#### Answer

The smallest distance between two copper atoms is $2.29\times 10^{-10}~m$

#### Work Step by Step

Let's assume that we have 1 mole of copper atoms. We can find the mass.
$m = (6.02\times 10^{23})(64)(1.66\times 10^{-27}~kg)$
$m = 0.06396~kg$
We can find the volume of the copper cube.
$V = \frac{m}{\rho}$
$V = \frac{0.06396~kg}{8.9\times 10^3~kg/m^3}$
$V = 7.187\times 10^{-6}~m^3$
We can find the length $L$ of each side of the cube.
$L = V^{1/3}$
$L = (7.187\times 10^{-6}~m^3)^{1/3}$
$L = 0.0193~m$
We can find the number of atoms along each side of the cube.
$(6.02\times 10^{23})^{1/3} = 8.44\times 10^7~atoms$
We can find the distance $d$ between the atoms along the side of the cube.
$d = \frac{0.0193~m}{8.44\times 10^7}$
$d = 2.29\times 10^{-10}~m$
The smallest distance between two copper atoms is $2.29\times 10^{-10}~m$.