The decay constant of a radioactive sample is `lambda`. The half-life and the average-life of the sample are respectively

Options

A. `1//(lambda) and 1n2//(lambda)`
B. `(1n 2//(lambda)) and 1//(lambda)`
C. `lambda(1n 2) and 1//(lambda)`
D. `lambda//(1n 2) and 1//lambda)`

Answer

To solve the question regarding the half-life and average life of a radioactive sample with a decay constant \( \lambda \), we will follow these steps:

### Step 1: Understand the Definitions
- **Decay Constant (\( \lambda \))**: This is a probability rate at which a radioactive sample decays.
- **Half-Life (\( T_{1/2} \))**: The time required for half of the radioactive sample to decay.
- **Average Life (\( \tau \))**: The average time that a particle exists before decaying.

### Step 2: Formulas for Half-Life and Average Life
1. The formula for half-life \( T_{1/2} \) is given by:
\[ T_{1/2} = \frac{0.693}{\lambda} \]
Here, \( 0.693 \) is the natural logarithm of 2 (\( \ln 2 \)).

2. The formula for average life \( \tau \) is given by:
\[ \tau = \frac{1}{\lambda} \]

### Step 3: Substitute the Formulas
From the above formulas:
- Half-life \( T_{1/2} = \frac{\ln 2}{\lambda} \)
- Average life \( \tau = \frac{1}{\lambda} \)

### Step 4: Identify the Correct Option
Now, we need to compare the derived values with the provided options:
1. Option 1: \( \frac{1}{\lambda}, \frac{\ln 2}{\lambda} \)
2. Option 2: \( \frac{\ln 2}{\lambda}, \frac{1}{\lambda} \)
3. Option 3: \( \lambda \ln 2, \frac{1}{\lambda} \)
4. Option 4: \( \frac{\lambda}{\ln 2}, \frac{1}{\lambda} \)

From our calculations:
- Half-life \( T_{1/2} = \frac{\ln 2}{\lambda} \)
- Average life \( \tau = \frac{1}{\lambda} \)

This matches with **Option 2**: \( \frac{\ln 2}{\lambda}, \frac{1}{\lambda} \).

### Conclusion
Thus, the half-life and average life of the radioactive sample are respectively:
- Half-life: \( \frac{\ln 2}{\lambda} \)
- Average life: \( \frac{1}{\lambda} \)

### Final Answer
**Option 2** is correct.

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