A binary crystalline solid forms a body-centered cubic-like unit cell where atoms of element P occupy the corners and an atom of element Q occupies the body center. The atoms touch each other along the body diagonal. If the packing efficiency of the crystal is 3 6 and the radius of Q is greater than the radius of P, what is the ratio of the radius of Q to the radius of P?

Step-by-step solution

Let the radius of atom P be r and the radius of atom Q be xr . Since the atoms touch along the body diagonal, the length of the body diagonal is equal to the sum of the diameters of the atoms: 3 a = 2r + 2(xr) = 2r(1+x) a = 2r(1+x) 3 The volume of the unit cell is: V_ cell = a^3 = 8r^3(1+x)^3 3 3 The effective number of atoms per unit cell is: For P (corners): 8 1 8 = 1 For Q (body center): 1 The total volume occupied by the atoms is: V_ occupied = 4 3 r^3 + 4 3 (xr)^3 = 4 3 r^3(1+x^3) The packing efficiency is giv