### Video Transcript

In this video, we’ll be talking about the Bohr model of the atom. The Bohr model is a simplified description of the atom which describes electrons as occupying circular orbits around the nucleus in much the same way that planets orbit the sun. In this video, we’ll be talking about why the Bohr model is useful in physics and looking at some of the equations that come from it. But first, let’s have a quick history lesson.

The Bohr model describes atoms as small, dense, and positively charged nuclei surrounded by negatively charged electrons which orbit in a circular path. This way of describing the atom was first presented by the physicists Niels Bohr and Ernest Rutherford in 1913. In fact, its full name is the Rutherford–Bohr model. But unfortunately for Rutherford, this isn’t as catchy. And today, it’s mostly referred to as the Bohr model.

Physicists developed models as ways of explaining physical systems. They then test these models using experiments. And if experimental evidence disproves the model, then we need to either change the model or develop a completely new one that can explain the experimental result. The Bohr model of the atom is one such model.

Prior to the development of the Bohr model, physicists used a cubic model for atoms consisting of a positively charged cube with negatively charged electrons at the corners. This model was later replaced by the so-called plum pudding model, which described atoms as balls of positively charged material with little electron plums embedded in them. However while this model does sound tasty, experimental results at the time forced physicists to reconsider it.

In the following years, a few more models for the atom were developed each one improving on the last until, eventually, the Bohr model was developed. Since then, new physical evidence has forced us to change our description of the atom further. And today physicists use a quantum mechanical model which describes electrons using statistics. So, we know today that the Bohr model is really a simplification of how atoms behave.

However, development of the model was incredibly influential and useful to scientists at the time. In fact, it’s still useful to us as a first-order approximation of how atoms behave, in particular, when they only have one electron. So let’s 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, because the nucleus is positively charged and the electrons are negatively charged, the electrons experience an electrostatic attraction toward the nucleus. And it’s this electrostatic force which causes the electrons to orbit the nucleus. This is similar to how the attractive gravitational force between the Earth and the Sun causes the Earth to orbit the Sun.

Now, some earlier models of the atom actually treated electrons in this way, too. But the Bohr model makes one important additional claim. It theorizes that the angular momentum of an orbiting electron is quantized; that is, it can only take certain values. Specifically, the Bohr model says that electrons can only have angular momentum equal to a whole number multiple of a constant, known as the reduced Planck constant.

This idea is summed up by this equation, where the uppercase 𝐿 represents the angular momentum of an electron, which we usually express in units of kilogram meter squared per second. 𝑛 represents the electron’s principal quantum number, which denotes its energy level. And this symbol represents the reduced Planck constant. The symbol that we’re using here is just a lowercase ℎ with a bar through it, and we call this ℎ bar.

As we might guess, the reduced Planck constant is closely related to the ordinary Planck’s constant, which we represent with a lowercase ℎ. In fact, the reduced Planck constant, ℎ bar, is exactly equal to Planck’s constant ℎ divided by two 𝜋. So if we wanted to, we could rewrite this equation like this, with the ordinary Planck’s constant divided by two 𝜋. However, this factor of ℎ over two 𝜋 crops up in so many different physical equations that it’s easier to just write ℎ bar as a shortcut.

The reduced Planck constant has a value of 1.05 times 10 to the power of negative 34, and the units are the same as that of the normal Planck’s constant, which we usually express as joule seconds. It’s worth noting that joule seconds are actually equivalent to kilogram meter squared per second. And because the principal quantum number is dimensionless — that is, it’s just a number without any units — this means that the units on the left of the equation match the units on the right of the equation.

This simple equation is the fundamental basis of the Bohr model, and it makes it incredibly easy to work out the angular momentum of an electron in an atom. But one really important thing to note here is that the limitations of the Bohr model mean that it’s only really accurate for atoms with one electron, which is why we only really talk about the Bohr model in the context of hydrogen atoms, which just have one proton and one electron. So to calculate the angular momentum of an electron in a hydrogen atom using the Bohr model, all we need to do is multiply the principal quantum number of that electron by the reduced Planck constant.

Because the lowest possible value of the principal quantum number is one, this means that the lowest possible amount of angular momentum that an electron in an atom can have is given by one times the reduced Planck constant. In other words, it’s equal to ℎ bar or 1.05 times 10 to the negative 34 joule seconds. We could call this amount of angular momentum 𝐿 sub one to signify that it’s the angular momentum of an electron in the lowest possible energy level. That is, it has principal quantum number one.

The next highest energy level is denoted by a principal quantum number of two. And the equation on the left tells us that the angular momentum of an electron in this energy level, which we could call 𝐿 two, is given by two times ℎ bar. And two times ℎ bar is equal to 2.10 times 10 to the power of negative 34 joule seconds. So we can use this type of calculation to determine the angular momentum of an electron in a hydrogen atom.

In the Bohr model, the quantization of the angular momentum leads to another really useful equation. This equation enables us to calculate the orbital radius — that is, the radius of the circular path that the electron follows around the nucleus — of an electron in a given energy level of a hydrogen atom. This equation looks like this. 𝑟 𝑛 represents the orbital radius of an electron in a given energy level of the hydrogen atom. So, for example, the orbital radius of an electron in the lowest possible energy level of the hydrogen atom — that is, with principal quantum number equal to one — this radius would be denoted by 𝑟 one. Because the orbital radius is a distance, we measure it in meters.

