Question

Which of the following products in a hydrogen atom are independent of the principal quantum number n? The symbols have their usual meanings.

(a) vn
(b) Er
(c) En
(d) vr

Answer 1

(a) vn
(b) Er
Relations for energy, radius of the orbit and its velocity are given by
$$\begin{gathered} E=-\frac{\mathrm{m}{Z}^2{\mathrm{e}}^4}{8{{\in }_0}^2{\mathrm{h}}^2{n}^2} \\ r=\frac{{\in }_0{\mathrm{h}}^2{n}^2}{{\mathrm{\pi mZe}}^2} \\ v=\frac{Z{\mathrm{e}}^2}{2{\in }_0\mathrm{h}n} \end{gathered}$$
Where
Z : the atomic number of hydrogen like atom
e : electric charge
h : plank constant
m : mass of electron
n : principal quantam number of the electron
${\in }_0$ : permittivity of vacuum
From these relations, we can see that the products independent of n are vn, Er.