First slide

Bohr Model of the Hydrogen atom.

Question

If energy and angular momentum of electron in the ground state of hydrogen atom E₀ and L₀ respectively, then in a He⁺ ion, energy of the electron whose angular momentum is 3L₀, will be

Moderate

A

$\frac{{3E}_{0}}{16}$
B

$\frac{{9E}_{0}}{64}$
C

$\frac{{27E}_{0}}{64}$
D

$\frac{{4E}_{0}}{9}$

Solution

In the nth energy state of a hydrogen-like ion, $L_{n} = n \frac{h}{2 π} and E_{n} = - \frac{13 .6 z^{2}}{n^{2}} eV$
$∴ E_{n} = - \frac{13 .6 z^{2}}{{(2 π L_{n} / h)}^{2}} = - (\frac{13 .6 h^{2}}{4 π^{2}}) \frac{z^{2}}{L_{n}^{2}}$
$∴ \frac{E^{1}}{E} = {(\frac{z^{1}}{z})}^{2} . {(\frac{L_{n}}{{L^{1}}_{n}})}^{2} \Rightarrow \frac{E^{1}}{E_{0}} = {(\frac{2}{1})}^{2} {(\frac{1}{3})}^{2} \Rightarrow E^{1} = \frac{4 E_{0}}{9}$

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