Question

Consider two radioactive materials A and B. When A decays into 6.25 percent of its original amount, B decays into 25 percent of its original amount. The ratio of their decay constants \( \lambda_A \) and \( \lambda_B \) is:

• A. 1:4
• B. 4:1
• C. 1:2
• D. 2:1

Hint:

When dealing with decay problems, use the decay formula \( N = N_0 e^{-\lambda t} \) and apply logarithms to simplify the equations and find the decay constants.

Answer 1

The Correct Option is B

The decay of radioactive materials can be described by the equation: \[ N = N_0 e^{-\lambda t} \] Where: - \( N \) is the remaining amount of the substance, - \( N_0 \) is the original amount, - \( \lambda \) is the decay constant, - \( t \) is the time elapsed. For material A, it decays to 6.25 percent of its original amount. Therefore, the fraction remaining is \( 0.0625 \), and we can express this decay process as: \[ 0.0625 = e^{-\lambda_A t} \] Taking the natural logarithm on both sides: \[ \ln(0.0625) = -\lambda_A t \] For material B, it decays to 25% of its original amount. Therefore, the fraction remaining is \( 0.25 \), and we can express this decay process as: \[ 0.25 = e^{-\lambda_B t} \] Taking the natural logarithm on both sides: \[ \ln(0.25) = -\lambda_B t \] Dividing the two equations gives: \[ \frac{\ln(0.0625)}{\ln(0.25)} = \frac{\lambda_B}{\lambda_A} \] Simplifying this ratio: \[ \frac{\ln(0.0625)}{\ln(0.25)} = \frac{-\lambda_B}{-\lambda_A} = 4 \] Thus, the ratio of the decay constants is \( \lambda_A : \lambda_B = 4:1 \).