The Lyman series of emission lines of the hydrogen atom are those for which nf = 1. (b) Calculate the wavelengths of the first three lines in the Lyman series—those for which ni = 2, 3, and 4.

Verified step by step guidance

1. Identify the formula to calculate the wavelength of the emission lines in the hydrogen spectrum, which is given by the Rydberg formula: \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \), where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant (approximately 1.097 x 10^7 m^-1), \( n_f \) is the final energy level, and \( n_i \) is the initial energy level.
2. Set \( n_f = 1 \) for the Lyman series as given in the problem statement.
3. Calculate the wavelength for the first line in the series where \( n_i = 2 \). Plug these values into the Rydberg formula and solve for \( \lambda \).
4. Repeat the calculation for the second line in the series where \( n_i = 3 \).
5. Repeat the calculation for the third line in the series where \( n_i = 4 \).

Key Concepts

Lyman Series

The Lyman series refers to a set of spectral lines corresponding to transitions of electrons in a hydrogen atom from higher energy levels (ni) to the lowest energy level (nf = 1). These transitions result in the emission of ultraviolet light, and the wavelengths of these lines can be calculated using the Rydberg formula.

Rydberg Formula

The Rydberg formula is an equation used to predict the wavelengths of spectral lines in hydrogen and other hydrogen-like atoms. It is expressed as 1/λ = R_H (1/nf^2 - 1/ni^2), where R_H is the Rydberg constant, λ is the wavelength, nf is the final energy level, and ni is the initial energy level. This formula is essential for calculating the wavelengths of the Lyman series.

Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, which are quantized. The energy levels are defined by the principal quantum number n, where n = 1 is the ground state. When an electron transitions between these levels, energy is emitted or absorbed in the form of photons, with the energy difference corresponding to the wavelength of the emitted light.