Question

Radium decomposes at a rate proportional to the amount present. If half the original amount disappears in 1600 years, then the percentage loss in 100 years is

• A. 3.24 %
• B. 5.24 %
• C. 2.24 %
• D. 4.24 %

Hint:

For decay problems, remember to use the formula \( A(t) = A_0 e^{-kt} \) and solve for the decay constant first before calculating the percentage loss.

Answer 1

The Correct Option is D

Step 1: Use the exponential decay formula.
The rate of decomposition is governed by the equation \( A(t) = A_0 e^{-kt} \), where \( A(t) \) is the amount of radium remaining after time \( t \), \( A_0 \) is the initial amount, and \( k \) is the decay constant. We are given that half the original amount disappears in 1600 years, so we can find \( k \).

Step 2: Finding \( k \).
Since \( A(1600) = \frac{A_0}{2} \), we have: \[ \frac{A_0}{2} = A_0 e^{-k \times 1600} \] Solving for \( k \), we get: \[ k = \frac{\ln 2}{1600} \approx 0.000432 \]
Step 3: Calculating the percentage loss after 100 years.
Now, we can find the percentage loss after 100 years using the formula: \[ A(100) = A_0 e^{-k \times 100} \] Substitute \( k \approx 0.000432 \), and calculate the percentage loss: \[ \text{Percentage loss} = \left( 1 - e^{-0.000432 \times 100} \right) \times 100 \approx 4.24 % \]
Step 4: Conclusion.
The correct answer is 4.24 %.