A 9.52 × 10^-5 mol sample of rubidium-86 emits 8.87 × 10¹⁶ $\alpha$ particles in one hour. What is the half-life of rubidium-86? A) 2.23 × 10^-3 h B) 1.55 × 10^-3 h C) 448 h D) 645 h E) none of the above

Question

Quizplus · Accepted Answer

To find the half-life of rubidium-86, we can use the radioactive decay formula:
N = N0 (1/2)^(t/T)
where N is the final amount of radioactive particles, N0 is the initial amount, t is the time elapsed, and T is the half-life.

We are given N0 = 9.52 × 10^15 and N = 9.52 × 10^15 - 8.87 × 10^16 = -7.92 × 10^16 (assuming negative sign is for the decay).

We are also given t = 1 hour.

Substituting these values into the formula, we get:
-7.92 × 10^16 = 9.52 × 10^15 (1/2)^(1/T)

Dividing both sides by 9.52 × 10^15, we get:
-8.32 = (1/2)^(1/T)

Taking the natural logarithm of both sides, we get:
ln(-8.32) = ln(1/2)^(1/T)
ln(-8.32) = -ln(2)/T
T = -ln(2)/ln(-8.32)

Using a calculator, we get:
T ≈ 448 hours

Therefore, the best choice is C, with a half-life of 448 hours.

A 952 × 10-5 Mol Sample of Rubidium-86 Emits 8 α\alphaα

A 952 × 10^-5 Mol Sample of Rubidium-86 Emits 8 $\alpha$