Derivation of Radius of the n th Bohr’s Orbit for Hydrogen-Like Atoms Assumptions for s imple a tom To derive an expression for radius, consider a hydrogen atom (or hydrogen-like atoms such as He + , Li 2+ , Be 3+ , B 4+ , C 5+ ) with atomic number equal to z consisting of a single electron with charge –e and mass m revolving around the nucleus of charge +Ze (+e is charge of proton) with a tangential velocity v in the orbit whose radius is r. Now revolving electron is being acted upon simultaneously by the following two types of forces ; (i) Electrostatic force of Attraction / Centripetal Force According to Coulomb’s law, the electrostatic force of attraction (F e ) between the nucleus of charge ‘+Ze’ and electron of charge ‘–e’ separated by a distance ‘r’ is given by: Where ‘K’ is proportionality constant. It is equal to 1/4 π ε o r 2 Hence attractive force between nucleus and electron can be written as...
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