Table of Contents
What Is a Hydrogen-Like Atom?
A hydrogen-like atom (or hydrogenic atom) is any atomic species with only one electron orbiting a nucleus with Z protons. This includes neutral hydrogen (Z=1), singly ionized helium He+ (Z=2), doubly ionized lithium Li2+ (Z=3), and so on. The Bohr model provides exact analytical solutions for these systems.
The importance of hydrogen-like atoms in physics cannot be overstated. They serve as the starting point for understanding atomic structure, quantum mechanics, and spectroscopy. The energy levels scale as Z squared, meaning highly ionized atoms have much tighter electron orbits and much higher ionization energies than hydrogen.
Bohr Model Formulas
Energy Levels for Hydrogen (Z=1)
| n | Energy (eV) | Radius (Å) | Name |
|---|---|---|---|
| 1 | -13.60 | 0.529 | Ground state |
| 2 | -3.40 | 2.116 | First excited |
| 3 | -1.51 | 4.761 | Second excited |
| 4 | -0.85 | 8.464 | Third excited |
| ∞ | 0 | ∞ | Ionized |
Spectral Series
- Lyman series: Transitions to n=1 (ultraviolet).
- Balmer series: Transitions to n=2 (visible light).
- Paschen series: Transitions to n=3 (near infrared).
- Brackett series: Transitions to n=4 (infrared).
- Pfund series: Transitions to n=5 (far infrared).
Frequently Asked Questions
Why does the Bohr model work exactly for hydrogen-like atoms?
The Bohr model works because hydrogen-like atoms have only one electron, so there are no electron-electron interactions to complicate the problem. The single electron experiences a pure Coulomb potential from the nucleus, and the Bohr model's quantization condition gives exact energy levels that match the full quantum mechanical solution.
What is the ionization energy of He+?
For He+ (Z=2), the ground state energy is -13.6 x 4 = -54.4 eV. So the ionization energy is 54.4 eV, four times that of hydrogen. This demonstrates the Z-squared scaling of energy levels in hydrogen-like atoms.