Question

Question (1) Consider the following radioactive decay chain of fission products, where nuclide D is stable. AO, BO, and Do are the constant fission rates of production (nuclides/s), and λΑ, AB and AC are the l decay constants. None of the fission products undergo any nuclear reactions other than the radioactive decay shown. (a) Write the set of differential equations describing the radioactive decay chain in terms of the above parameters, and the number of nuclei NA, NB, NC and ND at time t. (Do not solve the differential equations.) (b) Assume that at some time t, the fission process was terminatéd; that is, the constant fission rates of production became zero. At that time, the number densities of the fission products A, B, C, and D were NTA, NTB, NTC, and NTD respectively. At a very long time (assume infinity) in the future, in terms of NTA, NTB, NTC, and NTD, what would the final number densities of the four nuclides be?

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Answer 1

To solve this problem the details calculations are given below.
Consider a radioactive nuclide P, to decay into another radioactive nuclide Q, and so on to a radioactive chain of any number of membranes the so-called general case: P(1) rightarrow^lambda_1 Q(2) rightarrow^lambda_2 R(3) rightarrow^lambda_3 S(4) rightarrow^lambda_4 where P(1) is the parent nuclei, and Q(2), R(3), S(4) are the successive products of disintegration \r\n if lambda_1, lambda_2, lambda_3 ... be decay constants and N_1, N_2, N_3, N_4 ... be the number of atoms at any instant t of nuclides 1, 2, 3 .... etc we have the following set of equations: dN_1/dt = -lambda_1 N_1 dN_2/dt = lambda_1 N_1 - lambda_2 N_2 dN_3/dt = lambda_2 N_2 - lambda_3 N_3 dN_4/dt = lambda_3 N_3 - lambda_4 N_4 . . . . dN_n/dt = lambda_n-1 N_n-1 - lambda_n N_n These are the differential equation governed by the decreasing process. \r\n (b) Now the case when the fission process was terminated: Let us first consider the simple case of three successive disintegrations, the last (fourth) nuclide being stable i.e., the disintegration (fission) process was terminated. \r\n