A hydrogen-like ion has a nucleus of charge +Z e and a...

A hydrogen-like ion has a nucleus of charge $+Z e$ and a single electron outside this nucleus. The energy levels of these ions are $-Z^{2} R_{\mathrm{H}} / n^{2}$ (where $Z=$ atomic number). Calculate the wavelength of the transition from $n=3$ to $n=2$ for $\mathrm{He}^{+}$, a hydrogen-like ion. In what region of the spectrum does this emission occur?

Key Concepts

Electromagnetic Spectrum Regions

The electromagnetic spectrum is divided into various regions such as infrared, visible, ultraviolet, and others based on wavelength. Determining the region in which a particular transition's wavelength falls is important for understanding the type of radiation involved. For transitions with high energy differences, the resulting wavelengths often fall in the ultraviolet range.

Photon Energy and Wavelength Relation

Photon energy, related to the frequency or wavelength of light, is given by the equation E = hc/?, where h is Planck's constant and c is the speed of light. This relation allows for the conversion of energy differences between electron levels into the corresponding wavelength of the emitted photon, thereby linking atomic transitions to observable spectral lines.

Hydrogen-like Ions

Hydrogen-like ions are atomic species that consist of a nucleus and a single electron. Their energy levels resemble that of the hydrogen atom but are modified by the nuclear charge, which results in energy levels that depend on the square of the charge. This concept is fundamental in atomic physics as it simplifies the calculation of spectral lines by reducing the system to one electron interacting with the nucleus.

Quantized Energy Levels

In atomic physics, electrons occupy discrete energy levels. For hydrogen-like ions, these levels are given by a formula that involves the principal quantum number and the nuclear charge. This quantization explains why atoms absorb and emit light only at specific wavelengths corresponding to transitions between these levels.

Rydberg Constant and Formula

The Rydberg constant is a key physical constant used in the calculation of the wavelengths of spectral lines. The Rydberg formula, which relates the wavelengths of emitted or absorbed light to the electron transitions between energy levels, is central to understanding the spectral emissions of hydrogen-like ions. This formula is modified by the square of the nuclear charge in the case of hydrogen-like ions.

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