 Solution

The first balance treats the fertile parent as being removed by neutron absorption. The second has a production contribution associated with capture in the parent and a removal contribution from absorption in the fissile population. Capture and total absorption are not interchangeable: in general absorption includes capture and fission, and a capture channel is the one that feeds the specified transmutation chain.

The lumped description assumes spatially averaged populations and flux, fixed effective reaction coefficients, no transport of material into or out of the system, and a shortened chain in which intermediate decay delays are neglected. It omits explicit precursor accumulation, competing reaction chains and the changing neutron spectrum. It also prescribes flux rather than solving a coupled power-and-composition feedback problem. Neglecting intermediate lifetimes requires them to be short relative to the variation being modelled; that assumption can fail during a rapid change in irradiation.