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Use Bohr's postulate to prove that the radius of nth orbit in a hydrogen atom is proportional to n2.

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Best answer

As the electron revalues in an stationary orbit, 

Thus centrifugal force acting on the electron to balanced by the coulombic force

Fcentrifugal = Fcoulomb

\(\frac{mv^2}{r_n}=\frac{kl^2}{{r_n}^2}\)

\(r_n=\frac{kl^2}{m{v_n}^2}\)

We take 

\(m\,v_nr_n= \frac{nh}{2\pi}\)

\(\because V_n = \frac{nh}{2\pi mr_n}\)

Then,

\(r_n = \frac{kl^2}{m{v_n}^2}\)

\(r_n = \frac{kl^2}{m}.\frac{(2\pi mr_n)^2}{n^2h^2}\)

\(r_n = \frac{kl^2 4\pi^2m^2{r_n}^2}{m\,n^2h^2}\)

\(r_n = \frac{n^2h^2}{kl^24\pi^2m}\)

\(r_n \propto n^2\)

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