A radioactive nuclei with decay constant 0.5 / s is being produced at a constant rate of 100 nuclei/s. If at t=0 there were no nuclei, the time when there are 50 nuclei is:

Step-by-step solution

Let N be the number of nuclei at any time t then, aligned & d N d t =100- N & or ₀^N d N (100- N) = ₀^t d t &- 1 [ (100- N)]₀^N=t & (100- N)- 100=- t & 100- N 100 =- t & 100- N 100 =e^ - t &1- N 100 =e^ - t aligned aligned &N= 100 (1-e^ - t ) & As, N=50 and =0.5 / sec & 50= 100 0.5 (1-e^ -0.5 t ) aligned Solving we get, t=2 ( 4 3 ) sec