Hard

Bohr's Hydrogen Atom & Atomic Spectra

Simple Explanation

Bohr's model of the hydrogen atom proposes that electrons can only orbit the nucleus at specific, fixed (quantised) energy levels — not at any arbitrary distance. When an electron jumps between levels, it absorbs or emits a photon with an energy exactly equal to the difference between those levels, producing the characteristic line spectra observed for hydrogen.

Why Do We Need It?

Bohr's model was a landmark step toward modern quantum mechanics, and directly explains why atoms emit and absorb light only at specific, discrete wavelengths (their atomic 'fingerprint'), rather than a continuous range.

Formula

Bohr Model Energy Levels (Hydrogen)

Eₙ = −13.6 / n² eV

In Bohr's model of the hydrogen atom, electrons can only occupy specific, quantised energy levels, numbered by n = 1, 2, 3, ... — this formula gives the energy of each allowed level.

Eₙ: — Energy of the electron in level n, in electron-volts (eV)
n: — Principal quantum number — a positive integer (1, 2, 3, ...) labelling the energy level

When to use it: Use to find the energy of an electron in a specific level of a hydrogen atom, or to find the energy released/absorbed when an electron jumps between levels.

Worked Example

Finding the energy released in an electron transition

An electron in a hydrogen atom drops from level n=3 to level n=1. Find the energy of the photon released.

Why Does This Work?

Because only specific, quantised energy levels are allowed, an electron can only jump between them by absorbing or emitting a photon with EXACTLY the right energy to match the gap between two levels — since these gaps are fixed and specific to hydrogen's structure, only certain specific photon energies (and therefore specific wavelengths of light) are ever absorbed or emitted, producing hydrogen's characteristic line spectrum rather than a continuous rainbow.

Real-Life Example

Identifying elements in stars using their spectra

Astronomers determine what elements distant stars are made of by analysing the specific wavelengths of light they emit or absorb.

Since each element has its own unique set of quantised energy levels (and therefore its own unique spectral 'fingerprint' of specific wavelengths), astronomers can identify elements in stars millions of light-years away simply by matching observed spectral lines to those known from laboratory measurements of each element.

Practice

Find the energy of an electron in the n=2 level of hydrogen using Eₙ = −13.6/n² eV.

Hard

Common mistake

Forgetting the negative sign in the energy formula — energies are conventionally negative because they are measured relative to a free (unbound) electron at zero energy; a MORE negative value means the electron is more tightly bound (lower energy level, e.g. n=1 is the most tightly bound).

Quick Review

Eₙ = −13.6/n² eV gives the energy of each allowed hydrogen electron level.
Electrons emit/absorb photons only when jumping between specific, quantised levels.
This produces atomic line spectra — each element has a unique spectral fingerprint.