Q. 5.30 - An Introduction to General, Organic, and Biological Chemistry

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 $F-18$ with a half-life of $110 \min$ after $330 \min$

c. a sample of $Au-198$ with a half-life of $2.7$ days after $5.4$ days

Step-by-Step Solution

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 1: 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 $\left(1\right)$.

Step 2: 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.

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

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 $110 min$ after $330 min$ matches subpart $\left(3\right)$.

Step 4: Explanation (part b)

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

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

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

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

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

Step 5: 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 $\left(2\right)$.

Step 6: 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.

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

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 $\left(2\right)$.