The half-life of ¹⁹⁸Au is 3 days. If atomic weight of ¹⁹⁸Au is 198 g/mol then the activity of 2 mg of ¹⁹⁸Au is [in disintegration/second] :

Solution

Physics sir · 3 steps

Activity is just λ N: find how many nuclei are in 2 mg, then multiply by λ in per second.

1. Count the nuclei. 2 mg is (2 × 10⁻³)/198 mol, so N = (2 × 10⁻³)/198 × 6.022 × 10²³ ≈ 6.08 × 10¹⁸ nuclei.
2. Decay constant. The answer must be in disintegrations per second, so convert the half-life first: T_(1/2) = 3 × 86400 = 2.592 × 10⁵ s. Then λ = 0.693/(2.592 × 10⁵) ≈ 2.67 × 10⁻⁶ s⁻¹.

   A = λ N = (2.67 × 10⁻⁶)(6.08 × 10¹⁸) ≈ 1.62 × 10¹³ s⁻¹

3. So. The activity is about 16.2 × 10¹² disintegrations per second. The small difference from the listed value comes only from rounding, so the answer is (d).

The idea to keep. For activity, always go grams to moles to nuclei, then use A = λ N with λ = 0.693/T_(1/2). Put the half-life in seconds before you divide.

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