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  • 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 nth level from levels higher than n, are approximate harmonics (i.e. in the ratio 1: 2: 3...) when n >>1

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Frequency of emitted radiation:

\Delta E=hf\Rightarrow f=\frac{\Delta E}{h}=\frac{E_{2}-E_{1}}{h}=RcZ^{2}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right ) 

Wavenumber:

Wavenumber is the number of waves in unit length

\bar{v}=\frac{1}{\lambda}=\frac{f}{c}\Rightarrow \frac{1}{\lambda}=RZ^{2}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right )=\frac{13.6Z^{2}}{hc}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right ) 

The number of spectral lines: If an electron jumps from higher energy orbit to lower energy orbit, it emits radiation with various spectral line.

If electron falls from n2 to n1, then the number of spectral lines emitted is given by

N_{E}=\frac{\left ( n_{2}-n_{1}+1 \right )\left ( n_{2}-n_{1} \right )}{2}

If electron falls from nth orbit to ground state (i.e n2=n and n1=1), then number of spectral lines emitted N_{E}=\frac{\left ( n(n-1) \right )}{2}

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

f_{min}=cRZ^{2}\left [ \frac{1}{\left ( n+p \right )^{2}}-\frac{1}{n^{2}} \right ]

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

f_{min}=cRZ^{2}\left [ \frac{1}{n^{2}}\left ( 1+\frac{p}{n} \right )^{-2}-\frac{1}{n^{2}} \right ]\\ f_{min}=cRZ^{2}\left [ \frac{1}{n^{2}}-\frac{2p}{n^{3}}-\frac{1}{n^{2}}\right]\\ \left [ by \; Binomial\; theorem \left ( 1+x \right )^{2}=1+nx\; \; If |x|<1 \right ]\\ f_{min}=cRZ^{2}\frac{2p}{n^{3}}=\left ( \frac{2cRZ^{2}}{n^{3}} \right )p

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

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