Doubly ionized lithium (Li^2+) is a singleelectron species, like hydrogen except for the greater

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Doubly ionized lithium (Li^2+) is a singleelectron species,...

Doubly ionized lithium $\left(\mathrm{Li}^{2+}\right)$ is a singleelectron species, like hydrogen except for the greater charge on its nucleus. That results in a Rydberg constant that's greater than hydrogen's by a factor of the square of the ratio of the nuclear charges. Find the wavelength of the spectral line associated with the transition from $n=3$ to $n=2$ in doubly ionized lithium.

Key Concepts

Hydrogen-like Atoms

These are atoms or ions that have only one electron orbiting the nucleus, making their energy levels and spectral lines similar in pattern to hydrogen. The simplified structure allows the use of models like the Bohr model or quantum mechanical solutions tailored for single electron systems.

Rydberg Formula

The Rydberg formula is an equation that predicts the wavelengths of spectral lines emitted or absorbed by electrons transitioning between energy levels in a hydrogen-like atom. It relates the inverse wavelength (or wavenumber) to the difference between the reciprocals of the squares of the principal quantum numbers, scaled by the Rydberg constant.

Energy Level Transitions

This concept deals with the movement of an electron between different quantized energy states in an atom. When an electron transitions from a higher energy level to a lower one, it emits a photon with energy equal to the difference between the levels, and hence, a specific wavelength. This fundamental process underpins the formation of spectral lines.

Nuclear Charge Effects

In hydrogen-like ions, the effective nuclear charge plays a critical role in determining the energy levels of the electron. A higher nuclear charge increases the attraction between the nucleus and the electron, resulting in lower energy states and, consequently, shifts in the spectral lines. This effect is incorporated into the Rydberg constant, which scales with the square of the nuclear charge.

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