Consider the emission spectrum of singly ionized helium (He...

Consider the emission spectrum of singly ionized helium (He $^{+}$ ). Find the longest three wavelengths for the series in which the electron makes a transition from a
higher excited state to the first excited state (not the ground state).

Key Concepts

Series of Electron Transitions

A transition series refers to a group of related electron transitions that all end at the same energy level. In this context, transitions ending at the first excited state (n = 2) form such a series. Understanding the ordering of the wavelengths in a series, where the smallest energy difference yields the longest wavelength, is crucial for analyzing and identifying spectral lines.

Electron Transitions

When an electron moves from a higher energy level to a lower one, it emits a photon whose energy is exactly equal to the energy difference between these levels. In the context of emission spectra, each such transition produces a spectral line, and the specific transition determines the wavelength observed.

Rydberg Formula

The Rydberg formula mathematically relates the wavelengths of spectral lines to the principal quantum numbers of the involved energy levels. In hydrogen-like ions, the formula is modified by the square of the nuclear charge. This formula is key to calculating and predicting the wavelengths of the emitted photons during electron transitions.

Hydrogen-like Ions

Hydrogen-like ions are atoms that have only one electron, such as the singly ionized helium (He?). Their energy levels can be modeled similarly to hydrogen, but the higher nuclear charge leads to energy levels that are more tightly bound. This foundational concept is essential for understanding the quantized nature of atomic spectra in such systems.

Quantum Energy Levels

Electrons in atoms occupy discrete energy levels characterized by principal quantum numbers. The differences between these energy levels determine the energy, and thus the wavelength, of photons emitted or absorbed during electron transitions. This concept underpins the entire field of atomic spectroscopy.

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