Question

Sterling silver,an alloy of copper, contains 7.5% copper by mass and has a density of 10.3g/cm^3. How many copper atoms are in a sterling silver sphere whose radius is 2.0mm?

Answer 1

I get approximately `2.45*10^20` copper atoms.

Explanation:

NOTE: The answer is kind of long!

We have a sphere of sterling silver with a radius of `2 \ "mm"`. First, we need to find the volume of the sphere.

The volume of a sphere is given by: `4/3pir^3`

where `r` is the radius of the sphere.

Plugging that into the equation, we get

`V=4/3*pi*(2 \ "mm")^3`

`=4/3*pi*8 \ "mm"^3`

`~~33.5 \ "mm"^3`

So, the volume of this alloy in the shape of a sphere is `33.5 \ "mm"^3`. To find the mass, we need to use the density equation. But since the density is in `"g/cm"^3`, we might as well convert `"mm"^3` into `"cm"^3`.

We know that `1 \ "cm"=10 \ "mm", :.1 \ "cm"^3=1000 \ "mm"^3`.

So, `33.5 \ "mm"^3=0.0335 \ "cm"^3`.

The density of this alloy is `10.3 \ "g/cm"^3`, so the mass of this alloy will be

`0.0335color(red)cancelcolor(black)("cm"^3)*(10.3 \ "g")/(color(red)cancelcolor(black)("cm"^3))=0.34505 \ "g"`

So, this alloy's mass is `0.34505 \ "g"`.

Since `7.5%` of its mass is copper, then the mass of copper in this alloy is

`0.34505 \ "g"*7.5%=0.02587875 \ "g"`

To find the amount of copper atoms in this alloy, we first need to find the number of moles of copper, and we can do that by dividing the total mass of copper atoms in here by copper's molar mass.

Copper `(Cu)` has a molar mass of `63.546 \ "g/mol"`.

Therefore, there are

`(0.02587875color(red)cancelcolor(black)"g")/(63.546color(red)cancelcolor(black)"g""/mol")=0.000407244358 \ "mol"`

To convert between moles to atoms, we multiply the number of moles by Avogadro's number `(N_A)`, which is `6.02*10^23`.

So we have in total:

`0.000407244358*6.02*10^23~~color(red)underline(color(black)(2.45*10^20))` copper atoms.