## Related questions with answers

Consider an electron in its first excited state in a hydrogen atom. Determine its (a) orbital speed, (b) angular speed, (c) linear momentum, and (d) angular momentum.

Solution

VerifiedBohr considered that an electron in a hydrogen atom described circular orbits of radius $r$ around the nuclei. Using classical mechanics, we know that the angular momentum is

$\begin{align} L_n = mv_nr_n \end{align}$

and the linear momentum is defined as

$\begin{align} P_n = mv_n \end{align}$

where the orbital speed $v_n$ of the electron is

$\begin{align} v_n = \frac{nh}{2\pi m r_n } \end{align}$

and the radius of the orbit is

$\begin{align} r_n = \left( \frac{h^2}{4\pi^2ke^2m} \right) n ^2 = 0.0529n^2 \text{ nm} \end{align}$

The angular speed is

$\begin{align} \omega_n = \frac{v_n}{r_n} \end{align}$

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