A bcc lattic is made up of hollow spheres of X. spheres of soldid 'Y,' are present in hollow spheres of X. The radius of 'Y'is half of the radius of 'X' . Calculate the ratio of the total volume of spherees of 'X' unoccupied by Y in a unitcell and volume of the unit cell ?

A bcc lattic is made up of hollow spheres of X. spheres of soldid 'Y,'

1.	A bcc lattic is made up of hollow spheres of X. spheres of soldid 'Y,' are present in hollow spheres of X. The radius of 'Y'is half of the radius of 'X' . Calculate the ratio of the total volume of spherees of 'X' unoccupied by Y in a unitcell and volume of the unit cell ?
Answer» Solution :Let the radius of hollow SPHERE X be r. As spheres, X have bccstructure, edge lenth `(a)= (4R)/(sqrt3)` Volume of the unit CELL = ` a^(3) = ( (4r)/(sqrt3))^(3)` Radius of shphere Y =` r/2 ` (Given ) Volume of Sphere X =` 4/3 PIR^(3)` volume of sphere Y ` 4/3 pi (r/2)^(3)` Volume of X unoccupied by Y in unit cell = `2 xx [ 4/3 pi r^(3)- 4/3 pi ( r/2)^(3) ]` ( bcc lattice has 2 spheres per unit cell ) ` 2xx 4/3 pi r^(3) (1-1/8) = 2xx 4/3 pir^(3) xx 7/8` `(" Volume of X unoccpied by Y in unit cell")/( " volume of unit cell")= (2xx 4/3 pir^(3) xx7/8)/(((4r)/sqrt3)^(3))= ( (7 pisqrt3))/64`