The ratio of the longest wavelength to the shortest wavelength \((\frac{\lambda_L}{\lambda_S})\) in Paschen series of hydrogen spectrum is:

Concept:

Hydrogen spectrum:

• The hydrogen spectrum is an important piece of evidence to show the quantized electronic structure of an atom.
• The hydrogen atoms of the molecule dissociate as soon as an electric discharge is passed through a gaseous hydrogen molecule.
• It results in the emission of electromagnetic radiation initiated by the energetically excited hydrogen atoms.
• The hydrogen emission spectrum comprises radiation of discrete frequencies. ​

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• First three spectral series in the hydrogen spectrum: Lyman, Balmer, and Paschen series
• The formula for the wavelength, \(\frac{1}{λ}=R(\frac{1}{p^2}-\frac{1}{n^2})\) where, p =  higher energy level, n = lowest energy level, λ = wavelength
• Here R = 109677 is called the Rydberg constant for the hydrogen atom

Calculation:

The longest wavelength in the Paschen series was obtained when p = 3, n = 4

For the longest wavelength, 

\(\frac{1}{λ_l}=R(\frac{1}{p^2}-\frac{1}{n^2})\)

\(\frac{1}{λ_l}=R(\frac{1}{3^2}-\frac{1}{4^2})\)

\(\frac{1}{λ_l}=R(\frac{1}{9}-\frac{1}{16})\)

\(λ _l=\frac{144}{7R}\)

The shortest wavelength in the Paschen series was obtained when p = 3, n = ∞ 

For the shortest wavelength, 

\(\frac{1}{λ_s}=R(\frac{1}{p^2}-\frac{1}{n^2})\)

\(\frac{1}{λ_s}=R(\frac{1}{3^2}-\frac{1}{∞^2})\)

\(λ _s=\frac{9}{R}\)

Ratio, \(\frac{λ_l}{λ _s}=\frac{144/7R}{9/R}\)

\(\frac{λ_l}{λ _s}=\frac{16}{7}\)