What is the ratio of the frequencies of the long wavelength limits of…

The correct answer is Option (1) → 27 : 5

The lines of the hydrogen spectrum follow the Rydberg formula \( \frac{1}{\lambda} = R\left(\frac{1}{n_f^{2}} - \frac{1}{n_i^{2}}\right) \). The long wavelength limit of a series is its longest wavelength, the first line of the series, where \( n_i = n_f + 1 \).

Lyman series (\( n_f = 1,\ n_i = 2 \)):

    \( \frac{1}{\lambda_L} = R\left(1 - \frac{1}{4}\right) = \frac{3R}{4} \Rightarrow \lambda_L = \frac{4}{3R} \)

Balmer series (\( n_f = 2,\ n_i = 3 \)):

    \( \frac{1}{\lambda_B} = R\left(\frac{1}{4} - \frac{1}{9}\right) = \frac{5R}{36} \Rightarrow \lambda_B = \frac{36}{5R} \)

The frequency is \( \nu = c/\lambda \), so the ratio of the frequencies is the inverse of the ratio of the wavelengths:

    \( \displaystyle \frac{\nu_L}{\nu_B} = \frac{\lambda_B}{\lambda_L} = \frac{36}{5R}\times\frac{3R}{4} = \frac{27}{5} \)

So the ratio of the frequencies of the long wavelength limits of the Lyman and Balmer series is \( 27 : 5 \).

Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.

NCERT (Pre-rationalised edition, 2022-23):
Spectral Series – “… Balmer found a simple empirical formula for the observed wavelengths

    \( \displaystyle \frac{1}{\lambda} = R\left(\frac{1}{2^2} - \frac{1}{n^2}\right) \qquad (12.5) \)

where \( \lambda \) is the wavelength, R is a constant called the Rydberg constant, and n may have integral values 3, 4, 5, etc.”

Spectral Series – “… These are known, after their discoverers, as Lyman, Paschen, Brackett, and Pfund series. These are represented by the formulae:

Lyman series:

    \( \displaystyle \frac{1}{\lambda} = R\left(\frac{1}{1^2} - \frac{1}{n^2}\right) \qquad n = 2,3,4\ldots \qquad (12.6) \)”