The shortest wavelength of the Lyman series in the hydrogen spectrum is 912 Å.
Understanding the Lyman Series
The Lyman series is a specific collection of spectral lines emitted by the hydrogen atom. These lines are produced when an electron transitions from a higher energy level (represented by n > 1) down to the first, or ground, energy level (n=1). Because these transitions involve a return to the lowest possible energy state, the photons released are typically very energetic, placing the Lyman series in the ultraviolet (UV) region of the electromagnetic spectrum.
Calculating the Shortest Wavelength
The wavelength of light emitted during electron transitions in the hydrogen atom can be determined using the Rydberg formula. For the Lyman series, the final energy level for the electron is always n = 1. The shortest wavelength corresponds to the highest energy transition. This occurs when an electron falls from an infinitely high energy level (n = ∞) to the ground state (n=1).
When an electron undergoes this specific transition from n = ∞ to n = 1, the energy difference is at its maximum. This maximal energy release results in the emission of the highest energy photon possible for the series, which in turn corresponds to the shortest possible wavelength. This fundamental transition yields the precise wavelength of 912 Å.
Significance of the 912 Å Wavelength
This particular wavelength, often referred to as the Lyman limit, holds significant importance in physics and astrophysics:
- Ionization Energy: It directly represents the minimum energy required to ionize a hydrogen atom from its ground state. Any photon with a wavelength shorter than 912 Å (or, equivalently, higher energy) possesses enough energy to remove the electron from a hydrogen atom.
- Astrophysical Applications:
- Astronomers utilize the Lyman limit and other lines from the Lyman series to study distant celestial objects, such as quasars and the intergalactic medium. The absorption or emission of these photons by neutral hydrogen provides crucial insights into the composition and structure of cosmic environments.
- The observation that light with wavelengths shorter than 912 Å is often absent from the spectra of distant astrophysical sources is a strong indicator of intervening neutral hydrogen gas, which absorbs these high-energy photons.
- Fundamental Constant: The value of 912 Å is derived from fundamental physical constants and serves as a key experimental validation of quantum mechanics principles applied to atomic structure.
Hydrogen Spectral Series Overview
To provide context for the Lyman series, here's a brief overview of the principal spectral series for hydrogen, categorized by their final energy levels and the type of light they emit:
| Series Name | Final Energy Level ($n_f$) | Spectral Region | Example Transition ($n_i \to n_f$) |
|---|---|---|---|
| Lyman | 1 | Ultraviolet | $2 \to 1, 3 \to 1, \dots, \infty \to 1$ |
| Balmer | 2 | Visible | $3 \to 2, 4 \to 2, \dots, \infty \to 2$ |
| Paschen | 3 | Infrared | $4 \to 3, 5 \to 3, \dots, \infty \to 3$ |
| Brackett | 4 | Infrared | $5 \to 4, 6 \to 4, \dots, \infty \to 4$ |
This table highlights that the Lyman series involves electron transitions to the lowest energy level, accounting for its characteristic high-energy photons and shortest wavelengths.
