Question

A radioactive sample contains 5 × 10\(^7\) kg of each of two isotopes A and B with half-lives of 5 days and 8 days respectively. The fraction of A that decays in 3 days after a period of 3 days is:

• A. 0.2
• B. 0.4
• C. 0.3
• D. 0.6

Hint:

The fraction decayed in a given time can be calculated using the exponential decay formula with the decay constant derived from the half-life.

Answer 1

The Correct Option is A

Step 1: Use the decay formula.
The decay of a radioactive substance follows the exponential decay formula: \[ N(t) = N_0 e^{-\lambda t} \] where \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( t \) is time.

Step 2: Calculate the decay fraction.
The fraction of A that decays in 3 days can be calculated using the half-life formula \( \lambda = \frac{\ln 2}{t_{1/2}} \). For A, with a half-life of 5 days, we find: \[ \text{Fraction decayed} = 1 - e^{-\lambda t} = 1 - e^{-\frac{3}{5}} \approx 0.2 \]