A metal crystallises into two cubic faces namely face centered (F… - Vidyadip

Question

A metal crystallises into two cubic faces namely face centered (FCC) and body centered (BCC), whose unit cell edge lengths are 3.5 Å and 3.0 Å respectively. Find the ratio of the densities of FCC and BCC.

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Answer

Edge length of FCC unit cell (a1) = 3.5 Å
Edge length of BCC unit cell (a2) = 3.0 Å
Density of unit cell $=d=\frac{Z \times M}{N_A \times a^3} gcm ^{-3}$
Density of FCC unit cell $=d_1=\frac{4 \times M}{N_A \times(3.5 A)^3}$
Density of BCC unit cell $=d_2=\frac{2 \times M}{N_A \times(3.0 A)^3}$
Ratio of densities of FCC and BCC unit cell is,
$\frac{d_1}{d_2}=\frac{4 \times M}{N_A \times(3.5 A)^3} \times \frac{N_A \times(3.0 A)^3}{2 \times M}$
$\frac{d_1}{d_2}=\frac{54}{42.875}=1.259 \approx 1.26$

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