Q. 5.29 - 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 $Pd-103$ with a half-life of $17 days$ after $34 days$

b. a sample of $C-11$ with a half-life of $20 mins$ after $20 min$

c. a sample of $At-211$ with a half-life of $7 \mathrm{h}$ after $21 h$

Step-by-Step Solution

Answer

(part a) The subpart $\left(a\right)$ a sample of $Pd-103$ with a half-life of $17$days after $34$ days matches with subpart $\left(2\right)$.

(part b) The subpart $\left(b\right)$ a sample of $C-11$ with a half-life of $20$min after $20$min matches with subpart $\left(1\right)$.

(part c) The subpart $\left(c\right)$ a sample of $At-211$ with a half-life of $7$h after $21$h, matches with subpart $\left(3\right)$.

Step 1: Given information (part a)

The subpart $\left(a\right)$ a sample of $Pd-103$ after $34$ days with a half-life of $17$ days fits subpart $\left(2\right)$.

Step 2: Explanation (part a)

(part a) $Pd-103$has a half-life of $17$day in a sample.

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

After $34$ days, calculate the half-lives of $Pd - 103$.

$Pd - 103 \stackrel{1halflife}{\to } 17 days Pd - 103 \stackrel{2halflives}{\to } 34 days$

$Pd - 103$ has two half-lives after $34$ days.

As a result, subpart $\left(a\right)$ a $Pd-103$ sample with a half-life of $17$ days after $34$days fits subpart $\left(2\right)$.

Step 3: Given information (part b)

The subpart $\left(b\right)$ a sample of $C-11$ with a half-life of $20$ minutes after $20$ minutes corresponds to subpart $\left(1\right)$.

Step 4: Explanation (part b)

(part b) $C-11$ having a $20$-minute half-life.

Calculate the elapsed half-life for $C-11$ after $20 min$.

$\mathrm{C}-11\stackrel{1\text{ half-life }}{⟶}20 \min$

After $20 min$, $C-11$ has one half-life.

As a result, subpart $\left(b\right)$ a sample of $C-11$ with a half-life of $20$ minutes after $20$ minutes corresponds to subpart .

Step 5: Given information (part c)

The subpart $\left(c\right)$, a sample of $At-211$ with a half-life of $7 h$ after $21 h$, corresponds to subpart $\left(3\right)$.

Step 6: Explanation (part c)

(part c) $At-211$ sample with a $7h$ half-life.

After using $21 h$, calculate the half-lives for $At-211$.

$At - 211 \stackrel{1\text{ half-life }}{⟶} 7h At - 211 \stackrel{\text{2 half-lives }}{⟶} 14h At - 211 \stackrel{\text{3 half-lives }}{⟶} 21h At - 211$

After $21 h$, $At-211$ has three half-lives.

As a result, subpart $\left(b\right)$ a sample of $C-11$ with a half-life of $20$ minutes after $20$ minutes corresponds to subpart $\left(1\right)$.