Show that 74% space is occupied in case of face centered cube.

Asked by Topperlearning User | 16th Jun, 2016, 03:38: PM

Expert Answer:

In the face diagonal  three spheres touch each other.

 

In Δ ABC Using the Pythagorean Theorem...

AC2 =b2 =BC2 + AB2

=a2 + a2 = 2a2 or

b =√2a

If r is the radius of the sphere,

b = 4r = √2a

or a = 4r/√2 = 2√2r

volume of the cube is a3 or (2√2r)3.

Packing efficiency = Volume occupied by four spheres in the unit cell x 100 / Total volume of the unit cell.  %

= 4x(4/3)pr3 x 100/(2√2r)3   %

= (16/3) p r3x100 /16√2r%

=74%.

Answered by  | 16th Jun, 2016, 05:38: PM