The Bohr Model and Modern Atomic Theory

The Bohr model, proposed by Niels Bohr in 1913, describes the atom as a nucleus surrounded by electrons that move in fixed circular orbits with specific energies, while the modern quantum mechanical model describes electrons as wave-like and found in regions of probability called orbitals. In brief: in the Bohr model, electrons jump between energy levels by absorbing or emitting photons, which explains the colored lines in the hydrogen spectrum; the modern model replaces orbits with orbitals but keeps the idea of quantized energy levels.

The Bohr model solved the stability problem of the nuclear atom and explained atomic spectra, and it is still used as a first picture of the atom. It works well for hydrogen and similar single-electron ions, but it fails for atoms with more electrons. This article explains the postulates of the Bohr model, the energy levels of hydrogen, how spectral lines arise, why the model was replaced, the main ideas of the modern model, a worked calculation and a short history.

What are the ideas of the Bohr model?

1. Electrons move around the nucleus in circular orbits without radiating energy. Each allowed orbit has a fixed energy and is called an energy level or shell, labeled by the quantum number n = 1, 2, 3, and so on.
2. Only certain orbits are allowed. The energy of the electron is quantized.
3. An electron can jump from one level to another by absorbing or emitting a photon whose energy equals the difference between the two levels: E_photon = |E_final − E_initial| = hf.
4. The lowest level (n = 1) is the ground state, and higher levels are excited states.

These postulates did not follow from classical physics. Bohr introduced them to explain observations, and they were later justified by quantum mechanics.

What are the energy levels of hydrogen?

For hydrogen, Bohr’s model gives the energy of level n as E_n = −13.6 eV ÷ n², where 1 eV = 1.602 × 10⁻¹⁹ J. The negative sign shows that the electron is bound to the nucleus. The energy rises toward zero as n increases, and at n = ∞ the electron is free (ionization).

The ionization energy of hydrogen from the ground state is therefore 13.6 eV, which agrees with experiment.

How does the model explain spectral lines?

When a hydrogen gas in a tube is excited by electricity, it emits light, and a prism separates this light into a few colored lines, not a continuous rainbow. This is an emission spectrum. Each line corresponds to an electron falling from a higher level to a lower level. In the Bohr model, the wavelength follows 1 ÷ λ = R(1 ÷ n_f² − 1 ÷ n_i²), where R = 1.097 × 10⁷ m⁻¹ is the Rydberg constant. The visible lines of hydrogen, known as the Balmer series, come from transitions that end at n = 2.

Worked example: the red line of hydrogen

1. Energy difference. E₃ = −13.6 ÷ 9 = −1.511 eV and E₂ = −13.6 ÷ 4 = −3.40 eV. ΔE = −1.511 − (−3.40) = 1.889 eV.
2. In joules. 1.889 × 1.602 × 10⁻¹⁹ = 3.026 × 10⁻¹⁹ J.
3. Frequency. f = E ÷ h = 3.026 × 10⁻¹⁹ ÷ 6.626 × 10⁻³⁴ = 4.567 × 10¹⁴ Hz.
4. Wavelength. λ = c ÷ f = 2.998 × 10⁸ ÷ 4.567 × 10¹⁴ = 6.564 × 10⁻⁷ m = 656 nm, which is red light.
5. Check with the Rydberg formula. 1 ÷ λ = 1.097 × 10⁷ × (1 ÷ 4 − 1 ÷ 9) = 1.097 × 10⁷ × 0.13889 = 1.5236 × 10⁶ m⁻¹, so λ = 6.563 × 10⁻⁷ m. The two methods agree.

Why was the Bohr model replaced?

• It works only for hydrogen-like atoms with one electron. It cannot predict the spectra of helium or heavier atoms.
• It does not explain why some lines are brighter than others, or the splitting of lines in a magnetic field or at high resolution.
• It treats the electron as a particle on a defined path, which conflicts with the wave nature of electrons.
• It does not explain chemical bonding.

What is the modern atomic theory?

In the 1920s, Louis de Broglie suggested that electrons have wave properties, and in 1926 Erwin Schrödinger developed an equation for the electron’s wave function. Werner Heisenberg’s uncertainty principle (1927) states that the position and momentum of an electron cannot both be known precisely. As a result the electron does not have a definite orbit. Instead, the square of the wave function gives the probability of finding the electron at a point in space. The region where the probability is high is called an orbital. Orbitals have characteristic shapes and are described by quantum numbers.

Bohr model and quantum mechanical model compared

How did the theory develop?

Max Planck introduced the quantum of energy in 1900, and Albert Einstein explained the photoelectric effect with photons in 1905. Bohr’s model, published in 1913, applied quantization to the atom and earned him the Nobel Prize in Physics in 1922. Arnold Sommerfeld refined the model with elliptical orbits in 1916. Wolfgang Pauli proposed his exclusion principle in 1925, and the full quantum mechanics of Heisenberg and Schrödinger appeared in 1925 and 1926.

Which mistakes are common?

Mistake 1: Treating orbits and orbitals as the same thing

An orbit is a path. An orbital is a region of space with a probability of finding the electron.

Mistake 2: Saying that the Bohr model is useless

It is a good approximation for hydrogen, and it introduces the key idea of energy levels.

Mistake 3: Thinking that electrons orbit like planets

Electrons do not follow definite paths. Their position can only be described by probability.

Mistake 4: Forgetting that emission corresponds to falling to a lower level

Absorbing a photon raises the electron to a higher level, and emitting a photon lowers it.