The half life of a radioactive substance is 20s, the time taken for the sample to decay by `7//8^(th)` of its initial value is

Options

A. 20 s
B. 40 s
C. 60 s
D. 80 s

Explanation

To solve the problem, we need to determine the time taken for a radioactive substance to decay by \( \frac{7}{8} \) of its initial value, given that its half-life is 20 seconds.

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. In this case, the half-life is given as 20 seconds.

2. **Determine the Remaining Nuclei After Decay**:
If the sample decays by \( \frac{7}{8} \) of its initial value, this means that only \( \frac{1}{8} \) of the initial amount remains.
- Let \( N_0 \) be the initial number of nuclei.
- After decaying \( \frac{7}{8} \), the remaining number of nuclei is:
\[ N = N_0 - \frac{7}{8}N_0 = \frac{1}{8}N_0 \]

3. **Calculate the Number of Half-Lives Required**:
We know that after each half-life, the remaining amount of the substance is halved:
- After 1 half-life: \( N_0 \to \frac{1}{2}N_0 \)
- After 2 half-lives: \( N_0 \to \frac{1}{4}N_0 \)
- After 3 half-lives: \( N_0 \to \frac{1}{8}N_0 \)

From the above, we can see that it takes 3 half-lives for the amount of substance to reduce to \( \frac{1}{8}N_0 \).

4. **Calculate the Total Time Taken**:
Since each half-life is 20 seconds, the total time taken for 3 half-lives is:
\[ \text{Total time} = 3 \times \text{half-life} = 3 \times 20 \text{ seconds} = 60 \text{ seconds} \]

### Final Answer:
The time taken for the sample to decay by \( \frac{7}{8} \) of its initial value is **60 seconds**.

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