Radioactive decay: Fermium-253 has a half-life of 3.00 d. A sample of fermium contains 7.37 × nuclei. How long will it take for there to be only 3.36 × fermium nuclei in this sample?

Question

Radioactive decay: Fermium-253 has a half-life of 3.00 d. A sample of fermium contains 7.37 ×  nuclei. How long will it take for there to be only 3.36 ×  fermium nuclei in this sample?

Quizplus · Accepted Answer

We can use the radioactive decay formula:
N(t) = N0*(1/2)^(t/T)
where:
N(t) = number of nuclei at time t
N0 = initial number of nuclei
t = time elapsed
T = half-life of the sample

We are given:
N0 = 7.37 × 10^11 nuclei
N(t) = 3.36 × 10^11 nuclei
T = 3.00 d

Substituting these values into the formula, we get:
N(t)/N0 = (1/2)^(t/T)
(3.36 × 10^11)/(7.37 × 10^11) = (1/2)^(t/3)
Taking the natural logarithm of both sides, we get:
ln((3.36 × 10^11)/(7.37 × 10^11)) = (t/3) * ln(1/2)
Solving for t, we get:
t = (-3/ln(1/2)) * ln((3.36 × 10^11)/(7.37 × 10^11))
t ≈ 13.4 d

Therefore, the answer is (C) 13.4 d.

Radioactive Decay: Fermium-253 Has a Half-Life of 3