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I'm having trouble understanding why I can't get the correct answer from this homework problem:

Solid aluminum forms a face-centered cubic unit cell. Aluminum has a density of $\pu{2.70 g/cm^3}$. Determine the edge length of the $\ce{Al(s)}$ unit cell in $\pu{cm}$.

Usually if I enter the wrong answer the it will show all the calculations to determine the correct answer. However this one does not show the calculations.

Here is the work I've done:

  1. $\ce{Al }= \pu{26.98 g/mol}$
  2. $\text{Mass} = \frac{26.98}{6.022\times10^{23}} = \pu{4.480E-23 g//atom}$
  3. $\text{Mass of one atom} = \text{mass of one unit cell}$
  4. $\text{Mass of one unit cell} = \pu{4.480E-23 g//unit cell}$
  5. $D = \frac{M}{V}$, thus $V = \frac{M}{D}$
  6. $V = \frac{\pu{4.480E-23}}{2.70} = \pu{1.66E-23 cm^3//unit cell}$
  7. For a cube $V = l^3$, thus $l = V^{1/3}$
  8. $\text{Unit cell edge length} = V^{1/3} = {\pu{1.66E-23 cm^3}}^{1/3} = \pu{1.18E-8 cm}$

The correct answer that the book gives is $\pu{4.049E-8 cm}$.

The only example I have to go off of in my book is for a simple cubic unit cell. Since $\ce{Al}$ is a face-centered cubic unit cell and has 4 particles per unit cell I tried multiplying my final answer by $4$, but that gives me $4.72$. I also tried multiplying the unit cell mass by $4$ which gave me a final answer of $1.87$. I just can't seem to get $4.049$. Any help understanding this would be greatly appreciated.

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  • $\begingroup$ Who said that mass of one atom equals mass of one unit cell? $\endgroup$ – Ivan Neretin Feb 20 '18 at 13:17
  • $\begingroup$ My book said that in the explanation for the simple cubic unit cell. $\endgroup$ – Haruku Feb 20 '18 at 13:24
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    $\begingroup$ True, but your cell is not simple. It is fcc. $\endgroup$ – Ivan Neretin Feb 20 '18 at 13:42
  • $\begingroup$ Hey @haruku, check this out. Your step "I also tried multiplying the unit cell mass by 4 which gave me a final answer of 1.87." should give you the correct answer instead. Are you making a calculation mistake? $\endgroup$ – Gaurang Tandon Feb 20 '18 at 13:56
  • $\begingroup$ You're right, I went back and redid the calculations and it turns out my calculator wasn't grouping numbers properly. $\endgroup$ – Haruku Feb 20 '18 at 14:28
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Correct approach is to do two things:

1) Put four atomic masses into one unit cell.

2) Properly extract the cube root of the volume you obtain. That volume should now be about $6.6×10^{-23}\text{cm}^3$ from part 1. To take the cube root properly, render the exponent on 10 as a multiple of 3, thus $6.6×10^{-23}$ = $66×10^{-24}$ where $10^{-24}$, not $10^{-23}$, is $(10^{-8})^3$.

Then your numbers should work out.

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