In a radioactive disintegration, the ratio of initial number of atoms to the number of atoms present at an instant of time equal to its mean life is

Options

A. `1/e^2`
B. `1/e`
C. e
D. `e^2`

Answer

To solve the problem, we need to find the ratio of the initial number of atoms (N₀) to the number of atoms present at an instant of time equal to its mean life (τ). Here are the steps to derive the solution:

### Step-by-Step Solution:

1. **Understand the Mean Life (τ)**:
The mean life (τ) of a radioactive substance is defined as the average time that a particle exists before it decays. It is given by the formula:
\[ \tau = \frac{1}{\lambda} \]
where λ is the decay constant.

2. **Radioactive Decay Formula**:
The number of atoms remaining (N) after time t can be expressed using the radioactive decay formula:
\[ N = N_0 e^{-\lambda t} \]
where:
- \( N_0 \) = initial number of atoms
- \( N \) = number of atoms remaining after time t
- \( e \) = base of the natural logarithm
- \( \lambda \) = decay constant
- \( t \) = time elapsed

3. **Substituting Mean Life into the Formula**:
Since we are interested in the case where \( t = \tau \), we substitute \( t \) with \( \tau \):
\[ N = N_0 e^{-\lambda \tau} \]
Now, substituting \( \tau = \frac{1}{\lambda} \):
\[ N = N_0 e^{-\lambda \left(\frac{1}{\lambda}\right)} = N_0 e^{-1} \]

4. **Finding the Ratio**:
Now, we need to find the ratio of the initial number of atoms to the number of atoms remaining after time \( \tau \):
\[ \frac{N_0}{N} = \frac{N_0}{N_0 e^{-1}} = e \]

5. **Conclusion**:
Therefore, the ratio of the initial number of atoms to the number of atoms present at an instant of time equal to its mean life is:
\[ \frac{N_0}{N} = e \]

### Final Answer:
The answer is \( e \).