Question

Sodium is a monovalent metal having a density of 0.971 g/cm3 and a molar
mass of 23.0 g/mol. Use this information to calculate (i) the density of charge
carriers and (ii) the Fermi energy of sodium.

Answer 1

a) ρ = 0.971 g/cm³

M = 23.0 g/mol

$V = \frac{M}{ρ} \\ = \frac{23.0}{0.971} = 23.68 \;cm^3/mol \\ = \frac{23.68}{6.02 \times 10^{23}} = 3.934 \times 10^{-23} \;cm^3$

The density of the charge carriers is

$n_e = \frac{1}{V} \\ = \frac{1}{8.934 \times 10^{-23}} \\ = 2.54 \times 10^{22} \;cm^{-3} \\ = 2.54 \times 10^{28} \;m^{-3}$

b) Fermi energy of sodium is

$E_F = (3.65 \times 10^{-19})n_e^{\frac{2}{3}} \;eV \\ = (3.65 \times 10^{-19})(2.54 \times 10^{28})^{\frac{2}{3}} \;eV \\ = 6.797 \times 0.4645 \;eV \\ = 3.157 \;eV$