Question

Fermium-253 has a half life of 3 days. What percent of the sample is still Fermium-253 after 15 days?

Answer 1

Answer:

$3.125\,\%$.

Explanation:

After every half-life, the quantity of the sample will become $1/2$ of the quantity at the beginning of the half-life. If the half-life for Fermium-253 is 3 days, it can be deduced that:

$\begin{array}{|c|c|c|}\cline{1-3}\text{Number of days} & \text{Number of half-lives} & \text{\%\text{ of Sample that's still Fermium-253}} \\ \cline{1-3} 0 & 0 & 100\% \\ \cline{1-3}3 & 3/3 = 1 & (1/2) \times 100\% = 50\%\\ \cline{1-3} 6 & 6/3 = 2 & (1/2) \times ((1/2) \times 100\%)= 25\% \\ \cline{1-3} \vdots & \vdots & \vdots \\\cline{1-3}\end{array}$.

Let $n \ge 0$. In general, after $n$ half-lives (for the fermium-253 in this question, that would be $3\, n$ days,) the percentage of the sample that's still the initial substance would be:

$\displaystyle 100\% \times \left(\frac{1}{2}\right)^{n}$.

There are $15 / 3 = 5$ half lives in 15 days. (In other words, $n = 5$.) Therefore, the percentage of this sample that's still fermium-253 would be:

$\begin{aligned}&100\% \times \left(\frac{1}{2}\right)^5 \\ &= 100\%\times \frac{1}{32} = 100\% \times 0.03125 \\&= 3.125\%\end{aligned}$.