Question 27: Atoms & Nuclei

Question

What is the frequency ' $
u$ ' of the electron in Bohr's first orbit of radius ' $r$ ' of the hydrogen atom?

Accepted Answer

Start from the two Bohr conditions for the first orbit (n = 1):
AngularÃ¢ÂÂmomentum quantisation
$$m v r = \frac{h}{2\pi}\,. $$
Coulomb force provides centripetal acceleration
$$\frac{e^2}{4\pi\varepsilon_0\,r^2} = \frac{m\,v^2}{r}\,. $$
StepÃ¢ÂÂ1: from (1)
$$v = \frac{h}{2\pi m r}\,. $$
StepÃ¢ÂÂ2: orbital frequency
$$
u = \frac{	ext{revolutions}}{	ext{second}} = \frac{v}{2\pi r} = \frac{1}{2\pi r}\,\frac{h}{2\pi m r} = \frac{h}{4\pi^2 m r^2}\,. $$
StepÃ¢ÂÂ3: eliminate $m$ using (2).  Rearranged, (2) gives
$$m = \frac{e^2}{4\pi\varepsilon_0}\;\frac{r}{v^2}.$$
Plugging into the expression for $
u$ (or solving the two equations simultaneously) yields
$$\boxed{
u = \frac{e^2}{4\pi\varepsilon_0\,h\,r}}\,. $$
That matches Option A.

Answer

$v=\frac{e^2}{4 \pi \varepsilon_0 h r}$

Answer

$v=\frac{e^2}{2 \pi \varepsilon_0 h r^2}$

Answer

$v=\frac{e^2}{4 \pi \varepsilon_0 h r^2}$

Answer

$v=\frac{e^2}{2 \pi \varepsilon_0 h r}$

Atoms & Nuclei

What is the frequency ' \nu ' of the electron in Bohr's first orbit of radius ' r ' of the hydrogen atom?

Choose the correct answer: