Question

Potassium crystallizes in a body centered cubic lattice.Calculate the approx no. of unit cells in 1gm of potassium?

Answer 1

     In a body centered cubic lattice structure, called bcc or bl,  we have atoms present in the center of the unit cube or cell.  one atom is present at the center and 8 atoms at the 8 centers of the unit cell of the crystal lattice.

   Thus, if we consider the atoms to be spherical with radius r, and the edge or side of the cube to be a units, then we have:
     √3 a = r + 2r + r = 4 r,  as the atoms at the diagonally opposite atoms of the cube are in line with the atom in the middle at the center of the cube.

  As each of the atoms at the corners contributes to 1/8 of a sphere into the cube, the cube consists of  1 atoms + 1/8 * 8 atoms = 2 atoms. remaining space is void.

  So amount of volume occupied by the 2 atoms = 2 * 4π/3  r³ = 8π/3 * (√3/4a)³

 size of the cube = a³

The ratio = space utilization of phosphorous lattice bcc = √3π/8 = 68%
so 32 % of the space is empty in the bcc lattice.

Phosphorous atomic weight = 19  so 19 gms consist of 6.023 * 10^23 atoms.

   1 gm consists of  3.17 * 10^22 atoms.

  there are two atoms in a unit cell.  Hence there are  1.585 * 10^22 units cells in 1 gm of Phosphorous.