Lesson Explainer: The Bohr Model of the Atom Physics

In this explainer, we will learn how to calculate the orbital radius of an electron in different energy levels of a hydrogen atom.

The Bohr model is a simplified description of the atom. It states—among other things—that electrons in atoms occupy circular orbits around the nucleus, similar to the way planets orbit the sun. In this explainer, we will look at the Bohr model in detail and see how we can use it to accurately calculate the angular momentum and orbital radius of an electron in an atom.

Physicists use models to describe how physical systems behave. A model can be put to the test by experiments, which compare the predictions of the model to the observed behavior of a physical system. If an experiment reveals that a model is inaccurate, then physicists need to either change the model or develop a new one that can accurately describe the observed experimental results.

Historically, scientists have used a variety of models to describe atoms. Note that these diagrams are not to scale.

• The cubic model, proposed in 1902, described atoms as cubes of positively charged matter with negatively charged electrons at the vertices.

• The plum pudding model, proposed in 1904, describes electrons as balls of positively charged matter studded with negatively charged electrons. This model improved on the predictions made by the cubic model before it.

• In 1913, physicists Niels Bohr and Ernest Rutherford proposed the Rutherford–Bohr model, which is now commonly referred to simply as the Bohr model. The model describes electrons as small negatively charged particles orbiting a dense, positively charged nucleus.

• Today, physicists use a quantum mechanical model that describes the positions of electrons in terms of probabilities. Quantum mechanics enables us to describe the behavior of atoms and electrons with great accuracy. However, it is possible that in the future our quantum mechanical model may be refined or replaced to account for new experimental observations.

So, today we recognize the Bohr model is really a simplification of how atoms behave. However, the development of the Bohr model was hugely influential and useful to scientists at the time as it captured some of the key features of atoms and enabled scientists to make accurate predictions about certain atoms. In fact, the Bohr model is still useful to us as an approximation of how atoms behave—in particular, when they only have one electron.

Let’s now take a closer look at how the Bohr model works and how we can use it to make predictions about how atoms behave.

In the Bohr model, electrons are described as negatively charged particles orbiting a positively charged nucleus. Because the electrons are negatively charged, they experience an electrostatic attraction to the nucleus, which causes them to orbit it. This is similar to how the attractive gravitational force between Earth and the Sun causes Earth to orbit the Sun.

In addition to this, the Bohr model goes one step further: it states that the angular momentum of an orbiting electron is quantized, meaning it can only take specific values. Specifically, the Bohr model tells us that the angular momentum of an electron in an atom must be equal to a multiple of a constant known as the reduced Planck constant, . We can write this as where is the angular momentum of the electron and (pronounced “h bar”) is the reduced Planck constant. is just some positive integer, but we give this a special name: the principal quantum number. The principal quantum number denotes the energy level of an electron, where corresponds to the lowest possible energy state (also known as the ground state).

The “reduced” Planck constant is equal to the ordinary Planck constant, , divided by , and it has a value of J⋅s. It is worth noting that the unit of can also be expressed as kg⋅m²⋅s⁻¹, which is the usual unit we would use to represent angular momentum.

The equation makes it easy to calculate the angular momentum of an electron in an atom. We just need to take the electron’s principal quantum number (which denotes its energy level) and multiply this by .

Earlier we mentioned that the Bohr model is only really accurate for atoms with just one electron. Since the most common single-electron atom is the hydrogen atom, we will generally only hear about the Bohr model in the context of hydrogen atoms.

Consideration of electrostatic forces in a hydrogen atom makes it possible to derive a useful equation from the Bohr model: this expresses the orbital radius of the electron in a hydrogen atom.

As we can see, this equation contains many different quantities! However, most of these are just constants. The equation actually only contains two variables: the orbital radius and the principal quantum number .

If we factor out the constants in the equation we can see that is proportional to the square of .

Because the lowest possible value of is 1, the smallest possible atomic radius allowed by the Bohr model is

We give this “lowest possible” radius a special name: the Bohr radius. We represent the Bohr radius with the symbol :

is a convenient way of grouping together all the constants in our equation for , which allows us to express the equation like this:

The two equations we have looked at can help us build up a picture of the Bohr model. Low-energy electrons (with low values of ) orbit closer to the nucleus than high-energy electrons (with high values of ) that have relatively large orbits.