In an experiment, a radioactive isotope is produced at a constant rate dN/dt = R . The isotope has a half-life t_ 1/2 . After a time t t_ 1/2 , the number of active nuclei becomes constant. Determine the value of this constant.

Step-by-step solution

The rate of change of the number of active nuclei is given by the difference between the rate of production and the rate of decay. dN dt = R - N where R is the constant production rate and is the decay constant. After a long time ( t t_ 1/2 ), the number of active nuclei becomes constant, which means the system reaches a steady state. At steady state, dN dt = 0 R - N = 0 N = R The decay constant is related to the half-life t_ 1/2 by the equation: = 2 t_ 1/2 Substituting the value of into the expression for N : N =