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).
Answers
In simple cubic,edge length or side of the cube = a,the radius of each particle = r.
We know: a = 2r:
The volume of the cubic unit cell = side3, = a3, = (2r)3, = 8r3.Number of atoms in unit cell = 8 x 1 /8; = 1.
The volume of the occupied space = (4/3)πr3.P.E = 4/3πr3/8r3*100=52.4%.
Body-centred;edge length or side of the cube = a,the radius of each particle = r.We know a√3 = 4r.The volume of the cubic unit cell = side3 = a3 = (4r/√3)3.Number of atoms in unit cell =2 .
The volume of the occupied space = (4/3)πr3.P.E= 2* 4/3πr3 *100 / (4r/√3)r3-68%. Face centered:edge length or side of the cube = a,radius of each particles = r. We know: a√2 = 4r.
The volume of the cubic unit cell = side3 = a3 = (4r/√2)3 = (2√2 r)3;number of atoms at the corner = 8 x 1/8 = 1;number of atoms at the face = 6 x1/2 = 3;
total number of atoms = 4;
The volume of the occupied space = (4/3)πr3; P.E = 4/3πr3/8r3*100=52.4%