# Question ee4f4

Jul 8, 2017

$\text{0.407 nm}$

#### Explanation:

The idea here is that you need to use the configuration of a face-centered cubic unit cell to find a relationship between the length of the cell, which is usually labeled $a$, and the radius of an atom of gold, which is usually labeled $r$.

So, a face-centered cubic unit cell looks like this

Now, you should know that the diagonal of a square is equal to

$\text{diagonal" = "side} \times \sqrt{2}$

In this case, the diagonal of a face has a length of

$\text{diagonal} = r + 2 r + r = 4 r$

Here $r$ represents the radius of the two quarter of an atom that occupy the two corners of the face and $2 r$ represents the diameter of the atom that occupies the middle of the face.

This means that the length of the side of the unit cell

$\frac{\text{side" = "diagonal}}{\sqrt{2}}$

will be equal to

$a = \frac{4 r}{\sqrt{2}} = \frac{4 r \sqrt{2}}{2} = 2 \sqrt{2} \cdot r$

Plug in your value to find

a = 2 sqrt(2) * "0.144 nm" = color(darkgreen)(ul(color(black)("0.407 nm")))#

The answer is rounded to three sig figs.