Fundamental E3

[Jacinda Wollenweber 1D]

Question: In a nanotechnology lab you might have the capability ti manipulate individual atoms. The atoms on the left are gallium atoms (molar mass 70 g/mol), those on the right are atoms of astatine (molar mass 210 g/mol). How many astatine atoms would the pan on the right have to contain for the masses on the two pans to be equal?

So what I did was convert the grams per mol of each element into atoms. However, I was confused when I checked my answer with the solution manual because the solution stated that 3 atoms of At were needed to balance it evenly. When I solved for the atoms, I got 4.22x10^25 At atoms and 1.27x10^26 Ga atoms.

What I want to know is:
a) Am I on the right track for solving the problem?
and
b) Why is the solution that 3 atoms of At are needed instead of Ga? Does it have something to do with their molar mass?

Thanks!

[Kyle Sheu 1C]

Hmm, in the picture below the question, there are 9 Ga atoms displayed on the left hand side and another huge mass of At atoms on the right side, but I can totally understand how this was confusing because I was at first confused by it as well! I believe the 9 Ga atoms are given, while the chunk of At atoms is meant to represent the number of At atoms needed to balance the scales. With that said, we can formally work through the problem:

Gallium: 9 atoms * (1mol/6.023*10²³atoms) * (70g/1mol)=1.04599*10^-11g

Astatine (work in reverse, since we want the masses to be balanced):
1.04599*10^-11g* (1mol/210g) * (6.023*10²³atoms/1mol)=3 atoms of At

Low definition (sorry) picture attached is attached--let me know if there's anything unclear!

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Another way to look at this is to note that the molar masses of Ga and At are in a 1:3 ratio, so 3 moles of Ga have the same mass as 1 mole of At. Since the number of atoms in each mole are the same, 3 atoms of Ga have the same mass as 1 atom of At, and 9 atoms of Ga (given) would be balanced on a set of scales by 3 atoms of At.