Question

Copper has a face-centered cubic unit cell. the density of copper is 8.96 g/cm3. calculate a value for the atomic radius of copper.

Answer 1

Given That copper has a face centered cubic unit cell. that is, copper is present on the corners and the face center. Thus, the number of Cu atoms in one unit cell is 4.
For FCC, the relation between edge length and radius of atom is given as follows:
a = 2 √2r
Here, a is the edge length and r is the radius of copper atom.
The density of the unit cell is 8.96 g/cc. Density can be calculated as follows:
d = $\frac{Z * M}{a^{3}* N_{A} }$
Here, d is the density of the unit cell, Z is the number of Cu atoms in the unit cell, M is the molar mass of copper, $N_{A}$ is Avogadro's number, and a is the edge length.
Substitute the expression for a as shown below:
d = $\frac{Z x M}{(2 \sqrt{2r} )^{3} x N_{A} }$
d = $\frac{Z x M}{16 \sqrt{2 r^{3} xN_{A}} }$

8.96 g /$cm^{3}$ = $\frac{4 x (63.5)}{16 \sqrt{2 r ^{3} x (6.022 x 10^{23} )} }$
$r^{3} = 2.08 x 10^{-24} cm^{-3} r = 1.28 x 10^{-8} cm$