Question

Using Bohr’s postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number n_i) to the lower state, (n_f).

When electron in hydrogen atom jumps from energy state n_i = 4 to n_f = 3, 2, 1, identify the spectral series to which the emission lines belong.

Answer 1

In the hydrogen atom,

Radius of electron orbit, r =`(n^2h^2)/(4pi^2kme^2)`................. (1)

Kinetic energy of electron, `E_k = 1/2 mv^2 = (ke^2)/(2x)`

Using eq (1), we get

`E_k = (ke^2)/2 (4pi^2kme^2)/(n^2h^2)`

`=(2pi^2k^2me^4)/(n^2h^2)`

Potential energy, `E_p = (k(e)xx (e))/r  = -(ke^2)/r`

Using eq (1),we get

`E_p = -ke^2 xx (4pi^2kme^2)/(n^2h^2)`

`= - (4pi^2k^2 me^4)/(n^2h^2)`

Total energy of electron, `E = (2pi^2k^2me^4)/(n^2h^2) - (4pi^2k^2me^4)/(n^2h^2)`

`= (2pi^2k^2me^4)/(n^2h^2)`

`=(2pi^2k^2me^4)/(h^2)  xx (1/n^2)`

Now, according to Bohr’s frequency condition when electron in hydrogen atom undergoes transition from higher energy state (quantum number n_i) to the lower state (n_f) is,

`hv = e_(n_i) - E_(n_i`

`or hv = - (2pi^2k^2me^4)/h^2  xx 1/n_f^2  - ((2pi^2k^2me^4)/h^2  xx 1/n_f^2)`

`or hv= (2pi^2k^2me^4)/h^2  xx (1/n_f^2 -1/n_f^2)`

`or v= (2pi^2k^2me^4)/h^3  xx (1/n_f^2 -1/n_f^2)`

`or v= (c2pi^2k^2me^4)/(ch^3)  xx (1/n_f^2 -1/n_f^2)`

`(2pi^2 k^2me^4)/(ch^3) =`R = Rydherg constant = `1.097 xx 10^7m^-1`

Thus,

`v = Rc xx (1/n_f^2 -1/n_f^2)`

Now, higher state n_i = 4, lower state, n_f = 3, 2, 1

For the transition,

n_i = 4 to n_f = 3:→ Paschen series

n_i = 4 to n_f = 2:→ Balmer series

n_i = 4 to n_f = 1:→ Lyman series