# Question 73929

Sep 9, 2016

$\text{1.55 cm}$

#### Explanation:

Your strategy here will be to find the mass of the sphere first, then use copper's density to find its volume.

So, the problem provides you with the number of atoms of copper present in the sphere. To find the mass of the sphere, convert the number of atoms to moles by using Avogadro's number

1.33 * 10^(24) color(red)(cancel(color(black)("atoms Cu"))) * "1 mole Cu"/(6.022 * 10^(23)color(red)(cancel(color(black)("atoms Cu"))))

$= \text{ 2.209 moles Cu}$

Next, use the molar mass of copper to find the number of grams that would contain that many moles of copper

2.209 color(red)(cancel(color(black)("moles Cu"))) * "63.546 g"/(1color(red)(cancel(color(black)("mole Cu")))) = "140.4 g Cu"

Now, you know that copper has a density of ${\text{8.96 g/cm}}^{3}$. This tells you that ${\text{1 cm}}^{3}$ of copper has a mass of $\text{8.96 g}$.

In your case, the sphere will have a volume of

140.4 color(red)(cancel(color(black)("g Cu"))) * "1 cm"^3/(8.96color(red)(cancel(color(black)("g Cu")))) = "15.67 cm"^3

All you have to do now is use the formula for the volume of the sphere to calculate its radius.

Rearrange to solve for $r$

$V = \frac{4}{3} \cdot \pi \cdot {r}^{3} \implies {r}^{3} = \frac{3}{4 \cdot \pi} \cdot V$

This will get you

$r = \sqrt[3]{\frac{3}{4 \pi} \cdot V}$

Plug in your value to find

r = root(3)(3/(4 * pi) * "15.67 cm"^3) = color(green)(bar(ul(|color(white)(a/a)color(black)("1.55 cm")color(white)(a/a)|)))#

The answer is rounded to three sig figs.