# Question #39370

Jun 20, 2017

$\frac{1}{256}$

#### Explanation:

You know that the nuclear half-life of a radioactive nuclide, ${t}_{\text{1/2}}$, tells you the time that must pass in order for half of an initial sample of said to nuclide to undergo radioactive decay.

This means that with every passing half-life, the mass of the sample gets reduced by half.

If you take ${A}_{0}$ to be the initial mass of the sample, you can say that you will be left with

${A}_{t} = {A}_{0} \cdot {\left(\frac{1}{2}\right)}^{\textcolor{red}{n}}$

Here

• ${A}_{t}$ is the mass of the sample that remains undecayed after a period of time $t$
• $\textcolor{red}{n}$ represents the number of half-lives that pass in a given time period $t$

In your case, you know that

${t}_{\text{1.2" = "3 hours}}$

You also know that the total time that passes is equal to

$\text{1 day = 24 hours}$

You can thus say that you have

$\textcolor{red}{n} = \left(24 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{hours"))))/(3color(red)(cancel(color(black)("hours}}}}\right) = \textcolor{red}{8} \to$this means that eight half-lives pass in a $\text{24-hour}$ period.

Therefore, the mass of the sample that remains undecayed is equal to

${A}_{t} = {A}_{0} \cdot {\left(\frac{1}{2}\right)}^{\textcolor{red}{8}}$

${A}_{t} = {A}_{0} \cdot \frac{1}{256}$

To find the fraction that remains undecayed, simply divide the amount that remains undecayed by the initial amount

${A}_{t} / {A}_{0} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{{A}_{0}}}} \cdot \frac{1}{256}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{{A}_{0}}}}} = \frac{1}{256}$