Consider the following three transitions in a hydrogen atom:...

Consider the following three transitions in a hydrogen atom: (A) $n_{1}=5, n_{f}=2$ (B) $n_{1}=7, n_{f}=2 ;(C) n_{1}=7, n_{f}=6 .$ Rank the transitions in order of increasing (a) wavelength and (b) frequency of the emitted photon. Indicate ties where appropriate.

Key Concepts

Hydrogen Atom Energy Levels

The hydrogen atom is one of the simplest atomic systems, featuring discrete energy levels that electrons occupy. These levels are quantified by the principal quantum number, and the energy associated with each level decreases (becomes more negative) as the quantum number increases. This concept is fundamental in atomic physics and quantum mechanics, as transitions between these levels lead to discrete spectral lines.

Quantum Electron Transitions

Whenever an electron in an atom transitions between energy levels, it either absorbs or emits a photon whose energy corresponds exactly to the difference between the two levels. In the context of the hydrogen atom, the magnitude of the energy difference is governed by the inverse square of the principal quantum numbers involved, making it essential for calculating the properties of the emitted photon.

Photon Energy, Wavelength, and Frequency Relationship

The energy of a photon emitted or absorbed during an electron transition is directly related to its frequency and inversely related to its wavelength, as described by the Planck-Einstein relation (E = hf) and the speed of light (c = ?f). This relationship is crucial for understanding how energy differences in atomic transitions are manifested as different spectral lines in the electromagnetic spectrum.

Rydberg Formula

The Rydberg formula provides a mathematical relationship that predicts the wavelengths of the spectral lines of hydrogen by relating the inverse wavelength to the difference in the reciprocals of the squares of the principal quantum numbers. This formula is instrumental in linking the quantum transitions in the hydrogen atom to observable spectral properties such as wavelength and frequency.

Transcript

Please give Ace some feedback