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 of determining the time taken for a radioactive substance to decay by \( \frac{7}{8} \) of its initial value, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Decay Process**:
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. Given that the half-life is 20 seconds, we can use this information to determine how many half-lives it takes for the substance to decay by \( \frac{7}{8} \) of its initial amount.

2. **Determine the Remaining Fraction**:
If a sample decays by \( \frac{7}{8} \), then the remaining amount of the substance is:
\[ 1 - \frac{7}{8} = \frac{1}{8} \]
This means that after a certain time, only \( \frac{1}{8} \) of the initial amount remains.

3. **Use the Half-Life Formula**:
The fraction of undecayed nuclei remaining after \( n \) half-lives is given by:
\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^n \]
We need to find \( n \) such that:
\[ \left( \frac{1}{2} \right)^n = \frac{1}{8} \]

4. **Solve for \( n \)**:
We know that:
\[ \frac{1}{8} = \left( \frac{1}{2} \right)^3 \]
Therefore, \( n = 3 \). This indicates that it takes 3 half-lives for the substance to decay to \( \frac{1}{8} \) of its initial value.

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

### 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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