question_answer

State Bohr's postulate for the permitted orbits for the electron in a hydrogen atom. Use this postulate to prove that the circumference of the nth permitted orbit for the electron can contain exact n wavelengths of the de-Broglie wavelength associated with the electron in that orbit.

Answer:

Bohr's postulate of permitted orbits is only those circular orbits which are permitted for electron in which angular momentum of electron is an integral multiple of \[h/2\pi .\] i.e.        \[mvr=n\frac{h}{2\pi }\]                         ?(i) where, n is an integer. From de-Broglie hypothesis, wavelength associated with electron, \[\lambda =\frac{h}{mv}\Rightarrow mv=\frac{h}{\lambda }\] Substituting this value in Eq.(i), we get \[\frac{h}{\lambda }r=n\frac{h}{2\pi }\]             or         \[2\pi r=n\lambda \] i.e. circumference \[(S=2\pi r)\] of nth permitted orbit for the electron can contain exactly n wavelength of de-Broglie wavelength associated with electron in that orbit,