In a radioactive sample, the fraction of initial number of redioactive nuclie, which remains undecayed after n mean lives is

Options

A. `(1)/(e^(n))`
B. `e^(n)`
C. `1-(1)/(e^(n))`
D. `((1)/(e-1))^(n)`

Explanation

To solve the problem of finding the fraction of the initial number of radioactive nuclei that remains undecayed after \( n \) mean lives, we can follow these steps:

### Step 1: Understand the concept of mean life
The mean life (or average lifetime) of a radioactive nucleus, denoted as \( \tau \), is the average time a nucleus exists before it decays. It is related to the decay constant \( \lambda \) by the formula:
\[ \tau = \frac{1}{\lambda} \]

### Step 2: Relate the number of undecayed nuclei to the decay constant
The number of undecayed nuclei \( N \) at time \( t \) can be expressed as:
\[ N = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of radioactive nuclei.

### Step 3: Substitute time with mean lives
Since we are interested in the number of undecayed nuclei after \( n \) mean lives, we can substitute \( t \) with \( n \tau \):
\[ N = N_0 e^{-\lambda (n \tau)} \]
Substituting \( \tau = \frac{1}{\lambda} \) into the equation gives:
\[ N = N_0 e^{-\lambda \left(n \cdot \frac{1}{\lambda}\right)} = N_0 e^{-n} \]

### Step 4: Find the fraction of undecayed nuclei
The fraction of the initial number of radioactive nuclei that remains undecayed after \( n \) mean lives is given by:
\[ \text{Fraction} = \frac{N}{N_0} = \frac{N_0 e^{-n}}{N_0} = e^{-n} \]

### Step 5: Final expression
Since we need the fraction that remains undecayed, we can express it in terms of \( e^{-n} \):
\[ \text{Fraction remaining} = \frac{1}{e^n} \]

### Conclusion
Thus, the fraction of the initial number of radioactive nuclei that remains undecayed after \( n \) mean lives is:
\[ \frac{1}{e^n} \]

### Answer
The correct option is:
**Option A: \( \frac{1}{e^n} \)**

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