The Bohr Model of the Hydrogen Atom — Question 4

Question 4

A spectral series has a longest line and a shorter limiting wavelength

Consider every hydrogen emission transition $n\to2$ with integer $n\ge3$. 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 longest wavelength in the series.

2. Find the short-wavelength limit as $n\to\infty$ and identify which endpoint is approached rather than attained by a finite n.

Question 4 - Solution

A spectral series has a longest line and a shorter limiting wavelength

(a) Method and calculation

$$E_\gamma=13.6\left(\frac14-\frac1{n^2}\right)\mathrm{eV};\quad E_\gamma\text{ smallest at }n=3.$$$$\boxed{\lambda_{\max}=\frac{1240}{13.6(1/4-1/9)}=656.471\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$n\to\infty:\quad E_\gamma\to3.40\,\mathrm{eV}\Rightarrow\boxed{\lambda_{\mathrm{limit}}=364.706\,\mathrm{nm}}.$$$$\boxed{364.706\,\mathrm{nm}<\lambda\le656.471\,\mathrm{nm}}\quad\text{for finite bound-state indices}.$$

Physical check

The series limit corresponds to the continuum threshold as the initial binding tends to zero. Finite upper levels produce wavelengths strictly above the limit.