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The Bohr Model

The Bohr Model

11 min read Updated Mar 26, 2026

In 1910, Ernest Rutherford demonstrated that atoms have a dense, positively charged nucleus with electrons surrounding it. But Rutherford’s model had a problem - classical physics predicted that orbiting electrons should continuously radiate energy and spiral into the nucleus within nanoseconds. Atoms should not be stable, yet here we are.

Three years later, Niels Bohr solved this problem by borrowing an idea from Max Planck: energy is quantized. It comes in discrete packets, not a continuous stream.

Planck’s Quantum Theory

Planck proposed that energy emitted as electromagnetic radiation comes in bundles called quanta. The energy of a single quantum is:

Since the speed of light relates frequency and wavelength (c = f x lambda), we can also write:

Bohr’s Key Postulates

Bohr proposed a model for the hydrogen atom with these critical features:

Hydrogen energy levels and the lines they emit

Bohr model
energy (eV) electron free · ionised n = ∞ 0 eV n = 6 -0.38 eV n = 5 -0.54 eV n = 4 -0.85 eV n = 3 -1.51 eV n = 2 -3.40 eV n = 1 -13.60 eV 10.2 eV: the n = 1 → 2 gap alone That single step is three quarters of the whole 13.6 eV it takes to strip the electron off. Every rung above it shares the last quarter, which is why they pile up. Lyman lands on n = 1 · ultraviolet Balmer lands on n = 2 · visible Paschen lands on n = 3 · infrared What the eye actually sees A hot hydrogen tube emits only these four visible lines, on an otherwise black field. Hα 656 nm Hβ 486 nm Hγ 434 nm Hδ 410 nm 400 nm wavelength 700 nm line drop ΔE λ Hα 3 → 2 1.89 eV 656 nm Hβ 4 → 2 2.55 eV 486 nm Hγ 5 → 2 2.86 eV 434 nm Hδ 6 → 2 3.02 eV 410 nm Absorption is this in reverse Shine white light through cold hydrogen and the same four wavelengths go missing: dark lines on a continuous background, at exactly the positions of the bright emission lines above.
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Why the rungs crowdEnergy goes as 1/n², so the n = 1 to n = 2 gap is 10.2 eV while n = 5 to n = 6 is under 0.2 eV. Every question that asks which transition emits the shortest wavelength is really asking which gap is biggest, and the biggest gaps are always at the bottom.

Reading a series off the ladderA series is named for where the electron lands, not where it starts. Landing on n = 1 means falling the whole way, so Lyman lines are ultraviolet. Landing on n = 2 gives the four visible Balmer lines. Landing on n = 3 gives infrared.

The sign conventionEnergies are negative because the electron is bound: zero is the free electron at n = ∞. Absorbing a photon moves the electron up toward zero, and it takes the full 13.6 eV to ionise hydrogen from the ground state.

The ladder and the spectrum are the same fact. Each coloured line on the right is drawn at the wavelength its arrow on the left produces, through λ = hc/ΔE. A bigger drop makes a bluer photon, which is why H-α from n = 3 is red and H-δ from n = 6 is violet.
  1. Electrons orbit the nucleus in circular paths at fixed distances - called orbits or energy levels.
  2. Each orbit has a specific energy that depends on the principal quantum number n.
  3. Electrons do not radiate energy while in a stable orbit - this was the radical break from classical physics.
  4. Electrons can only jump between orbits by absorbing or emitting exactly the right amount of energy.

Energy of an Electron in the Bohr Model

The energy of an electron in the nth orbit of hydrogen is:

Key insights from this equation:

  • The ground state (n = 1) has the most negative energy - the electron is most tightly bound
  • As n increases, energy becomes less negative (increases toward zero) - the electron is less tightly bound
  • At n = infinity, E = 0: the electron has escaped the atom completely (ionization)
  • The energy levels get closer together as n increases (the gap between n = 1 and n = 2 is much larger than between n = 5 and n = 6)

Ground State vs. Excited State

The ground state is when all electrons occupy the lowest possible energy levels. This is the default state for atoms at room temperature.

An excited state occurs when one or more electrons have absorbed energy and jumped to a higher energy level. Excited states are unstable - the electron will quickly fall back to its ground state, releasing the excess energy as a photon.

Energy of a Transition

When an electron transitions between energy levels, the energy of the emitted or absorbed photon equals the difference between the two levels:

Limitations of the Bohr Model

The Bohr model works beautifully for hydrogen and other one-electron systems (He⁺, Li²⁺), but it fails for atoms with more than one electron. Why? Because it ignores the repulsion between multiple electrons orbiting the same nucleus. The quantum mechanical model, which we will cover next, handles multi-electron atoms correctly.

Despite its limitations, the Bohr model remains essential for understanding quantized energy levels, emission spectra, and the fundamental idea that electrons can only exist at specific energies - not anywhere they please.

An electron in a hydrogen atom jumps from n = 2 to n = 4. Is energy absorbed or emitted? Is this an excited state transition?
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Energy is absorbed. The electron is moving to a higher energy level (n = 2 to n = 4), so it must absorb a photon with energy exactly equal to the difference between these levels. Yes, the atom is now in an excited state because the electron is no longer in its lowest possible energy level.

Why does the Bohr model fail for multi-electron atoms?
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It ignores electron-electron repulsion. The Bohr model only accounts for the attraction between one electron and the nucleus. In atoms with multiple electrons, the repulsion between electrons significantly affects their energies and positions, making the simple circular orbit model inaccurate.

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