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This question comes straight from a worksheet we were assigned in chemistry, though it’s more math focused of a question I hope that this is an appropriate question to ask here.

Calculate the edge length of a cube containing an Avogadro’s number ($6.02 \times 10^{23}$) of hydrogen atoms stacked to form a cube. Assume a hydrogen atom is equivalent to a tiny cube with edge length $\mathrm{1\times 10^{-10}\ m}$.

I calculated this two ways getting different answers for both.

First: I found the length of stacking these cubes in a straight line. $\mathrm{((6.02\times 10^{23})\times(1\times10^{-10}) = 6.02\times10^{13}\ m)}$ Next I took the cube root of this thinking that it would be edge length of a cube just as a square root would give the edge length of a square? $\mathrm{( = 39192.1267 \space m )}$

Second: I took the edge length of the hydrogen atom and cubed it, thinking that the volume of each atom would be useful, and multiplied it by the number of atoms. $((6.02\times 10^{23})(1\times 10^{-30}) = 6.02\times 10^{-7})$ So here I thought that I had the volume for the entire cube or at least I think. Now to get the length of a single edge we take a cube root of this number? $\mathrm{( = 0.0084\ m )}$

Sorry it’s long winded but couldn’t find anything to shed any light on this question, and thank you for your time.

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Stacking the cubes in a straight line doesn't give you a big cube, it gives you a long cuboid so when you take the square root of its length you don't get anything meaningful.

The second method is correct because you are finding the total volume of the cube and then cube rooting this to find the side length of the cube.

In addition, does 39km seem like a sensible length for a mole of hydrogen atoms in a solid cube? Whenever you are doing calculations of this type, or even any calculations, always stop to consider whether the answer you have got is actually sensible, or whether it is orders of magnitude out. This is usually a good way to spot silly mistakes such as this.

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  • $\begingroup$ Thank you very much for the response! Yea I thought that was nonsensical, but at the same time who knows! Too new to chemistry haha I knew they wouldn't ask us unless it was really large or really small though. $\endgroup$ – DanBaba May 22 '15 at 13:37
  • $\begingroup$ Ugh still can't edit and didn't mean to hit enter... Your answer was very enlightening because I realized my mistake! I was taking the number of "cubes" and getting the length of the cube then cube rooting it. I get the same .0084 m answer if I cube root the NUMBER of cubes, then multiply by the length. This makes so much sense to me now and thank you again. $\endgroup$ – DanBaba May 22 '15 at 13:40

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