In an experiment, a radioactive isotope is generated at a constant rate dN/dt = R . The isotope has a half-life t_ 1/2 . Assuming that the generation of the radioactive isotope begins at t = 0 , determine the number of active nuclei at time t .

Step-by-step solution

The rate of change of the number of active nuclei is the difference between the rate of generation and the rate of decay. dN dt = R - N where = 2 t_ 1/2 is the decay constant. Rearranging the variables: dN R - N = dt Integrating both sides from t = 0 to t , with N(0) = 0 : ₀^ N dN R - N = ₀^ t dt - 1 [ (R - N)]₀^ N = t ( R - N R ) = - t R - N R = e^ - t R - N = R e^ - t N = R (1 - e^ - t )