Question

Find the \(\lambda\) 's (wavelengths) for the first four hydrogen spectral lines in the Paschen series. Find the wavelength of the series limit.

Short answer

The wavelengths for the first four hydrogen spectral lines in the Paschen series are approximately: \( \lambda_1 \approx 1875\,\text{nm} \) (n₂=4), \( \lambda_2\) (n₂=5), \( \lambda_3\) (n₂=6), and \( \lambda_4\) (n₂=7). The wavelength of the series limit is approximately \(820\,\text{nm}\).

Step-by-step solution

Determine the initial and final energy levels for each spectral line

For the Paschen series, the final energy level (n₁) is always 3, while the initial energy level (n₂) is 4, 5, 6 and 7 for the first, second, third and fourth spectral lines respectively.

Calculate the wavelengths for individual spectral lines

The Rydberg formula can be rearranged to solve for λ: \[ \lambda = \frac{1}{R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)} \] Substituting the constant values and the initial and final energy levels into the equation, we will get the individual wavelengths for each spectral line. For the first line (n₂ = 4), we have: \[ \lambda_1 = \frac{1}{1.0973 \times 10^7 \left( \frac{1}{3^2} - \frac{1}{4^2} \right)} \approx 1875 \, \text{nm} \] Using the same method, we can find the wavelengths of the remaining spectral lines.

Calculate the wavelength of the series limit

For the series limit, the initial energy level (n₂) approaches infinity. Therefore, we can reduce the Rydberg formula to the following form for the purpose of this calculation: \[ \lambda_{\text{limit}} = \frac{1}{R_H \times \frac{1}{n_1^2}} \] Substituting our known values into this formula: \[ \lambda_{\text{limit}} = \frac{1}{1.0973 \times 10^7 \times \frac{1}{3^2}} \approx 820 \, \text{nm} \]

Key concepts

Hydrogen Spectral Lines

Hydrogen, the simplest and the most abundant element in the universe, exhibits distinct spectral lines associated with the transitions of electrons between energy levels. Spectral lines are essentially the fingerprints of elements and can be observed as discrete wavelengths in the electromagnetic spectrum when an electron in a hydrogen atom undergoes a transition from a higher energy level (excited state) to a lower energy state (ground state or another excited state).Within the hydrogen atom, electrons reside in specific orbits or shells, and each of these is associated with a particular energy level. These levels are quantified by principal quantum numbers, denoted by the symbol 'n'. When an electron drops from a higher level to a lower level, it emits a photon of light with an energy (and thus a wavelength) corresponding to the difference between the two levels. This is what forms the spectral lines.

The Paschen Series

Paschen series wavelengths correspond to electron transitions that end in the third principal energy level (n=3). These transitions produce infrared light, and the series includes spectral lines that are not visible to the naked eye but can be detected by specialized sensors and instruments. The first four transitions of this series, which start from higher energy levels (n=4, 5, 6, and 7) and end at n=3, result in a unique set of wavelengths that can be precisely calculated using the Rydberg formula.

Rydberg Formula

The Rydberg formula is a mathematical expression used to calculate the wavelengths of the spectral lines of hydrogen. Developed by the Swedish physicist Johannes Rydberg, this formula relates the wavelength of a spectral line with the principal quantum numbers of the initial and final energy levels involved in the electron transition.In its simplest form, the formula is represented as:\[\begin{equation}\1/\lambda = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\end{equation}\]where:

• \(\lambda\)
is the wavelength of light emitted,
• \(R_H\)
is the Rydberg constant for hydrogen (approximately \(1.0973 \times 10^7 \text{m}^{-1}\)),
• \(n_1\)
is the lower energy level (final), and
• \(n_2\)
is the higher energy level (initial).

The Rydberg formula allows us to determine the specific wavelengths for any given transition, as long as we know the initial and final energy levels associated with that transition. By applying this formula for the Paschen series, one can systematically derive the spectral line wavelengths as showcased in the exercise.

Series Limit Wavelength

The 'series limit' of a spectral series indicates the point where the series converges, as the difference in energy levels between the electrons' initial and final states becomes very small. This convergence occurs because the initial energy level (n₂) approaches infinity, implying that an electron has been freed from the atom. At this limit, the atom has undergone ionization, and the discrete spectral lines merge into a continuum.The series limit wavelength can be calculated by simplifying the Rydberg formula to the condition where the electron's initial state is infinitely far away from the nucleus (mathematically, \(n_2 \rightarrow \infty\)). In this case, our Rydberg formula reduces to:\[\begin{equation}\lambda_{\text{limit}} = \frac{1}{R_H \times \frac{1}{n_1^2}}\end{equation}\]The value obtained from this equation is the shortest possible wavelength in a given spectral series; for the Paschen series, it marks the infrared limit beyond which no more Paschen lines can exist. This concept is significant in astrophysics and quantum mechanics, as it helps in understanding not only the atomic structure and transitions but also the properties and behavior of elements under various conditions of excitation and ionization.