Hydrogen-Like Ions

Introduction

Hydrogen is not the only one-electron atomic system. A helium nucleus with one remaining electron, He⁺, and a lithium nucleus with one remaining electron, Li²⁺, are hydrogen-like ions. Their stronger nuclear charges change energy scales while preserving the one-electron structure of the idealized model.

Core explanation

A hydrogen-like species has one electron bound to a nucleus of charge +Ze. In a simple infinitely heavy nucleus approximation, its level energies are Eₙ ≈ −13.6 Z²/n² eV. Hydrogen has Z = 1, He⁺ has Z = 2, and Li²⁺ has Z = 3. Increasing Z strengthens attraction and makes bound levels more negative. The energy required to remove a ground-state electron scales approximately as 13.6Z² eV in this model.

For He⁺, the n = 1 level is about −54.4 eV, four times the magnitude of hydrogen's −13.6 eV. Its n = 2 level is about −13.6 eV. A transition from 2 to 1 therefore releases about 40.8 eV, not hydrogen's 10.2 eV for the corresponding indices. Photon frequency is proportionally larger and wavelength shorter. Nuclear charge changes spectral position even when the same n labels appear.

The simple orbital length scale also contracts as Z increases: one electron is drawn more tightly toward a more highly charged nucleus. Thus hydrogen-like ions are useful clean tests of how nuclear attraction affects one-electron quantum states. They must not be confused with neutral helium or lithium, which have two or three electrons and electron-electron repulsion.

The Z² formula is approximate when precise measured values are needed because the nucleus has finite mass. The electron and nucleus both move about their shared center of mass, producing a reduced-mass correction. High-resolution spectra also involve finer effects. Introductory problems generally specify a simple Bohr or hydrogenic model and suitable constants.

The one-electron condition matters more than an ion's charge sign alone. H⁻ contains two electrons and is not hydrogen-like under this one-electron formula. He⁺ contains one electron even though it is positively charged. Count electrons from atomic number and ionic charge before selecting the model.

Step-by-step reasoning

1. Determine Z and the number of bound electrons from the ionic charge. 2. Confirm the species has exactly one electron. 3. Use Eₙ ≈ −13.6Z²/n² eV for stated simple-model calculations. 4. Find transition energy and convert to photon frequency or wavelength if needed.

Visual explanation

Draw hydrogen, He⁺, and Li²⁺ each with one electron cloud. Make the energy-level spacings and typical cloud scale change as nuclear charge grows.

Real-world analogy

A stronger central attraction pulls the same single object into a more tightly bound arrangement. The analogy suggests the trend, although quantum orbitals are probability states rather than literal trajectories.

Real-world example

Spectral lines of He⁺ test one-electron theory at Z = 2. Their wavelengths differ from neutral helium's more complex spectrum because neutral helium has two interacting electrons.

Why?

Why does He⁺ require more ionization energy than H in the simple model? Its doubly charged nucleus attracts the sole electron more strongly, increasing binding-energy magnitude.

Common misconception

“Any ion follows hydrogen's level formula.” The simple hydrogen-like formula applies to exactly one-electron species; multi-electron ions require electron-interaction treatment.

Worked example

Calculate the approximate n = 2 → n = 1 photon for He⁺. With Z = 2, E₁ = −13.6(4) = −54.4 eV and E₂ = −13.6(4)/4 = −13.6 eV. The emitted photon carries 40.8 eV. This is four times the energy of hydrogen's matching transition in the same simplified model.

Quick check

1. Is Li²⁺ hydrogen-like? Answer: Yes. Lithium has Z = 3 and the 2+ ion retains one electron.

Exam focus

Count electrons before using the formula. Apply Z² to energy and energy gaps, then invert that scaling for corresponding photon wavelengths.

Advanced insight

Finite nuclear mass slightly changes the effective particle mass and thus the Rydberg scale. Comparing isotopes of one hydrogen-like ion can reveal this small mass-dependent spectral shift.

Summary

Hydrogen-like ions contain exactly one electron and have approximate level energies proportional to −Z²/n². Greater nuclear charge tightens binding and raises corresponding transition photon energies.

Practice questions

1. What is the simple-model ground-state energy of Li²⁺? Answer: −13.6(3²) ≈ −122.4 eV. 2. Is neutral helium hydrogen-like? Answer: No. It has two electrons that interact with each other. 3. How does a corresponding He⁺ transition photon compare in energy with H? Answer: About four times greater in the simple Z² model.