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 the fraction of the initial number of active nuclei that will remain undisintegrated after half of the half-life of a radioactive sample, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life (t_half) of a radioactive material is the time required for half of the radioactive nuclei in a sample to decay. After one half-life, 50% of the original nuclei remain.

2. **Define the Time Interval**:
The problem states that we need to find the fraction of nuclei remaining after half of the half-life. This means we will consider a time t = (1/2) * t_half.

3. **Use the Exponential Decay Formula**:
The number of remaining nuclei (N_t) at time t can be expressed using the formula:
\[ N_t = N_0 e^{-\lambda t} \]
where:
- \(N_0\) is the initial number of active nuclei,
- \(\lambda\) is the decay constant,
- \(t\) is the time elapsed.

4. **Relate the Decay Constant to Half-Life**:
The decay constant \(\lambda\) is related to the half-life by the equation:
\[ \lambda = \frac{\ln(2)}{t_{half}} \]

5. **Substitute Time into the Formula**:
Since we are considering \(t = \frac{1}{2} t_{half}\), we substitute this into the decay formula:
\[ N_t = N_0 e^{-\lambda \left(\frac{1}{2} t_{half}\right)} \]

6. **Substitute for \(\lambda\)**:
Now substitute \(\lambda\) into the equation:
\[ N_t = N_0 e^{-\left(\frac{\ln(2)}{t_{half}}\right) \left(\frac{1}{2} t_{half}\right)} = N_0 e^{-\frac{1}{2} \ln(2)} \]

7. **Simplify the Exponential Expression**:
Using the property of logarithms, we can simplify:
\[ N_t = N_0 e^{\ln(2^{-1/2})} = N_0 \cdot 2^{-1/2} = \frac{N_0}{\sqrt{2}} \]

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

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