A radioactive element disintegrates for a time interval equal to its mean life. The fraction that has disintegrated is

Options

A. (a) `1/e`
B. (b) `1-1/e`
C. (c) `0.693/e`
D. (d) `0.693(1-1/e)`

Explanation

To solve the problem of finding the fraction of a radioactive element that has disintegrated after a time interval equal to its mean life, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mean Life**: The mean life (τ) of a radioactive element is the average time that a nucleus exists before it decays. It is related to the decay constant (λ) by the equation:
\[ τ = \frac{1}{λ} \]

2. **Time Interval**: We are given that the time interval (t) is equal to the mean life (τ). Therefore, we can write:
\[ t = τ = \frac{1}{λ} \]

3. **Exponential Decay Formula**: The number of undecayed nuclei (N) at time t can be expressed using the exponential decay formula:
\[ N = N_0 e^{-λt} \]
where \(N_0\) is the initial number of nuclei.

4. **Substituting Time**: Since \(t = τ\), we can substitute \(t\) in the equation:
\[ N = N_0 e^{-λ \cdot τ} \]
Substituting \(τ = \frac{1}{λ}\):
\[ N = N_0 e^{-λ \cdot \frac{1}{λ}} = N_0 e^{-1} \]

5. **Finding the Fraction Disintegrated**: The fraction of the original nuclei that has disintegrated (D) is given by:
\[ D = 1 - \frac{N}{N_0} \]
Substituting \(N = N_0 e^{-1}\):
\[ D = 1 - e^{-1} \]

6. **Calculating the Numerical Value**: The value of \(e^{-1}\) is approximately 0.3679. Therefore:
\[ D = 1 - 0.3679 \approx 0.6321 \]

7. **Conclusion**: The fraction of the radioactive element that has disintegrated after a time interval equal to its mean life is approximately:
\[ D \approx 0.6321 \]

### Final Answer:
The fraction that has disintegrated is approximately 0.6321 or 63.21%.