The half life period of a radioactive substance is 10 year. The amount of the substance decayed after 40 years would be

Options

A. 0.25
B. 0.125
C. 0.9375
D. 0.0625

Explanation

To solve the problem of how much of a radioactive substance has decayed after 40 years given a half-life of 10 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Half-Life**:
The half-life (t₁/₂) of the radioactive substance is given as 10 years.

2. **Determine the Total Time**:
We need to find out how much substance has decayed after a total time of 40 years.

3. **Calculate the Number of Half-Lives**:
To find the number of half-lives (n) that have passed in 40 years, we use the formula:
\[ n = \frac{\text{Total Time}}{\text{Half-Life}} = \frac{40 \text{ years}}{10 \text{ years}} = 4 \]

4. **Use the Decay Formula**:
The remaining amount of substance after n half-lives can be calculated using the formula:
\[ N_t = N_0 \left(\frac{1}{2}\right)^n \]
where \(N_t\) is the remaining amount, \(N_0\) is the initial amount, and \(n\) is the number of half-lives. Substituting in our values:
\[ N_t = N_0 \left(\frac{1}{2}\right)^4 = N_0 \cdot \frac{1}{16} \]

5. **Calculate the Amount Decayed**:
The amount of substance that has decayed (D) can be calculated as:
\[ D = N_0 - N_t \]
Substituting \(N_t\) from the previous step:
\[ D = N_0 - \left(N_0 \cdot \frac{1}{16}\right) = N_0 \left(1 - \frac{1}{16}\right) = N_0 \cdot \frac{15}{16} \]

6. **Conclusion**:
Therefore, the amount of the substance that has decayed after 40 years is:
\[ D = \frac{15}{16} N_0 \]

### Final Answer:
The amount of the substance decayed after 40 years is \( \frac{15}{16} N_0 \).