Question

According to Bohr's theory, the electronic energy of hydrogen atom in the nth Bohr's orbit is given by :
$E_n =\frac{-21.7 \times 10^{-19}}{n^2} \, J$
Calculate the longest wavelength of electron from the third Bohr's orbit of the $He^+$ ion.

Answer 1

For H-like species, the energy of stationary orbit is expressed as
$E(X)=Z^{2} \times E(H)$
$\Rightarrow $ For $H e^{+}(Z=2)$
$E=-\frac{4 \times 21.7 \times 10^{-19}}{n^{2}} J$
For longest wavelength transition from $3^\text{rd}$ orbit, electron must jump to $4^\text{th}$ orbit and the transition energy can be determined as
$ \Delta ; E=+4 \times 21.7 \times 10^{-19}\left(\frac{1}{9}-\frac{1}{16}\right) J=4.22 \times 10^{-19} J$
Also, $\because \Delta ; E=\frac{h c}{\lambda}$
$\therefore \lambda=\frac{h c}{\Delta E}=\frac{6.625 \times 10^{-34} \times 3 \times 10^{8}}{4.22 \times 10^{-19}} m$
$=471 \times 10^{-9} m=471 \,n m$