Vilume occupied for a cubic close packed lattice of sphere is

Options

A. `60%`
B. `52.4%`
C. `68%`
D. `74%`

Answer

D

Explanation

In c.c.p. or f.c.c. arrangement, number of aphere (atoms) per unit cell is 4.
Let a be the length of the cube.
`therefore` Volume of the cube, `V=a^(3)`……(1)
Let r be the radius of the sphere
For fcc arrangement `r=(asqrt2)/4`
Volume occupied by 4 sphere, `V'=4times4/3pir^(3)`
=`(4times4pi)/3times(asqrt2/4)^(3)`
=`(4times4pitimesa^(3)times2sqrt2)/(3times4times4times4)=(22/7timesa^(3)timessqrt2)/(3times2)`
`therefore` Ratio of volume occupied by the spheres to the volume of the cube
`(V')/V=22/7times(a^(3)timessqrt2)/(3times2timesa^(3))=0.74 " or"74%`