15.2 The Bohr Model of Atomic Structure

At the center of every atom is a tiny nucleus with a positive charge, and negatively charged electrons fill the space around it. The nucleus is about 100,000 times smaller across than the atom, yet it holds nearly all the atom's mass.

The nucleus contains protons (charge +e) and neutrons (no charge). Together they're called nucleons. Each proton or neutron has about 1840 times the mass of an electron, so an atom's mass is dominated by its nucleons.

The number of protons, called the atomic number Z, defines the element: every carbon atom has 6 protons. The total number of protons and neutrons is the mass number A. Atoms of the same element with different numbers of neutrons are different isotopes, such as carbon-12 and carbon-14.

An atom with equal numbers of protons and electrons is neutral. If an atom gains or loses electrons, it ends up with a net charge and is called an ion. The number and arrangement of electrons decide how atoms interact and bond with each other.

A nucleus is written as ${}^{A}_{Z}\text{X}$: the element's symbol X, with the mass number A at the top left and the atomic number Z at the bottom left. For example, ${}^{14}_{6}\text{C}$ is carbon-14, with 6 protons and 14 − 6 = 8 neutrons.

The number of neutrons is always N = A − Z.

In 1913, Niels Bohr modeled hydrogen as an electron moving in a circular orbit around the nucleus. The electric force from the nucleus provides the centripetal force, so you set the Coulomb force equal to the net force needed for circular motion:

$$\frac{k e^2}{r^2} = \frac{m v^2}{r}$$

Bohr's key step was to allow only certain orbits. Classical physics says an orbiting charge should radiate energy and spiral into the nucleus. Bohr simply declared that electrons in allowed orbits don't radiate, and that each allowed orbit has a definite energy.

De Broglie's matter waves give a picture of why. Think of the electron as a wave wrapped around the orbit. For the wave to join up smoothly with itself, the circumference must hold a whole number of wavelengths:

$$2\pi r = n\lambda, \quad n = 1, 2, 3, \ldots$$

Orbits that don't satisfy this would make the wave interfere with itself destructively. So only certain radii are allowed, and each comes with its own energy. That's why an atom has a ladder of discrete energy levels instead of any energy at all.

The Bohr model works well for hydrogen and other one-electron atoms but fails for atoms with more electrons. Modern quantum theory replaced it, but you won't need orbitals, orbital shapes or probability clouds here; this course stays with energy levels. Physics 2 materials from before 2024–25 may include wave functions and probability, which are no longer tested.