Density of copper metal is `8.95"g cm"^(-3)`. If the radius of copper atoms is `127.8"pm"` predict the nature of its unit cell whether simple cubic, faced centred or body centred cubic. (Given atomic mass of Cu = 63.54 g `mol^(-1)` and `N_(o)=6.022xx10^(23)mol^(-1)`)

Explanation

If copper atom were simple cubic, a=2 r=2 x 127.8 pm =255.6 pm
Z=1 , `rho=(ZxxM)/(a^3xxN_0)=(1xx63.54 "g mol"^(-1))/((255.6xx10^(-10)cm)^3 xx(6.02xx10^23 "mol"^(-1)))=6.34 "g cm"^(-3)`
Actual `rho=8.95 "g cm"^(-3)`.
Hence, copper atom were body-centred ,
`a="4r"/sqrt3 =(4xx127.8)/1.732` pm =295.15 pm
Z=2,` rho=(ZxxM)/(a^3xxN_0)=(2xx63.54 "g mol"^(-1))/((295.15 xx10^(-10) cm)xx6.02xx10^23 "mol"^(-1))=8.21 "g cm"^(-3)`
hence, copper atom is not body-centred.
If copper atom were face-centred
`a=2sqrt2r =2xx1.414xx127.8` pm =361.4 pm
Z=4,` rho=(ZxxM)/(a^3xxN_0)=(4xx63.54 "g mol"^(-1))/((361.4 xx10^(-10) cm)^3xx6.02xx10^23 "mol"^(-1))=8.94 "g cm"^(-3)`
Hence, copper is face-centred cubic.