Ncert Exercises & Solutions

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

Hint: Use the formula of frequency for a particular transition.

Step 1: Find the frequency of any line series for a particular transition.

The frequency of any line in a series in the spectrum of hydrogen-like atoms corresponding to the transition of electrons from (n+p) level to nth level can be expressed as a difference of two terms:

ν_{mn} = {cRZ}^{2} [\frac{1}{{(n + p)}^{2}} - \frac{1}{n^{2}}]

where, m=n+p, (p=1,2,3,...) and R is Rydberg constant.

For p < < n

ν_{mn} = {cRZ}^{2} [\frac{1}{n^{2}} {(1 + \frac{p}{n})}^{- 2} - \frac{1}{n^{2}}]

ν_{mn} = {cRZ}^{2} [\frac{1}{n^{2}} - \frac{2 p}{n^{3}} - \frac{1}{n^{2}}]

[By binomial theorem {(1 + x)}^{n} = 1 + nx if |x| < 1]

ν_{mn} = {cRZ}^{2} \frac{2 p}{n^{3}} = (\frac{2 {cRZ}^{2}}{n^{3}}) p

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

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