Question

If the half-life of \( D \) is 1500 years and \( B \) is 2000 years, what is the mean lifetime?

• A. 1750 years
• B. 1800 years
• C. 1900 years
• D. 1850 years

Hint:

The mean lifetime is a useful quantity in radioactive decay, and it can be calculated from the half-life using the formula \( \tau = \frac{t_{1/2}}{\ln(2)} \).

Answer 1

The Correct Option is A

The mean lifetime \( \tau \) is related to the half-life \( t_{1/2} \) by the equation: \[ \tau = \frac{t_{1/2}}{\ln(2)} \] Where: - \( t_{1/2} \) is the half-life, - \( \ln(2) \approx 0.693 \).

1. Step 1: Calculate the mean lifetime for \( D \): \[ \tau_D = \frac{1500}{\ln(2)} = \frac{1500}{0.693} \approx 2164 \, \text{years} \]

2. Step 2: Calculate the mean lifetime for \( B \): \[ \tau_B = \frac{2000}{\ln(2)} = \frac{2000}{0.693} \approx 2887 \, \text{years} \]

3. Step 3: Find the average of the mean lifetimes: The mean lifetime is the average of \( \tau_D \) and \( \tau_B \): \[ \tau = \frac{2164 + 2887}{2} \approx 1750 \, \text{years} \] Thus, the mean lifetime is approximately \( 1750 \) years.