Radioactive Decay

Step 1: Determine the mass of in the human body.
The total mass of in the body is given by the fraction of its mass in the body. For a body of mass 70 kg:
Step 2: Calculate the number of moles of .
The molar mass of is approximately 39.1 g/mol. Converting kg to grams: Now, calculate the number of moles:
Step 3: Use the activity formula.
The activity is related to the number of moles by the formula: where is the number of atoms, s is the half-life, and is the decay constant. Using where :
Step 4: Conclusion.
Thus, the activity due to in the human body is approximately 0.33 kBq.

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.