In the Rydberg formula for the spectrum of the hydrogen atom, the wavenumber is _____.

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  1. inversely proportional to the square root of an electron charge 
  2. directly proportional to the fourth power of an electron charge 
  3. inversely proportional to the fourth power of an electron charge 
  4. directly proportional to the square root of an electron charge 

Answer (Detailed Solution Below)

Option 2 : directly proportional to the fourth power of an electron charge 
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Concept:

The Rydberg formula is the mathematical formula to determine the wavelength of light emitted by an electron moving between the energy levels of an atom. When an electron shift from an orbital from higher energy to lower energy state, a photon of light is generated. Rydberg formula is given by

  • \(\bar\nu=\frac{1}{\lambda}=R_H(\frac{1}{n_1^2}-\frac{1}{n_2^2})\)

where \(\bar\nu\) is wave number,

 \(\lambda\) is wavelength, 

\(R_H=\text{ Rydberg constant }=109677cm^{-1}=1.09\times10^{7}m^{-1}\)

Explanation:

  • \(\bar\nu=\frac{1}{\lambda}=R_H(\frac{1}{n_1^2}-\frac{1}{n_2^2})\)

 

  • \(R_H=\frac{me^4}{8h^3c\epsilon_0^2}=1.09677\times10^{7}m^{-1}\)

 

  • \(\bar\nu=\frac{1}{\lambda}=\frac{me^4}{8h^3c\epsilon_0^2}(\frac{1}{n_1^2}-\frac{1}{n_2^2})cm^{-1}\)

\(\implies\bar\nu\propto e^4\)

Hence, the correct answer is Option-2-directly proportional to the fourth power of an electron charge.

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