Question

Calculate the efficiency of packing in case of a metal crystal for (i) simple cubic (ii) body-centred cubic (iii) face-centred cubic (with the assumptions that atoms are touching each other).

Answer 1

for finding packing efficiency we have require :
1. area occupied by all atoms /ions in unit cell
2. total area of unit cell.
because $P.E=\frac{\text{area occupied by all atoms/ions in unit cell}}{\text{total area of unit cell}}$

(i) $\bf{simple\:cubic}$
if edge length = a
then, atmic radius , r = a/2
volume of unit cell , V = a³
volume of all atoms in a unit cell, v = 4/3π(a/2)³
= πa³/6
so, packing efficiency = v/V = π/6 = 0.52 or 52%

(ii)$\bf{body-centered\:cubic}$
if edge length = a
atomic radius , r = √3a/4
volume of unit cell, V = a³
number of atoms per unit cell = 2
volume of all atoms in a unit cell = 2 × 4/3 π(√3a/4)³ = 4/3 πa³× 3√3/64 = √3πa³/8
so, packing efficiency = √3π/8 = 0.68 or 68%

(iii) $\bf{face-centered\:cubic}$
if edge length = a
atomic radius , r = a/2√2
volume of unit cell, V = a³
number of atoms per unit cell = 4
volume of all atoms in a unit cell , v = 4 × 4/3π(a/2√2)³ = πa³/3√2
so, packing efficiency = π/3√2 = 0.74 or 74 %

Answer 2

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