Question

Consider free electron gas model in metal and an atom can be viewed as a number of electrons moving around a positively charged nucleus. Use the model of a particle in a box to estimate the energy (in eV) required to raise an atomic electron from the state n=1 to the state n=3, assuming the atom has a radius of 2Å.

Answer 1

$\! \! \! \! \! \! \! \! \! $Energy of a particle in the n-th state in particle in a box, $ E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\\\\ $where, L is the length of the box. Here the equivalent of L is the radius of atom, R.$\\\\ $Energy required here, $\\\\ E = E_3-E_1 = \frac{3^2\pi^2\hbar^2}{2mR^2}- \frac{1^2\pi^2\hbar^2}{2mR^2} = \frac{8\pi^2\hbar^2}{2mL^2}\\\\\\ E = \frac{8\times \pi^2\times (1.05\times 10^{-34})^2}{2\times 9.11\times 10^{-31}\times (2\times 10^{-10})^2}=1.19\times 10^{-17}$J$\\\\\\ E = \frac{1.19\times 10^{-17}}{1.6\times 10^{-19}} = 74.38$eV$$