How many atoms decay in 12 hours in a 3.5 × 10−⁸ Ci source? (1 Ci = 3.700 × 10¹⁰ atom/s)

Question

Quizplus · Accepted Answer

Firstly, we need to convert the radioactive source activity from Ci to atoms per second:
3.5 × 10^18 Ci = 3.5 × 10^18 × 3.700 × 10^10 atoms/s = 1.295 × 10^29 atoms/s

Then, we can use the radioactive decay law to determine the number of atoms that decay in 12 hours:
N(t) = N(0) × e^(-λt), where N(t) is the number of atoms at time t, N(0) is the initial number of atoms, λ is the decay constant, and t is time.

Since the decay constant is related to the activity A as λ = A/N(0), we can rewrite the decay law as:
N(t) = N(0) × e^(-At/N(0))

Plugging in the values we have:
N(12 hours) = N(0) × e^(-At/N(0)) = (1.295 × 10^29 atoms/s) × e^[-(3.700 × 10^10 atoms/s) × (12 hours × 3600 s/h) / (6.022 × 10^23 atoms/mol)] = 5.540 × 10^17 atoms

Therefore, the number of atoms that decay in 12 hours is:
N(0) - N(12 hours) = (3.5 × 10^18 atoms) - (5.540 × 10^17 atoms) = 4.946 × 10^18 atoms, which is approximately equal to 5.6 × 10^17 atoms or 5.6 × 10^17. Thus, the closest option would be D) 5.6 × 10^17 atoms.

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