Obtain an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n–1). For large n, show that this frequency equals the classical frequency of revolution of the electron in the orbit.

The energy of radiation at level n is given by the relation:

…(i)

Where v₁ is the frequency of radiation at level n

H is Planck’s constant

m is mass of hydrogen atom

e is charge on electron

ε₀ is permittivity of free space

The energy of radiation at level (n-1) is given by the relation:

……………(ii)

Where v₂ is the frequency of radiation at level n-1

Due to deexcitation of the electrons, the energy released is:

E =E₂- E₁

hv = E₂- E₁…………………..(iii)

Putting equation (i) and (ii) in equation (iii), we get

v =

v =

For large values of n, (n-1)≈n

v =

Frequency of revolution of an electron is given by

V_c = v/2πr……………..(vi)

Velocity of electron in nth orbit is given by the following relation

…………………….(iv)

The radius of nth orbit is given by the following relation

r =………………….(v)

Substituting the values of equation (iv) and (v) in equation (vi), we get