Answer to Question #236523 in Physics for Arif

Question

Calculate the packing efficiency and density of sodium chloride from the following data: Radius of sodium ion = 0.98 Angstrom; Radius of chloride ion = Angstrom ; Atomic mass of sodium = 22.99 amu Atomic mass of chlorine = 35.45 amu

Answer 1

Sodium chloride has a face-centered cubic (FCC) crustal lattice. First, find the packing efficiency.

The volume of the cubic unit cell:

$$V=a^3=(\sqrt8r)^3=8\sqrt8r^3.$$

The volume of space that the atoms occupy:

$$v=4·\frac43\pi r^3=\frac{16}{3}\pi r^3.$$

The packing efficiency:

$$\epsilon=\frac vV=\frac{16\pi r^3/3}{8\sqrt8 r^3}=0.74.$$

Find the density:

$$\rho=\frac{n\Mu}{VN_A}=\frac{n(\Mu_\text{Na}+\Mu_\text{Cl})}{(8\sqrt8r^3)(N_A)}=\\\space\\ =\frac{4(23·10^{-3}+35.5·10^{-3})}{(0.563·10^{-9})^3·(6.023·10^{23})}=2177\text{ kg/m}^3.$$