Bohr model: interactive 3D Bohr models of all 118 elements

Bohr atom basics: nucleus, orbits and postulates

In the Bohr atom, a small, positively charged nucleus sits at the center and holds nearly all of the atom's mass. The nucleus contains protons, whose positive charge attracts the negatively charged electrons, and the electrons circle it the way planets circle the sun, which is why this picture is also called the planetary model or the solar system model. The difference is the force. Gravity keeps planets in orbit, while electrostatic force between opposite charges holds each electron in its orbit around the nucleus.

Every particle in this picture has a role. Each of them carries one unit of positive charge, and a neutral hydrogen atom has exactly one proton and one electron, so the charges cancel. A proton is about 1,836 times heavier than an electron, so the mass of the atom is almost entirely in the nucleus, and the light electrons are the parts that move. Bohr treated electrons as tiny charged particles following circular paths, and he added a few rules to make those paths stable.

The Bohr model rests on four postulates, and each one breaks a rule of classical physics:

1. Electrons move in circular orbits around the nucleus, and in those orbits they do not radiate energy.
2. Only certain orbits are allowed, and each has a fixed energy.
3. Electrons change orbit only by absorbing or emitting a photon whose energy equals the gap between two orbits.
4. The angular momentum of each electron in an allowed orbit is a whole-number multiple of Planck's constant divided by two pi.

Stationary states and quantized orbits

Bohr called the allowed orbits stationary states. In a stationary state the electron holds constant energy, so the classical rule that an accelerating charge must give off radiation is set aside. Because angular momentum is quantized, the orbits are discrete: electrons may occupy orbit one, two or three, but never an orbit in between. The rule can be written as:

$$L = m_e v r = n\,\frac{h}{2\pi}$$

Here \(L\) is the angular momentum, \(m_e\) the electron mass, \(v\) its speed, \(r\) the orbit radius and \(h\) Planck's constant. Fixing the angular momentum in whole steps is what fixes the allowed radii. Later wave ideas made the same point by treating each electron as a standing wave that must fit a whole number of wavelengths around the orbit, and that picture also explains why angular momentum comes in steps.

The ladder analogy for energy steps

A ladder explains the rule well. Climbing up stores potential energy, climbing down releases it, and you can stand only on a rung. Electrons work the same way: moving to a higher orbit needs an energy input, and falling back to a lower orbit returns that energy as a glow of a single color. In chemistry classes this comparison is often the first step toward understanding why atoms absorb and emit only specific colors.

Before Bohr: plum pudding and Rutherford atomic structure

Until the early twentieth century, atomic structure was mostly guesswork. Several planetary models had been suggested, but they ran into a serious problem. In 1897 Joseph Larmor showed that an accelerating charge should radiate energy, and electrons on circular paths are always accelerating. By classical theory, such electrons would spiral into the nucleus in a tiny fraction of a second. Any new theory had to explain why real atoms do not collapse, and why the electrons stay put.

Thomson's plum pudding model

J. J. Thomson, who discovered the electron, pictured the atom as a sphere of positive charge with electrons embedded in it, which is the plum pudding model. It was stable on paper, but Thomson later learned that an atom holds far fewer electrons than he first assumed, and the model could not account for the sharp lines in atomic spectra. His student Bohr spent time at Cambridge studying it before he left for Manchester.

Rutherford's nuclear model and its flaw

In 1911 Ernest Rutherford explained large-angle alpha scattering by placing almost all the mass in a compact, charged nucleus. The Rutherford model was a great step forward, yet it left the electrons unexplained. Classical theory predicted they would radiate away their energy, and the model said nothing about the colors atoms emit. Bohr's contribution was to repair exactly this gap. This experiment-driven history is why Thomson and Rutherford are always mentioned in the same breath as Bohr.

Atomic spectra and Balmer's clue

When hydrogen gas is excited, it glows in a handful of sharp colors rather than a smooth rainbow. These emission spectra had been catalogued by Johann Balmer and Johannes Rydberg, who found simple number patterns in them without knowing why they worked. Any explanation of these colors had to reproduce those patterns, and hydrogen became the test case for the new theory. The Bohr model was built to explain exactly these lines, which is why Bohr's 1913 paper starts from the hydrogen spectrum.

