# 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)**

**Answer:**

**Given Values**

Density (D) = 8.9 gm/cm^{3}

Radius of Atom (r)= 127.8 pm

Atomic mass (M) = 63.54 gm/mol

Avogadro Number (N_{A}) = 6.02 × 10^{23}

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/cm ^{3}**

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/cm ^{3}**

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/cm ^{3}**

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

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