At a certain time, a specific sample of a radioactive material decays at the rate of 500 per second. After 50 minutes , the count rate falls to 200 per second. Determine the half-life of the sample.

Step-by-step solution

The radioactive decay law is given by A = A₀ e^ - t . Given A₀ = 500 s ⁻¹ and A = 200 s ⁻¹ at t = 50 min . Substituting the values: 200 = 500 e^ -50 e^ 50 = 500 200 = 2.5 Taking the natural logarithm on both sides: 50 = 2.5 = 2.5 50 The half-life T_ 1/2 is given by: T_ 1/2 = 2 = 50 2 2.5 Using ₁₀ 2 0.301 and ₁₀ 2.5 = ₁₀ ( 10 4 ) = 1 - 2 ₁₀ 2 0.398 : T_ 1/2 = 50 0.301 0.398 37.8 min Rounding to the nearest integer, we get 38 min .