Question

What is the average life of a radioactive element whose half-life is 30 days ?

• A. 03
• B. 15.21
• C. 20.79
• D. 29

Hint:

To find the average life of any radioactive element, simply multiply its half-life by the constant factor 1.443 (since $\tau = 1.443 \times T_{1/2}$). For 30 days, $30 \times 1.443 = 29\text{ days}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Radioactive decay is a first-order kinetic process.
- Half-life ($T_{1/2}$) is the time required for half of the radioactive nuclei in a sample to decay.
- Average life (or mean life, $\tau$) is the average lifetime of all the radioactive nuclei in the sample before they decay.

Step 2: Key Formula or Approach:
The mathematical relationship between the decay constant ($\lambda$), half-life ($T_{1/2}$), and average life ($\tau$) is:
\[ T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda} \]
The average life ($\tau$) is the reciprocal of the decay constant:
\[ \tau = \frac{1}{\lambda} \]
Therefore:
\[ \tau = \frac{T_{1/2}}{\ln 2} \approx \frac{T_{1/2}}{0.693} \approx 1.443 \times T_{1/2} \]

Step 3: Detailed Explanation:
Let us perform the calculation using the values provided in the question:
- Given half-life ($T_{1/2}$) = $30\text{ days}$
Substitute this value into the average life formula:
\[ \tau = 1.443 \times T_{1/2} \]
\[ \tau = 1.443 \times 30 \]
\[ \tau = 29\text{ days} \]
Therefore, the average life of the radioactive element is $29\text{ days}$, matching Option (D).

Step 4: Final Answer:
The average life of the radioactive element is 29 days.