Sol. To calculate rate of release of energy at time t and total energy released upto time t, rate of decay at that instant and total number of decays upto that instant must be known.

Since, nuclei produced are radioactive, therefore, their decay starts as soon as their production is started. Let at some instant number of nuclei in the radionuclide be N. Then rate of its decay = λ N where λ is decay constant which is equal to .

Since, rate of production is q nuclei per second, therefore, at instant t, net rate of increase of nuclei

= q – λ N = q – or =

∴ = … (1)

Integrating above equation with limits at t = 0, and at t, N = ?

=

∴ N =

Hence, rate of decay, A = λ N = q

Since, energy E 0 releases during each decay, therefore, rate of release of energy at time t= AE 0 = qE 0

Total number of nuclei produced upto time t = q.t

But the number of nuclei remaining undecayed at that instant is N. Therefore, total number of nuclei which decayed upto time t = (qt –N)

Hence, total energy released upto this time = (qt – N) E 0 = qtE 0 –