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Calculate the efficiency of packing in case of a metal crystal for (i) simple cubic (ii) body-centred cubic (iii) face-centred cubic (with the assumptions that atoms are touching each other). |
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Answer» Solution :(i) Simple CUBIC : Number of spheres in a unit cell `=8xx(1)/(8)=1` Volume of the sphere `=(4)/(3)pir^(3)"(r is the radius of the sphere)"` Volume of the cube `=a^(3)=(2r)^(3)=8r^(3)` Effciency of packing `=(4//3pir^(3))/(8r^(3))=(pi)/(6)=0.524 or 52.4%`. (ii) Body - centred cubic structure : Assuming that body centre touches the spheres at the corner. Body diagonal AD = 4r FACE diagonal `AC=sqrt(AB^(2)+BC^(2))=sqrt(a^(2)+a^(2))=sqrt2a` Body diagonal `AD=sqrt(AC^(2)+CD^(2))=sqrt(2a^(2)+a^(2))=sqrt3a` `sqrt3a=4r or a=(4r)/(sqrt3)`, Volume of unit cell `=a^(3)=((4r)/(sqrt3))^(3)=(64r^(3))/(3sqrt3)`  Number of spheres in unit cell `=8xx(1)/(8)+1=2` Volume of two spheres `=2xx(4)/(3)pir^(3)=(8)/(3)pir^(3)` Efficiency of packing `=(8pir^(3)//3)/(64r^(3)//3sqrt3)=(pisqrt3)/(8)=0.68 or 68%` (iii) Face - centred cubic : `AC=sqrt(AB^(2)+BC^(2))=sqrt(a^(2)+a^(2))=sqrt2a` `therefore""sqrt2a=4r or a=(4)/(sqrt2)xxr` Volume of the unit cell `=a^(3)=(32)/(sqrt2)r^(3)` Number of sphere in a unit cell `=8xx(1)/(8)+6xx(1)/(2)=4` Volume of four spehres `=4xx(4)/(3)pir^(3)=(16)/(3)pir^(3)` Efficiency of packing `=(16pir^(3)//3)/(32r^(3)//sqrt2)=0.74 or 74%`. 
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