(b) Consider the multi-modal decay where nuclide 1 decays to nuclide 2 with a decay constant \(\lambda_2\) and nuclide 1 decays also to nuclide 3 with a decay constant \(\lambda_3\). Nuclides 2 and 3 are stable.
(b i) Write down differential equations for the numbers of three nuclides \(N_1(t)\), \(N_2(t)\) and \(N_3(t)\) (where \(N_1(t)\) is the number of nuclide 1 at time t etc.).
[2 marks]
(b ii) Let the initial condition be \(N_j(0) = N_{j,0}\) for \(j = 1, 2, 3\). Solve the differential equations for \(N_j(t)\) for \(j = 1, 2, 3\).
[6 marks]
(b iii) Let us assume that there is an equal number of all nuclides at \(t = 0\), i.e. \(N_{1,0} = N_{2,0} = N_{3,0}\). Calculate the time \(\bar{t}\) until the number of nuclide 2 has doubled, i.e. \(N_2(\bar{t}) = 2N_{2,0}\).
[3 marks]
