Derive an expression for the radius of nth orbit of hydrogen atom using Bohr's postulates. Interpret the nature of variation of the radius of orbit with the principal quantum number n.

Solution

According to Bohr's postulates,

$mvr = \frac{nh}{2 π} \Rightarrow v = \frac{nh}{2 πmr} . . . (1) Now, using the Bohr' s equation, \frac{{mv}^{2}}{r} = \frac{{KZ}^{2} e^{2}}{r^{2}} . . . (2) Putting (1) in equation (2), we have$

$\frac{m}{r} \frac{n^{2} h^{2}}{4 π^{2} m^{2} r^{2}} = \frac{K Z e^{2}}{r^{2}} \Rightarrow r = \frac{n^{2} h^{2}}{4 π^{2} m K Z e^{2}} F o r h y d r o g e n a t o m, Z = 1 ∴ r = \frac{n^{2} h^{2}}{4 π^{2} m K e^{2}} \Rightarrow r \propto n^{2}$

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Wave Optics

Derive an expression for the radius of nth orbit of hydrogen atom using Bohr's postulates. Interpret the nature of variation of the radius of orbit with the principal quantum number n.

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