If a sample initially contains 1.0 × 10^{12} atoms of the cadmium isotope 109Cd, which has a half-life of 462 days, approximately how many atoms of 109Cd will remain after 924 days?

• A. 2.0 × 10^{11}
• B. 2.5 × 10^{11}
• C. 2.5 × 10^{12}
• D. 2.5 × 10^{11}

Answer: (B)

Verified step by step guidance

1. Identify the given information: the initial number of atoms \(N_0 = 1.0 \times 10^{12}\), the half-life \(t_{1/2} = 462\) days, and the elapsed time \(t = 924\) days.
2. Determine the number of half-lives that have passed by dividing the total time by the half-life: \(n = \frac{t}{t_{1/2}} = \frac{924}{462}\).
3. Use the radioactive decay formula to find the remaining number of atoms: \(N = N_0 \times \left(\frac{1}{2}\right)^n\).
4. Substitute the values of \(N_0\) and \(n\) into the formula to express \(N\) in terms of known quantities.
5. Calculate the numerical value of \(N\) to find the approximate number of atoms remaining after 924 days.