In a sample of radioactive material , what fraction of the initial number of active nuclei will remain undisintegrated after half of the half life of the sample ?

Options

A. `1/4`
B. `1/(2sqrt2)`
C. `1/sqrt2`
D. `sqrt2-1`

Explanation

To solve the problem of determining what fraction of the initial number of active nuclei will remain undisintegrated after half of the half-life of a sample of radioactive material, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**:
The half-life of a radioactive material is the time required for half of the radioactive nuclei in a sample to decay.

2. **Define Variables**:
- Let \( N_0 \) be the initial number of active nuclei.
- Let \( N \) be the number of active nuclei remaining after a certain time.

3. **Half-Life Formula**:
The number of remaining nuclei after one half-life can be expressed as:
\[ N = N_0 \left(\frac{1}{2}\right)^1 = \frac{N_0}{2} \]

4. **Half of the Half-Life**:
We need to find the number of nuclei remaining after half of the half-life. This means we are looking at a time period of \( \frac{T_{1/2}}{2} \).

5. **Applying the Half-Life Formula**:
Using the half-life formula for the time period of \( \frac{T_{1/2}}{2} \):
\[ N = N_0 \left(\frac{1}{2}\right)^{\frac{1}{2}} = N_0 \frac{1}{\sqrt{2}} \]

6. **Finding the Fraction Remaining**:
The fraction of the initial number of active nuclei that remains undisintegrated is:
\[ \text{Fraction remaining} = \frac{N}{N_0} = \frac{N_0 \frac{1}{\sqrt{2}}}{N_0} = \frac{1}{\sqrt{2}} \]

### Final Answer:
Thus, the fraction of the initial number of active nuclei that will remain undisintegrated after half of the half-life of the sample is:
\[ \frac{1}{\sqrt{2}} \]