Question

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

Answer 1

After 15 days, approximately 3.125% of the Fermium-253 sample will still remain.

Step-by-step explanation:

To determine the percent of the Fermium-253 sample that is still present after 15 days, we need to consider the number of half-lives that have occurred. Since the half-life of Fermium-253 is 3 days, we divide the elapsed time (15 days) by the half-life:

Number of half-lives = Elapsed time / Half-life = 15 days / 3 days = 5 half-lives

Each half-life reduces the amount of Fermium-253 by half. Therefore, after 5 half-lives, the remaining percentage of the original sample is (1/2)^(number of half-lives) x 100% = (1/2)^5 x 100% = 3.125%

Answer 2

`3.125\,\%`.

Step-by-step 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\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) * 100\% = 50\%\\ \cline{1-3} 6 & 6/3 = 2 & (1/2) * ((1/2) * 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\% * \left((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\% * \left((1)/(2)\right)^5 \\ &= 100\%* (1)/(32) = 100\% * 0.03125 \\&= 3.125\%\end{aligned}`.