Question

The half-life periods of two radioactive materials A and B are 1500 years and 1200 years respectively. If their mean life periods are \( \tau_A \) and \( \tau_B \) respectively, then the value of the ratio \( \frac{\tau_A}{\tau_B} \) is

• A. \( \frac{5}{4} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{5}{7} \)
• E. \( \frac{2}{5} \)

Hint:

Mean life is directly proportional to half-life.

Answer 1

The Correct Option is A

Step 1: Understanding Half-life and Mean Life:
Half-life (\(T_{1/2}\)): The time taken for the number of radioactive nuclei in a sample to reduce to half its initial value.
Mean life (\(\tau\)): The average lifetime of a radioactive nucleus before it decays. It is the reciprocal of the decay constant (\(\lambda\)).
Step 2: Key Formula or Approach:
The relationship between mean life (\(\tau\)) and half-life (\(T_{1/2}\)) for a radioactive substance is given by:
\[ \tau = \frac{T_{1/2}}{\ln(2)} \] This equation shows that the mean life is directly proportional to the half-life.
\[ \tau \propto T_{1/2} \] Step 3: Detailed Calculation:
We are asked to find the ratio \(\frac{\tau_A}{\tau_B}\).
Since \(\tau\) is directly proportional to \(T_{1/2}\), the ratio of the mean lives will be the same as the ratio of their half-lives.
\[ \frac{\tau_A}{\tau_B} = \frac{T_{1/2, A}}{T_{1/2, B}} \] We are given the half-lives:
\(T_{1/2, A} = 1500\) years
\(T_{1/2, B} = 1200\) years
Substitute these values into the ratio:
\[ \frac{\tau_A}{\tau_B} = \frac{1500}{1200} \] Simplify the fraction:
\[ \frac{\tau_A}{\tau_B} = \frac{15}{12} = \frac{5 \times 3}{4 \times 3} = \frac{5}{4} \] Step 4: Final Answer:
The value of the ratio \(\frac{\tau_A}{\tau_B}\) is 5/4.