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

Options

A. (a) `1//lambda` and `(In2)//lambda`
B. (b) `(In2)//lambda` and `1//lambda`
C. (c) `lambda(In2)` and `1//lambda`
D. `lambda//(In2)` and `1//lambda`

Answer

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

### Step 1: Understanding Half-Life
The half-life (\( t_{1/2} \)) of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. The formula for half-life in terms of the decay constant \( \lambda \) is given by:

\[ t_{1/2} = \frac{\ln 2}{\lambda} \]

### Step 2: Understanding Average Life
The average life (or mean life) of a radioactive substance is the average time that a nucleus of the substance will exist before it decays. The formula for average life in terms of the decay constant \( \lambda \) is:

\[ \tau = \frac{1}{\lambda} \]

### Step 3: Conclusion
From the formulas derived, we can summarize:
- The half-life of the sample is \( t_{1/2} = \frac{\ln 2}{\lambda} \)
- The average life of the sample is \( \tau = \frac{1}{\lambda} \)

Thus, the half-life and average life of the sample are respectively:

\[ \text{Half-life} = \frac{\ln 2}{\lambda}, \quad \text{Average life} = \frac{1}{\lambda} \]