Question

If the initial decay rate of a radioactive sample is \( R_0 \), then the decay rate after a half-life time \( T_{1/2} \) is

• A. \( 2R_0 \)
• B. \( R_0 \)
• C. \( \sqrt{R_0} \)
• D. \( 3R_0 \)
• E. \( \frac{R_0}{2} \)

Hint:

The decay rate of a radioactive substance after one half-life is always half of its initial rate.

Answer 1

Answer: (E)

The decay rate \( R(t) \) of a radioactive sample is given by the equation: \[ R(t) = R_0 e^{-\lambda t} \] where \( R_0 \) is the initial decay rate, \( \lambda \) is the decay constant, and \( t \) is the time. After one half-life \( T_{1/2} \), the decay rate is reduced by half. 
Therefore, the decay rate after half-life is: \[ R(T_{1/2}) = \frac{R_0}{2} \] Thus, after one half-life, the decay rate of the sample is \( \frac{R_0}{2} \).