Step 1: Understanding secular equilibrium.
In the case of radioactive decay, secular equilibrium occurs when the parent nucleus (X) decays to the daughter nucleus (Y) at a slower rate than the daughter nucleus decays. This means that the activity of the parent nucleus equals the activity of the daughter nucleus, i.e., the number of decays per unit time for both nuclei becomes constant. For this to occur, the half-life of the parent nucleus must be much smaller than that of the daughter nucleus ( ).
Step 2: Analyzing the options.
(A) : Incorrect. If the half-lives are approximately equal, the equilibrium would not be secular because the daughter nucleus would not build up to a steady state.
(B) : Incorrect. While this condition is closer, the parent half-life must be much smaller than the daughter half-life for secular equilibrium to hold.
(C) : Correct. For secular equilibrium, the parent nucleus must decay much faster than the daughter nucleus. This ensures that the daughter nuclei accumulate at a constant rate.
(D) : Incorrect. If the parent nucleus has a much longer half-life than the daughter, the daughter would not accumulate at a steady rate, thus failing to reach secular equilibrium.
Step 3: Conclusion.
The correct answer is (C) because for secular equilibrium, the half-life of the parent must be much smaller than that of the daughter nucleus.