Using Bohr's postulates, derive the expression for the total energy of the electron revolving in `n^(th)` orbit of hydrogen atom. Find the wavelength of `H_(alpha)` line, given the value of Rydberg constant, `R = 1.1 xx 10^(7) m^(1)`

Explanation

T.E. of an electron in `n^(th)` orbit of hydrogen atom is sum of the KE and P.E. in the orbit.
i.e. `E_(n) = KE_(n) + PE_(n)`
`= .^(1)//_(2) mv_(n)^(2)` (Electrostatic pot at nth orbit due nucleus of H) `xx (-e)`
`= (1)/(2) (e^(2))/((4pi in_(0))r_(n)) + ((1)/(4pi in_(0)) (e)/(r_(n))) xx (-e) = (e^(2))/(8 pi in_(0)r_(n)) - (e^(2))/(4pi in_(0)r_(n)) = (-e^(2))/(8pi in_(0)r_(n)) " " [ :' r_(n) = (r^(2) h^(2) in_(0))/(pi me^(2))]`
So `E_(n) = (-me^(4))/(8h^(2)in_(0)^(2)n^(2)) = (-2(3.14)^(2) xx 9.1 xx 10^(-31) xx (1.6 xx 10^(-19))^(4)J)/((8.854 xx 10^(-12))^(2) (6.63 xx 10^(-34))^(2)n^(2))`
`= (-2.17 xx 10^(-18))/(1.6 xx 10^(-19) n^(2)) eV [1eV = 1.6 xx 10^(-19)J]`
So `E_(n) = (13.6)/(n^(2)) eV`