Question

For each of the following, indicate if the number of half-lives elapsed is:

1. one half-life

2. two half-lives

3. three half-lives

a. a sample of $Ce-141$with a half-life of $32.5$days after $32.5$days

b. a sample of localid="1652506143385" $F-18$with a half-life of localid="1652506148132" $110\min$after localid="1652506151432" $330\min$

c. a sample of localid="1652506155581" $Au-198$with a half-life of localid="1652506158916" $2.7$days after localid="1652506161934" $5.4$days

Short answer

(part a) The subpart $\left(a\right)$a sample of $Ce-141$with a half-life of $32.5$days after $32.5$days, matches with subpart $\left(1\right)$.

(part b) The subpart $\left(b\right)$a sample of $F-18$with a half-life of $110\min$after $330\min$, matches with subpart $\left(3\right)$.

(part c) Therefore, subpart $\left(c\right)$a sample of $Au-198$with a half-life of $2.7$days after $5.4$days matches with subpart $\left(2\right)$.

Step-by-step solution

Given information (part a)

The subpart $\left(a\right)$a $Ce-141$sample with a half-life of$32.5$days after $32.5$days fits subpart $\left(1\right)$.

Explanation (part a)

(part a) A $Ce-141$sample after $32.5$days having a half-life of $32.5$days.

Half-life: A radioisotope's half-life is the time it takes for half of a sample to decay.

Calculate the half-life elapsed for $Ce-141$after $32.5$days.

$Ce-141\stackrel{1half-life}{\to }32.5days$

After $32.5$days, $Ce-141$has one half-life.

As a result, subpart $\left(a\right)$a $Ce-141$sample with a half-life of $32.5$days after $32.5$days fits subpart $\left(1\right)$.

Step 3:Given information (part b)

The subpart $\left(b\right)$of a$F-18$sample with a half-life of $110min$after $330min$matches subpart$\left(3\right)$.

Explanation (part b)

(part b) sample with a half-life of $110$minutes.

After $330$minutes, calculate the half-life of $F-18$.

$\mathrm{F}-18\stackrel{\text{1half-life}}{⟶}110\min \mathrm{F}-18\stackrel{\text{2half-lives}}{⟶}220\min \mathrm{F}-18\stackrel{\text{3half-lives}}{⟶}330\min \mathrm{F}-18$

After $330min$, the $F-18$has three half-lives.

As a result, subpart $\left(b\right)$of a $F-18$sample with a half-life of $110min$after $330min$matches subpart $\left(3\right)$.

Given information (part c)

The subpart $\left(c\right)$of an Au-198 sample with a half-life of $2.7$days after $5.4$days matches subpart $\left(2\right)$.

Explanation (part c)

(part c) A sample of has a $2.7$day half-life.

Calculate the $Au-198$half-lives after $5.4$days.

$Au-198\stackrel{\text{1half-life}}{⟶}2.7daysAu-198\stackrel{\text{2half-lives}}{⟶}5.4days$

After $5.4$days, $Au-198$has two half-lives.

Therefore, subpart $\left(c\right)$a sample of $Au-198$with a half-life of $2.7$days after $5.4$days matches with subpart $\left(2\right)$.