Two radioactive materials A and B have decay constants 25 λ and 16 λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be "e" after a time 1/(a λ). The value of a is _________.

Solution

Physics sir · 3 steps

Divide the two decay laws: the same N₀ cancels and only the difference in decay constants survives.

1. Write both decays. With the same starting number N₀, N_A = N₀ e^(-25λ t) and N_B = N₀ e^(-16λ t). A decays faster, so B pulls ahead with time.

   N_B/N_A = e^(-16λ t)/e^(-25λ t) = e^((25-16)λ t) = e^(9λ t)

2. Set ratio to e. e^(9λ t) = e¹, so the exponents match: 9λ t = 1, which gives t = 1/9λ.
3. So. Comparing with t = 1/aλ gives a = 9.

The idea to keep. For two samples that start equal, the ratio of what is left is e^((λ₁-λ₂)t). Only the difference of the decay constants matters.

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