Question

The wavenumbers of the first three emission lines of Lyman series of hydrogen spectrum are respectively \( \nu_1, \nu_2, \nu_3 \). Similarly, the wavenumbers of first three emission lines of Balmer series of hydrogen spectrum are \( \nu_4, \nu_5, \nu_6 \), respectively. Identify the correct relationship:

• A. \( \nu_1 = 5 \nu_4 \)
• B. \( \nu_2 = 5 \nu_5 \)
• C. \( \nu_3 = 5 \nu_6 \)
• D. \( \nu_1 = 5 \nu_2 \)

Hint:

The wavenumbers of spectral lines are inversely proportional to the square of the energy levels involved in the transitions.

Answer 1

The Correct Option is B

Step 1:
The wavenumber \( \tilde{\nu} \) of a photon emitted by an electron transitioning between energy levels is given by the Rydberg formula: \[ \tilde{\nu} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \( R_H \) is the Rydberg constant, \( n_1 \) and \( n_2 \) are the principal quantum numbers of the two levels.
Step 2:
For the Lyman series, transitions occur to the ground state (\( n_1 = 1 \)), whereas for the Balmer series, transitions occur to the second energy level (\( n_1 = 2 \)). From the given options, the correct relation is between \( \nu_2 \) of the Lyman series and \( \nu_5 \) of the Balmer series, as they correspond to specific transitions. Thus, the correct answer is option (2).