The Bohr Model of the Hydrogen Atom — Question 2

Question 2

Equal binding energies in different hydrogenic ions need not imply equal radii

Compare hydrogen in its ground state with a one-electron helium ion in mode $n=2$. 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 both energies and radii.

2. Compare their speeds and orbital periods using $v_n\propto Z/n$.

Question 2 - Solution

Equal binding energies in different hydrogenic ions need not imply equal radii

(a) Method and calculation

$$\boxed{E_{\mathrm{H,1}}=-13.6\,\mathrm{eV}},\qquad\boxed{E_{\mathrm{He^+,2}}=-13.6(4/4)=-13.6\,\mathrm{eV}}.$$$$\boxed{r_{\mathrm{H,1}}=a_0=.0529\,\mathrm{nm}},\quad\boxed{r_{\mathrm{He^+,2}}=2a_0=.1058\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{v_{\mathrm{He^+,2}}/v_{\mathrm{H,1}}=(2/2)/(1/1)=1}.$$$$T=2\pi r/v\Rightarrow\boxed{T_{\mathrm{He^+,2}}/T_{\mathrm{H,1}}=2}.$$

Physical check

Equal energy fixes equal speed in this infinite-nuclear-mass comparison, but the stronger central charge permits that speed at a larger radius with higher angular momentum.