The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (b) Calculate the wavelengths of the first three lines in the Paschen series—those for which ni = 4, 5, and 6.

Verified step by step guidance

1. Identify the formula to use for calculating the wavelengths of the emission lines in the hydrogen atom. The Rydberg formula is appropriate here: \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \), 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 \( n_f = 3 \) for the Paschen series as given in the problem statement.
3. Calculate the wavelength for the first line in the series where \( n_i = 4 \). Plug these values into the Rydberg formula and solve for \( \lambda \).
4. Repeat the calculation for the second line where \( n_i = 5 \). Again, use the Rydberg formula with these values to find \( \lambda \).
5. Calculate the wavelength for the third line where \( n_i = 6 \) using the same method.

Key Concepts

Emission Spectra

Emission spectra are produced when electrons in an atom transition from a higher energy level to a lower one, releasing energy in the form of light. Each transition corresponds to a specific wavelength, resulting in distinct lines in the spectrum. For hydrogen, these lines are categorized into series based on the final energy level of the electron, such as the Paschen series, which ends at n=3.

Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of spectral lines in hydrogen and other hydrogen-like atoms. It is expressed as 1/λ = R_H (1/n_f^2 - 1/n_i^2), where λ is the wavelength, R_H is the Rydberg constant, n_f is the final energy level, and n_i is the initial energy level. This formula allows for the determination of wavelengths for transitions between specific energy levels.

Quantum Energy Levels

Quantum energy levels refer to the discrete energy states that electrons can occupy in an atom. In hydrogen, these levels are denoted by principal quantum numbers (n), where n=1 is the ground state and higher numbers represent excited states. The differences in energy between these levels dictate the wavelengths of light emitted during electron transitions, forming the basis for the observed emission spectra.