Question

Calculate the efficiency of paching in case of a metal crystal for bcc

Answer 1

Explanation:

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}}P.E=

total area of unit cell

area occupied by all atoms/ions in unit cell

(i) \bf{simple\:cubic}simplecubic

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}body−centeredcubic

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}face−centeredcubic

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 %