Following statements related to radioactivity are given below :
(A) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
(B) The number of un-decayed nuclei in the radioactive sample decays exponentially with time.
(C) Slope of the graph of logₑ (no. of undecayed nuclei) Vs. time represents the reciprocal of mean life time (τ).
(D) Product of decay constant (λ) and half-life time (T 1/2 ) is not constant.
Choose the most appropriate answer from the options given below :

Solution

Physics sir · 3 steps

Write N = N₀ e^(-λ t) first; every statement here can be checked straight from that one law.

1. The decay law. Each nucleus decays at random, and the chance of decay per second is a fixed λ. That gives an exponential fall in the number of undecayed nuclei.

   N = N₀ e^(-λ t) ⇒ ln N = ln N₀ - λ t

2. Two useful results. The ln N versus t graph is a straight line with slope -λ. Mean life is τ = 1/λ, and half-life is T_(1/2) = (ln 2)/λ.

   Statement	Check	Verdict
   (A)	Random and spontaneous, yes. But λ does not depend on temperature, pressure or chemical bonding	False
   (B)	N = N₀ e^(-λ t) is an exponential decay	True
   (C)	Slope is -λ = -1/τ, so its magnitude is the reciprocal of mean life	True
   (D)	λ T_(1/2) = ln 2 ≈ 0.693, the same for every nuclide	False

3. So. Only (B) and (C) are correct, which is option (c). The minus sign in the slope only shows that ln N falls with time. Its size is still 1/τ, so (C) is the most appropriate choice.

The idea to keep. Radioactive decay is a nuclear process, so heat, pressure or chemistry cannot change it. From N = N₀ e^(-λ t) remember: τ = 1/λ and λ T_(1/2) = ln 2 always.

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