Niels Bohr and Bohr's atomic model of 1913

Born in Copenhagen in 1885, Niels Bohr earned his doctorate in 1911 and worked with Thomson at Cambridge before moving to Rutherford's laboratory in Manchester. On March 6, 1913, he sent Rutherford a paper that tied the nuclear atom to the hydrogen spectrum. This is Bohr's atomic model, sometimes called the Rutherford–Bohr model because it keeps Rutherford's nucleus and adds quantum rules for the electron. The model of the atom he sketched was simple enough to draw on a blackboard in a minute.

Related quantum ideas had appeared earlier through Arthur Haas and John Nicholson, who also tried to attach Planck's constant to atomic structure. The Bohr model succeeded because it did not stop at a concept: it produced numbers for hydrogen that matched experiment. In 1922 the work earned Niels Bohr the Nobel Prize in Physics. This early hydrogen work remains his best-known contribution to physics.

Planck's quanta and the old quantum theory

Max Planck had shown in 1900 that energy is exchanged in discrete packets, and Albert Einstein extended the idea to radiation. Bohr applied the same logic to the atom. The period between Planck's discovery and the arrival of modern quantum mechanics in the mid-1920s is called the old quantum theory, a key chapter in atomic physics, and the Bohr model is its best-known product. The link to frequency is simple: the energy of one quantum equals Planck's constant times the frequency of the radiation, so a higher frequency means a larger packet of energy.

Electron energy levels in the Bohr model of the hydrogen atom

The Bohr model of the hydrogen atom, or of any hydrogen-like ion with a single electron, assigns every orbit a whole number n. Orbits farther from the nucleus carry more energy, and electrons are bound more weakly there. Bohr found the radii by balancing the electrostatic pull on the electron against the force needed to keep it moving in a circle, then applying the angular momentum rule. The result is a short set of formulas for the energy levels and orbit radii of hydrogen:

$$E_n = -\frac{13.6\ \text{eV}}{n^{2}} \qquad r_n = n^{2}\, a_0$$

Here \(a_0\) is the Bohr radius, about 0.0529 nanometers, and the minus sign means the electron is bound to the nucleus. For a hydrogen-like ion with nuclear charge \(Z\), the energy scales as \(Z^{2}\), so a helium ion with one electron has a ground-state energy near -54.4 eV. The table below lists the first four electron shells of hydrogen.

Notice how the radius grows as the square of \(n\) while the gaps between energy levels shrink. The orbits crowd together at higher levels, and the energy approaches zero, the point where the electron is free of the atom. A hydrogen atom therefore has a ladder of energy levels with wide steps near the nucleus and tiny steps far from it.

When an electron drops from a higher level to a lower one, the atom emits a photon. When the electron absorbs energy of exactly the right size, it jumps up. Absorption and emission use the same energy gaps, so the lines an atom absorbed match the lines it emitted, and the gaps are the same in both directions. Light or other radiation carries away the energy difference as electromagnetic energy, and a transition between two levels always gives the same wavelength. Electrons never stay in between, which is why the light comes in lines instead of a continuous band.

Principal quantum number and ground state

The whole number n is the principal quantum number. The lowest value, n = 1, is the ground state, the orbit where the hydrogen electron has the least energy. An atom left alone settles there. Larger values describe excited states that last only a short time before the electron falls back, releasing radiation. In modern quantum mechanics it survives as the label for an electron shell, which is one of the model's lasting legacies.

Bohr's model and the Rydberg formula for spectral lines

The biggest success of Bohr's model was explaining the Rydberg formula, an empirical rule for hydrogen's spectral lines that had no theoretical basis for decades. Setting the energy of the emitted quantum equal to the difference between two levels gives:

$$\frac{1}{\lambda} = R\left(\frac{1}{n_f^{2}} - \frac{1}{n_i^{2}}\right)$$

Here \(\lambda\) is the wavelength, \(R\) is the Rydberg constant (about \(1.097 \times 10^{7}\ \text{m}^{-1}\)), and \(n_i\) and \(n_f\) are the starting and ending levels. Bohr derived \(R\) from the mass and charge of the electron, Planck's constant and the speed of light, so a number that had only been measured was now predicted from basic constants. That match is what convinced many physicists to take the new quantum theory seriously.

The lines are grouped into series by the level the electron lands on. Ending on n = 1 gives ultraviolet radiation, ending on n = 2 gives visible color, and ending on n = 3 or higher gives infrared. Together the series account for the whole hydrogen spectrum that Balmer and Rydberg had measured.

