Question

Show that the radius of the orbit in hydrogen atom varies as n2 where n is the principal quantum number of the atom​

Answer 1

Answer:-

According to Bohr's second postulate,

$mvr = \frac{nh}{2\pi}$

where m be the mass of an electron, v be the velocity and r be the radius of orbit in which the electron revolves around the nucleus.

$So, v = \frac{nh}{2\pi \: mr}$

Here, the centripetal force is provided by the electrostatic force of attraction between electron and nucleus.

$Thus, \frac{m {v}^{2} }{r} = \frac{k(Ze)(e)}{ {r}^{2} }$

$\longrightarrow \: r = \frac{kZ {e}^{2} }{m {v}^{2} }$

$Substituting \: the \: value \: of  \: v:$

$r = \frac{ {n}^{2} {h}^{2} }{4 {\pi}^{2}mk Z {e}^{2} }$

$Thus, r \: ∝ \: {n}^{2}$