The probability of nucleus to decay in two mean lives is

Options

A. `1/4`
B. `(e^2-1)/e^2`
C. `3/4`
D. `1/e^2`

Answer

To find the probability of a nucleus to decay in two mean lives, we can follow these steps:

### Step 1: Understand Mean Life
The mean life (τ) of a nucleus is the average time it takes for a nucleus to decay. The relationship between the mean life and the decay constant (λ) is given by:
\[ \tau = \frac{1}{\lambda} \]
Thus, two mean lives can be expressed as:
\[ 2\tau = \frac{2}{\lambda} \]

### Step 2: Probability of Undecayed Nuclei
The probability \( P \) of a nucleus remaining undecayed after a time \( t \) is given by:
\[ P = e^{-\lambda t} \]
Substituting \( t = 2\tau \):
\[ P = e^{-\lambda \cdot \frac{2}{\lambda}} = e^{-2} \]

### Step 3: Probability of Decay
The probability of decay \( P' \) is the complement of the probability of remaining undecayed:
\[ P' = 1 - P \]
Substituting the value of \( P \):
\[ P' = 1 - e^{-2} \]

### Step 4: Simplifying the Probability of Decay
We can express \( P' \) in a different form:
\[ P' = 1 - \frac{1}{e^2} = \frac{e^2 - 1}{e^2} \]

### Conclusion
Thus, the probability of a nucleus to decay in two mean lives is:
\[ P' = \frac{e^2 - 1}{e^2} \]

### Final Answer
The correct option is:
\[ \text{Option 2: } \frac{e^2 - 1}{e^2} \]

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