In Paschen series, wavelength of first line is ' ₁ ' and for Brackett series, wavelength of first line is ' ₂ ' then ratio ₁ ₂ is

Step-by-step solution

The Rydberg formula gives the wavelength of spectral lines as 1 = R ( 1 n_f^2 - 1 n_i^2 ) where R is the Rydberg constant and n_i > n_f . For the first line of the Paschen series ( n_f = 3 , n_i = 4 ): 1 ₁ = R ( 1 9 - 1 16 ) = R ( 7 144 ) Hence ₁ = 144 7R . For the first line of the Brackett series ( n_f = 4 , n_i = 5 ): 1 ₂ = R ( 1 16 - 1 25 ) = R ( 9 400 ) Thus ₂ = 400 9R . The ratio becomes ₁ ₂ = 144 7R 9R 400 = 1296 2800 Simplifying by dividing numerator and denominator by 16 yields 81 175 . This corresponds to