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Question
Using an expression for energy of electron, obtain the Bohr’s formula for hydrogen spectral lines.
Solution
Suppose an electron jumps from nth higher orbit to pth lower orbit.
Let and En Ep be the energies of electron in nth and pth orbit respectively.
`therefore` Energy of electron in nth orbit is, En = `-((me^4)/(8ε_0^2h^2))=1/2.............(1)`
`therefore` Energy of electron in pth orbit , Ep =`-((me^4)/(8ε_0^2h^2))1/P^2..............(2)`
`therefore` According to Bohr’s 3rd postulate, energy emitted is given by,
`therefore` energy emitted, `hnu =E _n-E_p`
`therefore` `hnu = (me^4)/(8ε_0^2h^2)(1/p^2-1/n^2)`
Or `nu = (me^4)/(8ε_0^2h^3)(1/p^2-1/n^2)`
`therefore 1/lambda=(me ^4)/(8ε_0^2h^3C)(1/p^2-1/n^2)` `[because nu = C/lambda]`
`therefore 1/lambda="R"(1/p^2-1/n^2)`
Where, `"R" = (me ^4)/(8ε_0^2ch_3)` is called Rydberg’s constant.
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