_0)/8`
B. (b)`(N_0)/16`
C. (c)`(N_0)/2`
D. (d)`(N_0)/4`

Answer

To solve the problem, we need to determine the amount of a radioactive substance remaining after a certain period of time, given its half-life.

### Step-by-Step Solution:

1. **Identify the given values:**
- Original mass of the substance: \( N_0 \)
- Half-life period \( t_{1/2} = 5 \) years
- Total time elapsed \( t = 15 \) years

2. **Calculate the number of half-lives:**
- The number of half-lives \( n \) can be calculated using the formula:
\[ n = \frac{t}{t_{1/2}} \]
- Substituting the values:
\[ n = \frac{15 \text{ years}}{5 \text{ years}} = 3 \]

3. **Use the half-life formula to find the remaining amount:**
- The remaining amount of the substance after \( n \) half-lives is given by:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
- Substituting \( n = 3 \):
\[ N = N_0 \left( \frac{1}{2} \right)^3 \]
- This simplifies to:
\[ N = N_0 \cdot \frac{1}{8} \]

4. **Final expression:**
- Therefore, the amount of substance left after 15 years is:
\[ N = \frac{N_0}{8} \]

### Conclusion:
The amount of the substance left after 15 years is \( \frac{N_0}{8} \).