Fermium-253 has a half-life of 3.00 days. A sample of fermium has 7.37 × 10⁷ nuclei. How long will it take for there to be only 3.36 × 10⁶ fermium nuclei in this sample? A) 2.75 days B) 9.80 days C) 13.4 days D) 15.7 days E) 58.6 days

Question

Quizplus · Accepted Answer

We can find the decay constant ($\lambda$) using the half-life:

$$t_{1/2} = \frac{\ln 2}{\lambda} ightarrow \lambda = \frac{\ln 2}{t_{1/2}}$$

Plugging in $t_{1/2}=3.00$ days gives $\lambda = 0.231$ day$^{-1}$.

The number of nuclei at any time $t$ can be described by:

$$N(t) = N_0e^{-\lambda t}$$

where $N_0$ is the initial number of nuclei.

Letting $N(t) = 3.36	imes 10^{16}$ and $N_0 = 7.37	imes 10^{16}$, we can solve for $t$:

$$3.36	imes 10^{16} = 7.37	imes 10^{16}e^{-0.231t} ightarrow t = \frac{\ln (3.36/7.37)}{-0.231} \approx 13.4 	ext{ days}$$

Therefore, the best choice is C.

Fermium-253 Has a Half-Life of 3