Question

Which of the following products in a hydrogen atom are independent of the principal quantum number ${}^{\prime }n{}^{\prime }$? The symbols have their usual meanings.

• A. $vn$
• B. $Er$
• C. $En$
• D. $vr$

Answer 1

Answer: (A), (B)

$vn$
$Er$

The velocity of revolving electron is, $v=\frac{e^2}{2nh{ϵ}_0}$
Hence, $v_n=\frac{e^2}{2nh{ϵ}_0}\left(n\right)$
$v_n=\frac{e^2}{2h{ϵ}_0}$

The product thus obtained does not contain principal quantum number(n) Hence, $v_n$ is independent of $n$.
The energy of an electron is
$E=\frac{-m{e}^4}{8n^2{h}^2{ϵ}_0^2}$

The radius of an orbit of revolution of electron is
$r=\frac{n^2{h}^2{ϵ}_0}{\pi m{e}^2}$

Hence,
$Er=\frac{-m{e}^4}{8n^2{h}^2{ϵ}_0^2}\times \frac{n^2{h}^2{ϵ}_0}{\pi m{e}^2}$
$Er=-\frac{e^2}{8\pi {ϵ}_0}$

The product thus obtained does not contain principal quantum number(n).

Hence, $Er$ is independent of $n$.