# The density of copper metal is 8.95 g cm-3. If the radius of copper atom is 127.8 pm, is the copper unit cell a simple cubic or a body-centered cubic or a face-centered cubic? (Given At. Mass of Cu = 63.54 g/mol)

The density of copper metal is 8.95 g cm-3. If the radius of copper atom is 127.8 pm, is the copper unit cell a simple cubic or a body-centered cubic or a face-centered cubic? (Given At. Mass of Cu = 63.54 g/mol)

Given Values

Density (D) = 8.9 gm/cm3

Radius of Atom (r)= 127.8 pm

Atomic mass (M) = 63.54 gm/mol

Avogadro Number (NA) = 6.02 × 1023

The value of ‘Z’ depends on the type of unit cell: Simple unit cell (Z=1) , BCC (Z=2) , FCC (Z=4)

Calculations

Let us calculate the density for the simple unit cell:

$$D=\frac{Z\times M}{\left ( a \right )^{3}\times N_{A}}$$

a = 2r = 2 × 127.8 pm = 255.6 pm = 255.6 × 10-10 cm

$$D=\frac{1\times 63.54}{\left ( 255.6 \times 10^{-10} \right )^{3}\times 6.02 \times 10^{23}}$$ =6.3 gm/cm3

The calculated density does not match the given density, so copper does not have a simple unit cell.

Let us calculate the density for the BCC unit cell:

$$a=\frac{4.r}{\sqrt{3}}=\frac{4 \times 127.8~pm}{1.732}$$ = 295.15 pm = 295.15 × 10-10 cm

$$D=\frac{2\times 63.54}{\left ( 295.15 \times 10^{-10} \right )^{3}\times 6.02 \times 10^{23}}$$ =8.2 gm/cm3

The calculated density does not match the given density, so copper does not have a BCC unit cell.

Let us calculate the density for the FCC unit cell:

$$a=2\sqrt{2}.r$$ = 2 × 1.414 × 127.8 pm = 361.4 pm = = 361.4 × 10-10 cm

$$D=\frac{4 \times 63.54}{\left ( 361.4 \times 10^{-10} \right )^{3}\times 6.02 \times 10^{23}}$$ =8.9 gm/cm3

The calculated density corresponds to the given density, so copper has an FCC unit cell.