In a smaple of radioactive material, what fraction of the initial number of active nuclei will remain undisingrated after half of a half-life of the sample?
a. `(1)/(4)` b. `(1)/(2 sqrt(2))` c. `(1)/(sqrt(2))` d. `sqrt(2) - 1`

Options

A. a. `(1)/(4)`
B. b. `(1)/(2 sqrt(2))`
C. c. `(1)/(sqrt(2))`
D. d. `sqrt(2) - 1`

Explanation

To solve the problem, we need to determine the fraction of the initial number of active nuclei that will remain undisintegrated after half of a half-life of a radioactive sample.

### Step-by-Step Solution:

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

2. **Determine Half of Half-Life**:
Half of the half-life is given as \( \frac{1}{2} t_{half} \).

3. **Using the Half-Life Formula**:
The formula for the amount of substance remaining after \( n \) half-lives is:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
where \( N_0 \) is the initial quantity of the substance, and \( n \) is the number of half-lives that have passed.

4. **Calculate for Half of Half-Life**:
Since we are considering half of a half-life, we can express this as:
\[ n = \frac{1}{2} \]
Therefore, substituting \( n \) into the formula gives:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{1}{2}} = N_0 \cdot \frac{1}{\sqrt{2}} \]

5. **Finding the Fraction Remaining**:
To find the fraction of the initial number of active nuclei that remains, we can express it as:
\[ \text{Fraction remaining} = \frac{N}{N_0} = \frac{N_0 \cdot \frac{1}{\sqrt{2}}}{N_0} = \frac{1}{\sqrt{2}} \]

6. **Conclusion**:
Thus, the fraction of the initial number of active nuclei that will remain undisintegrated after half of a half-life is:
\[ \frac{1}{\sqrt{2}} \]

### Final Answer:
The correct option is **c. \( \frac{1}{\sqrt{2}} \)**.