In a sample of radioactive material, what fraction of initial number of active nuclei will remain undistintegrated after half of a half0life of the sample?

Options

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

Explanation

To solve the problem of determining the fraction of initial active nuclei that remain undisintegrated after half of a 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) of a radioactive material is the time required for half of the radioactive nuclei in a sample to decay.

2. **Define the Decay Constant (λ)**:
The decay constant (λ) is related to the half-life by the formula:
\[ T = \frac{\ln 2}{\lambda} \]
where \( \ln 2 \) is the natural logarithm of 2.

3. **Determine Half of the Half-Life**:
If we denote the half-life as \( T \), then half of the half-life is:
\[ t = \frac{1}{2} T = \frac{1}{2} \left(\frac{\ln 2}{\lambda}\right) = \frac{\ln 2}{2\lambda} \]

4. **Use the Exponential Decay Formula**:
The fraction of undecayed nuclei at time \( t \) is given by the formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of nuclei.

5. **Substitute for t**:
Substitute \( t = \frac{\ln 2}{2\lambda} \) into the decay formula:
\[ N(t) = N_0 e^{-\lambda \left(\frac{\ln 2}{2\lambda}\right)} \]

6. **Simplify the Exponential**:
The expression simplifies as follows:
\[ N(t) = N_0 e^{-\frac{\ln 2}{2}} = N_0 e^{\ln 2^{-1/2}} = N_0 \cdot 2^{-1/2} \]

7. **Calculate the Fraction**:
The fraction of undecayed nuclei is:
\[ \frac{N(t)}{N_0} = 2^{-1/2} = \frac{1}{\sqrt{2}} \]

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