A sample of a radioactive nucleus A disintegrates to another radioactive nucleus B, which in turn disintegrates to some other stable nucleus C. Plot of a graph showing the variation of number of atoms of nucleus B versus time is :
(Assume that at t = 0, there are no B atoms in the sample)

Solution

Physics sir · 4 steps

B is filled by A and drained by its own decay, so it starts at zero, rises, peaks smoothly and dies out.

1. Gain and loss. Each decay of A creates one B, so B gains atoms at the rate λ_A N_A. B also decays on its own, so it loses atoms at the rate λ_B N_B.

   dN_B/dt = λ_A N_A - λ_B N_B

2. Start and finish. At t = 0 there is no B, so N_B = 0 and the curve starts at the origin and goes up. After a long time A is used up, so B stops being made and keeps decaying, which means N_B → 0.
3. The maximum. In between, N_B is largest when gain equals loss, λ_A N_A = λ_B N_B. There dN_B/dt = 0, so the top of the curve is a smooth, rounded hump, not a sharp corner.
4. So. The graph must start at zero, rise, reach a rounded maximum and fall towards zero. That is option (d).

The idea to keep. For a parent-daughter chain, write daughter rate = rate in from parent minus rate out by own decay. Starting from zero, the daughter count always rises, peaks smoothly where the two rates match, then dies away.

Answer from the NTA final answer key. Solution by PrepWiser.