Question

If $Z$ is the atomic number of a one-electron species and $n$ is the orbit in which the electron is revolving, show that the number of revolutions made by the electron in that orbit in 1 second times is directly proportional to $\frac{Z^2}{n^3}$.

Answer 1

Radius of $nth$ orbit is given by,

$r=\frac{n^2{h}^2}{4{\pi }^{2}mZ{e}^2}$,

Velocity of $nth$ bohr orbit is given by,

$v=\frac{2\pi Z{e}^2}{nh}$

$v=2\pi rf$

$f=\frac{v}{2\pi r}$

$f=\frac{4{\pi }^2{Z}^2{e}^{4}m}{n^3{h}^3}$

$f\alpha \frac{Z^2}{n^3}$ $(\frac{4{\pi }^2{e}^{4}m}{h^3}$ is constant$)$