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⋅m^{2}⋅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 .

### Definition: The Bohr Model

The Bohr model is essentially summed up by these three “postulates” or assumptions:

- Electrons in atoms make circular orbits around the nucleus.
- The angular momentum of an orbiting electron is quantized: it can only be an integer multiple of the reduced Planck constant : In other words, electrons can only orbit the nucleus at the specific distances where their angular momentum obeys the above equation.
- In order to jump to a further (higher) orbit from the nucleus, an electron must absorb energy in the form of a photon. Conversely, an electron moving to a nearer (lower) orbit to the nucleus will emit energy in the form of a photon.

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.

### Example 1: Calculating the Angular Momentum of an Electron in a Hydrogen Atom

In the Bohr model of the atom, what is the magnitude of the angular momentum of an electron in a hydrogen atom in the ground state? Use a value of J⋅s for the reduced Planck constant.

### Answer

The Bohr model of the atom tells us that the angular momentum of an electron in an atom is quantized. Specifically, it tells us that can only be an integer multiple of the reduced Planck constant . This is expressed by the equation
where is a positive integer known as the **principal quantum number**.

In this question, we are looking for an electron in the **ground state** of a hydrogen atom. We can recall that the ground state of an atom is the lowest energy level, and that the principal quantum number of an electron in the ground state will take the lowest possible value, . The fact that this is a hydrogen atom is relevant because the Bohr model is generally only accurate for single-electron atoms, and hydrogen atoms have only one electron.

Now that we have established that the electron in this question has , we can just substitute this into the above equation to find the angular momentum:

Note that the unit J⋅s is equivalent to kg⋅m^{2}⋅s^{−1}, which is commonly used for angular momentum.

### Example 2: Calculating the Energy Level of an Electron from Its Angular Momentum

An electron in a hydrogen atom has an angular momentum of J⋅s. Under the Bohr model of the atom, what energy level is the electron in? Use a value of J⋅s for the reduced Planck constant.

### Answer

The question asks us for the “energy level” of an electron in a hydrogen atom. The key to answering this question is to recognize that the energy level of an electron is denoted by its **principal quantum number**, .

Recall that, according to the Bohr model, the angular momentum of an electron with principal quantum number is given by where is the reduced Planck constant. We want to calculate , so let’s start by rearranging this equation to make the subject:

Now we can substitute in the values of and given in the question:

In other words, the electron is in the third energy level of the hydrogen atom.

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.

### Equation: The Orbital Radius of an Electron in a Hydrogen Atom According to the Bohr Model

where

- is the orbital radius of an electron with principal quantum number ;
- is the permittivity of free space, equal to m
^{−3}⋅kg^{−1}⋅s^{4}⋅A^{2}; - is the reduced Planck constant, equal to J⋅s;
- is the mass of an electron, equal to kg;
- is the charge of an electron, equal to C.

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.

### Example 3: Calculating the Orbital Radius of an Electron Using the Bohr Model

Use the formula , where
is the orbital radius of an electron in energy level of a hydrogen atom, is the permittivity of free space, is the reduced Planck constant, is the mass of the electron, and is the charge of the electron, to calculate the orbital radius of an electron that is in energy level
of a hydrogen atom. Use a value of F⋅m^{−1} for the permittivity of free space, J⋅s for the reduced Planck constant, kg for the rest mass of an electron, and C for the charge of an electron. Give your answer to two decimal places.

### Answer

The formula given is derived from the Bohr model of the atom, and it describes the orbital radius of an electron in energy level of a hydrogen atom. In other words, it tells us the distance of an electron from the nucleus of a hydrogen atom based on its energy level.

In this question, we are given all the quantities we need; we just have to substitute them into the formula: or, equivalently, nm. Rounding this to 2 decimal places gives us a final answer of .

### Example 4: Calculating the Orbital Radius of an Electron Based on the Bohr Radius

Use the formula , where is the orbital radius of an electron in energy level of a hydrogen atom and is the Bohr radius, to calculate the orbital radius of an electron that is in energy level of a hydrogen atom. Use a value of m for the Bohr radius. Give your answer to three decimal places.

### Answer

The formula we are given in this question is derived from the Bohr model. It expresses the orbital radius of an electron in energy level of a hydrogen atom in terms of two quantities: and .

is a constant known as the Bohr radius. This is equal to the orbital radius of an electron in the lowest energy state of a hydrogen atom, as predicted by the Bohr model. We are told in the question to use a value of m for the Bohr radius.

is the principal quantum number of the electron. This is a number that denotes the energy level of an electron. In this case, we are considering an electron for which .

We can find the answer to this question simply by substituting and into the formula provided: or, equivalently, 0.4761 nm. Rounding to three decimal places gives us a final answer of 0.476 nm.

### Key Points

- The Bohr model states that the electrons in an atom make circular orbits around the nucleus.
- According to the Bohr model, electron angular momentum is quantized. This means that electron orbits are only possible where the angular momentum of an orbiting electron is an integer multiple of the reduced Planck constant : In this equation, is the principal quantum number: a positive integer denoting the energy level of the electron (where corresponds to the lowest energy state).
- In the Bohr model, the orbital radius of an electron in energy level is given by where is the permittivity of free space, is the mass of an electron, and is the charge of an electron.
- This equation can also be expressed as where is the Bohr radius. The Bohr radius is the lowest orbital radius according to the Bohr model, with a value of m.