Show that the first few frequencies of light that is emitted when electrons fall to the n th level from levels higher than n, are approximate harmonics (i.e. in the ratio 1: 2: 3...) when n 1.

Question

Show that the first few frequencies of light that is emitted when electrons fall to the n th level from levels higher than n, are approximate harmonics (i.e. in the ratio 1: 2: 3...) when n >>1.

Philoid · Accepted Answer

We know that the wavenumber of the emitted wavelength during transition from and of the electron is given as,

--------(1)

(where R is Rydberg’s constant)

Now putting

Putting this in equation (1), we get,

c= speed of light

and = the final and initial orbital level respectively

Now let us consider , where p in any integer 1,2,3….

Putting this in equation 1 we get,

∴

By binomial expansion the above equation becomes we get,

For p=1

For p=2

For p=3

∴ the ratios of frequencies,

Hence, we can say that the first few frequencies of light that is emitted when electrons fall to the n^th level from levels higher than n, are approximate harmonics (i.e. in the ratio 1: 2: 3...) when n >>1.

Show that the first few frequencies of light that is emitted when electrons fall to the nth level from levels higher than n, are approximate harmonics (i.e. in the ratio 1: 2: 3...) when n >>1.

Show that the first few frequencies of light that is emitted when electrons fall to the n^th level from levels higher than n, are approximate harmonics (i.e. in the ratio 1: 2: 3...) when n >>1.