Balmer series worked example

Take an electron falling from n = 4 to n = 2, which belongs to the Balmer series. The energy change is -3.40 eV minus (-0.85 eV), so the emitted energy is about 2.55 eV. Dividing 1,240 eV·nm by that energy gives roughly 486 nanometers, a blue-green line. The Rydberg formula agrees: \(1/\lambda = 1.097 \times 10^{7} \times (1/4 - 1/16)\), which yields 486 nm, the same value. A drop from n = 5 to n = 2 gives about 434 nm, a violet line.

Bohr's model in practice: predicting a lithium ion line

Say a lab group has a spectrograph reading from a lithium plasma, and a faint line near 73 nanometers has shown up. Before anyone blames a contaminant, I would test whether it fits a one-electron lithium ion, Li²⁺, because that ion is the one case besides hydrogen where Bohr's approach should work. The question is which transition produces it, and whether the answer tells the group what to do with the instrument.

I enter the nuclear charge first: Z = 3 for lithium. The ending level comes next, n = 2, then the starting level, n = 3. The energy gap is \(13.606\ \text{eV} \times 3^{2} \times (1/4 - 1/9)\), which is 122.45 eV times 0.1389, or 17.01 eV. Dividing 1,239.84 eV·nm by 17.01 eV gives 72.90 nm. As a cross-check I run the Rydberg formula with the constant multiplied by \(Z^{2}\), and it returns 72.90 nm too. The line fits.

The reading matters because 72.9 nm is extreme ultraviolet, far below the 380 nm edge of visible light. Ordinary glass optics absorb it, so the group needs a vacuum spectrograph, and that explains why the line looks so faint on the air-path instrument.

The sensitivity rerun shows what dominates: the number to watch is Z. Dropping it from 3 to 2 more than doubles the wavelength, because the energy scales with \(Z^{2}\), so a mix-up of ions would shift the line by 91 nm, far outside any calibration error. Changing the starting level moves it by about 19 nm.

One caveat is worth knowing. The calculation treats the nucleus as infinitely heavy. A real lithium-7 nucleus moves slightly, which lengthens the wavelength by about 0.007 nm, to roughly 72.907 nm. That is negligible for identifying the line but visible in a precise measurement. And the same arithmetic must not be trusted for neutral lithium: with three electrons repelling each other, the Bohr model gives the wrong answer there, and a lookup in a spectral database is the right next step.

Shell model: extending the solar system model to heavier atoms

Bohr extended his approach beyond hydrogen by treating orbits as shells that hold a limited number of electrons. In this shell picture, inner shells fill first, and the outermost electrons set chemical behavior. It is a rough early version of the valence shell idea that chemistry courses still use. Heavier atoms such as sodium or argon can be drawn with two, eight and then more electrons per shell, and the arrangement explains why elements fall into repeating groups. Many students first meet the Bohr model through these simple shell diagrams.

The same thinking explains X-ray lines. Henry Moseley showed that the X-ray frequency of an element rises with its atomic number, and Bohr's formulas, adjusted for the screening of the inner electrons, reproduced that trend. The model even offered a simple view of chemical bonds, with shared electrons tying atoms together. These were useful ideas, but they remain approximations for anything bigger than a single electron.

Shortcomings: why the model of the atom was replaced

For all its success, the model of the atom Bohr proposed had real limits. It matched the hydrogen atom and hydrogen-like ions, but it gave poor spectra for atoms with more than one electron, because it ignores the repulsion between electrons. It could not explain why some spectral lines are brighter than others, or why a strong magnetic field splits lines. Refinements by Arnold Sommerfeld added elliptical orbits, yet each fix felt like an added patch rather than a principle of the theory.

The deeper problem was conceptual. The Bohr model gives each electron a definite radius and speed at once, which the uncertainty principle forbids. By the mid-1920s, quantum mechanics replaced orbits with probability clouds called orbitals, and the new quantum mechanics treats electrons as both a particle and a wave. The older theory was left as a stepping stone. This is why textbooks call it an obsolete model of the atom even while they keep teaching it: it gives correct answers for hydrogen with very little math.

• Right for: hydrogen and one-electron ions, the main spectral series, and the size of the atom.
• Wrong for: multi-electron atoms, line intensities, and the true shape of electron clouds.

Why the planetary image persists

Even after it was replaced, the picture of electrons circling a nucleus became a cultural symbol of the atom, appearing on logos and in television shows. Its appeal is simple: anyone can sketch it, and it carries the central message that atoms have structure and energy comes in steps. Students who master it can move on to quantum mechanics and orbitals with a firm footing in modern physics.