Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as :
(where λ is the decay constant)

Solution

Physics sir · 4 steps

Turn 'decayed' into 'remaining' first, then find each time from the decay law and subtract.

1. Count what remains. The decay law tracks the nuclei that are left, N = N₀ e^(-λ t). When a quarter has decayed, N = 3/4N₀ remains. When half has decayed, N = 1/2N₀ remains.

   t = 1/λ ln N₀/N

2. Find both times. For a quarter decayed, t₁ = 1/λ ln 4/3. For half decayed, t₂ = 1/λ ln 2, which is the half-life.
3. Subtract. t₂ - t₁ = 1/λ(ln 2 - ln 4/3) = 1/λ ln(2 × 3/4) = 1/λ ln 3/2.
4. So. The time gap is (ln (3/2))/λ, which is option (d).

The idea to keep. The decay law counts the nuclei that are left, not the ones that have decayed. Convert first. The time between two remaining fractions is 1/λ ln of their ratio.

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