The Bohr Model of the Hydrogen Atom — Question 6

Question 6

Ionization from an excited Bohr level requires less than the ground-state binding

A hydrogen atom starts in mode $n=3$ and absorbs a photon of energy $2.00\,\mathrm{eV}$. Neglect nuclear recoil and assume ionization occurs. Use the nonrelativistic Bohr model with an infinitely heavy nucleus unless stated otherwise: $E_n=-13.6Z^2/n^2\,\mathrm{eV}$, $r_n=a_0n^2/Z$, $a_0=.0529\,\mathrm{nm}$, and $hc=1240\,\mathrm{eV\,nm}$. Here Z is the nuclear charge number.

1. Find the ionization threshold from that level.

2. Find the outgoing electron kinetic energy and explain the energy reference used.

Question 6 - Solution

Ionization from an excited Bohr level requires less than the ground-state binding

(a) Method and calculation

$$E_3=-13.6/9\,\mathrm{eV}=-1.51111\,\mathrm{eV}.$$$$\boxed{E_{\mathrm{threshold}}=0-E_3=1.51111\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$E_3+E_\gamma=K_{\mathrm{out}}\quad\Rightarrow\quad\boxed{K_{\mathrm{out}}=.488889\,\mathrm{eV}}.$$$$\boxed{E_{\mathrm{final}}=K_{\mathrm{out}}>0}\quad\text{relative to separated particles at rest}.$$

Physical check

Ionization reaches a continuum rather than another discrete bound level. The positive excess photon energy becomes outgoing kinetic energy in the stated recoil-free model.