question_answer 34)

Calculate the efficiency of packing (or packing fraction) in case of a metal crystal for (a) Simple cubic (b) Body centered cubic (c) Face centered cubic (with the assumptions that atoms are touching each other).  

Answer:

Packing efficiency is the percentage of the total space which is occupied by the particles in a certain packing. The fraction of the total space filled by spheres is called packing fraction. Packing fraction. (a) In a simple cubic unit cell, Suppose the edge length of the unit cell = a and radius of the sphere = r As spheres are touching each other so, a = 2r No. of spheres per unit cell   Volume of 1 sphere   Volume of the cube = a3 = (2r)3 = 8r3  Fraction occupied i.e. packing fraction or % of space occupied = 524% Percentage of unoccupied space = 100 ? 52 4 = 47 6% (b) In body centered cubic structure, As the sphere at the centre touches the spheres at the corner, body diagonal AD = 4r Further face diagonal,   and body diagonal, Volume of the unit cell   No. of spheres per unit cell Volume of 2 spheres    Fraction occupied i.e. packing fraction                 or % of space occupied = 68% Percentage of unoccupied space =100 ? 68 = 32% (c) In face-centred cubic structure As spheres on the face are touching, so AC = 4r But from the right angled triangle ABC, or Volume of the unit cell No. of spheres in the unit cell Volume of 4 spheres Fraction occupied i.e. packing fraction or efficiency of packing   or % of space occupied = 74% Percentage of unoccupied space = 100 ? 74 = 26%