Chapter 21, Problem 10PEB

To determine

The mass of Carbon-14 in $28,650\text{ year}$ old wood.

Answer to Problem 10PEB

Solution:

$7.5\text{ g}$

Explanation of Solution

Given data:

Mass of carbon-14 in the wood $mass=240\text{ g}$.

Half-life of carbon-14 $half-life=5730\text{ yr}$.

Age of the tree $age=28650\text{ yr}$.

Formula used:

the expression of number of half-lives iswritten as,

$numberofhalf-lives=\frac{age}{half-life}$

The expression of the mass after the half-lifeiswritten as,

$mas{s}_{i\left(half-life\right)}=\frac{mass}{2}$

Here, $i$ is the number of half-life.

Explanation:

Write the expression of number of half-lives.

$numberofhalf-lives=\frac{age}{half-life}$

Substitute $28650\text{ yr}$ for $age$ and $5730\text{ yr}$ for $half-life$.

$\begin{array}{c}numberofhalf-lives=\frac{28650\text{ yr}}{5730\text{ yr}}\\ =5\end{array}$

Since the number of half-life is 5, the mass will be divided 5 times.

The mass divided will be the mass of previous half-life divided by 2.

Write the expression of mass after the half-life.

$mas{s}_{i\left(half-life\right)}=\frac{mass}{2}$

Substitute $1$ for $i$ and $240\text{ g}$ for $mass$

$\begin{array}{c}mas{s}_{1\left(half-life\right)}=\frac{240\text{ g}}{2}\\ =120\text{ g}\end{array}$

Write the expression of mass after the half-life.

$mas{s}_{i\left(half-life\right)}=\frac{mass}{2}$

Substitute $2$ for $i$ and $120\text{ g}$ for $mass$

$\begin{array}{c}mas{s}_{2\left(half-life\right)}=\frac{120\text{ g}}{2}\\ =60\text{ g}\end{array}$

Write the expression of mass after the half-life.

$mas{s}_{i\left(half-life\right)}=\frac{mass}{2}$

Substitute $3$ for $i$ and $\text{60 g}$ for $mass$.

$\begin{array}{c}mas{s}_{1\left(half-life\right)}=\frac{60\text{ g}}{2}\\ =30\text{ g}\end{array}$

Write the expression of mass after the half-life.

$mas{s}_{i\left(half-life\right)}=\frac{mass}{2}$

Substitute $4$ for $i$ and $30\text{ g}$ for $mass$.

$\begin{array}{c}mas{s}_{1\left(half-life\right)}=\frac{30\text{ g}}{2}\\ =15\text{ g}\end{array}$

Write the expression of mass after the half-life.

$mas{s}_{i\left(half-life\right)}=\frac{mass}{2}$

Substitute $5$ for $i$ and $15\text{ g}$ for $mass$.

$\begin{array}{c}mas{s}_{1\left(half-life\right)}=\frac{\text{15 g}}{2}\\ =7.5\text{ g}\end{array}$

Conclusion:

The mass of carbon-14 is $7.5\text{ g}$ after the age of $28650\text{ yr}$.