a) Calculate the packing efficiency of particles in a body centred cube. 

Explanation

bcc diagram.
Edge length `= a = (4r)/(sqrt(3))`
Packing efficiency `= ("Volume of sphere" xx2xx100)/("Volume of unit cell")`
`=((4)/(3)pi r^(3)xx2xx100)/((4)/(sqrt(3)r))=68%`
Packing efficiency in body centred cubic (bcc) lattice.
The number of atoms per unit cell in bss is two.
So, the volume of two atoms (two spheres) `" " =2xx(4)/(3)pi r^(3)`
Let the edge length a, be the radius of the sohere.
The body diagonal `AF = y`
and face diagonal `FD = b`,
Then `AF=y=4r`
In `Delta EFD`
`FD^(2)=EF^(2)+ED^(2)`
`b^(2)=a^(2)+a^(2)=2a^(2)`
`b=sqrt(2)a`
Now in `Delta AFD`
`AF^(2)=AD^(2)+FD^(2)`
`y^(2)=a^(2)+b^(2)`
`= a^(2)+2a^(2)=3a^(2)`
`y=sqrt(3)a`
But `y=4r`
`therefore sqrt(3)a=4r`
`a=(4r)/(sqrt(3))`
Volume of unit cell `= a^(3)+((4r)/(sqrt(3)))^(3)`
`therefore` Packing efficiency. `= ("volume of two atoms in unit cell")/("volume of unit cell")xx100`
`=(2xx(4)/(3)pi r^(3))/(((4r)/(sqrt(3)))^(3))xx 100 % =((8)/(3)pi r^(3))/((64 pi r^(3))/(3sqrt(3)))xx 100%`
`= 68%`.