On the right-hand side of the equation, we can see some familiar symbols. 𝑛, once again, is the principal quantum number, and ℎ bar is the reduced Planck constant. As we can see, there are loads of other constants in this equation. 𝑚 e is the mass of an electron, equal to 9.11 times 10 to the negative 31 kilograms. And 𝑞 e is the charge of an electron given by negative 1.60 times 10 to the negative 19 coulombs. We also have a factor of four 𝜋 on the top of this fraction. And finally, we have 𝜀 naught, the permittivity of free space, which is equal to 8.85 times 10 to the negative 12 farads per meter.

Although there are clearly loads of different quantities in this equation, there are actually only two variables, the orbital radius and the principal quantum number. All of the other quantities in the equation are constants. Now, if we group all of these constants together, we can see that the orbital radius is simply given by a bunch of physical constants multiplied by the principal quantum number squared.

So let’s say we want to work out the orbital radius of an electron in the 𝑛 equals one energy level of a hydrogen atom. Because we’re dealing with a case where 𝑛 equals one, we would replace 𝑟 𝑛 on the left side of the equation with 𝑟 one, and 𝑛 on the right side of the equation would take a value of one. One squared is, of course, just one.

And all of these constants multiplied by one just leaves us with this expression. This tells us the orbital radius of an electron in the 𝑛-equals-one energy level of a hydrogen atom. In other words, it’s the orbital radius of the lowest energy electron in the simplest atom. For this reason, it’s been given a special name. We call it the Bohr radius. And this can be represented by the symbol 𝑎 naught.

If we substitute in the values of all of the constants in this equation, we can calculate that the Bohr radius has a value of 5.29 times 10 to the power of negative 11 meters. If we look again at the equation for the orbital radius of an electron in any energy level of a hydrogen atom, we can see that this part of the equation is simply equal to 𝑎 naught, which means it’s possible to write the equation like this. This makes it clearer that the orbital radius of an electron in a hydrogen atom is proportional to the square of that electron’s principal quantum number.

So here, we have two useful equations which we can use to calculate the angular momentum and the orbital radius of an electron in a hydrogen atom. Let’s have a go at putting these equations to use.

In the Bohr model of the atom, what is the magnitude of the angular momentum of an electron in a hydrogen atom for which 𝑛 equals two? Use a value of 1.05 times 10 to the negative 34 joule seconds for the reduced Planck constant.

So in this question, we’re considering a hydrogen atom, and we’ve specifically been asked to use the Bohr model of the atom. We can recall that a hydrogen atom just has one proton in the nucleus and one electron and the Bohr model describes atoms as consisting of electrons making circular orbits around the nucleus. So we can visualize this hydrogen atom like this. Here’s the nucleus, and here’s an electron making a circular path around it.

Let’s also recall that the Bohr model actually only makes accurate predictions for single-electron systems, which is why we’re being asked about hydrogen atom in this question. We’re being asked to find the angular momentum of an electron for which 𝑛 equals two. Let’s recall that 𝑛 is the principal quantum number of an electron. And it describes the energy level that the electron occupies. So an electron in the lowest possible energy level would have 𝑛 equals one. And in the Bohr model, this would refer to the innermost orbit around the nucleus.

In this question, we’re told that our electron has 𝑛 equals two, which would mean, according to the Bohr model, our electron occupies an orbit that’s further away from the nucleus. The Bohr model gives us a simple way of calculating the angular momentum of an electron in a hydrogen atom as long as we know what its principal quantum number is. In other words, we can find its angular momentum if we know which energy level it’s in.

This is given by the equation 𝐿 equals 𝑛ℎ bar, where 𝐿 represents the angular momentum of an electron, 𝑛 represents the principal quantum number, and ℎ bar is the reduced Planck constant. And we’re told in the question that this constant takes a value of 1.05 times 10 to the power of negative 34 joule seconds. We should note that even though it’s more common to express angular momentum in units of kilograms meters squared per second, these units are actually equivalent to the units of the reduced Planck constant.

Since the principal quantum number 𝑛 is dimensionless, this means that the units on the left and the right of the equation are equivalent. Since we’re looking to calculate the angular momentum and angular momentum is already the subject of this equation, all we need to do is multiply the principal quantum number of our electron by the reduced Planck constant. This gives us an angular momentum of two times 1.05 times 10 to the negative 34 joule seconds, which gives us a value of 2.10 times 10 to the negative 34 joule seconds. And this is the final answer to the question. In the Bohr model of the atom, the magnitude of the angular momentum of an electron in a hydrogen atom for which 𝑛 equals two is 2.10 times 10 to the negative 34 joule seconds.

Now, let’s take a look at another question.

Use the formula 𝑟 𝑛 equals four 𝜋𝜀 naught ℎ bar squared 𝑛 squared over 𝑚 e 𝑞 e squared, where 𝑟 is the orbital radius of an electron in energy level 𝑛 of a hydrogen atom, 𝜀 naught is the permittivity of free space, ℎ bar is the reduced Planck constant, 𝑚 e is the mass of the electron, and 𝑞 e is the charge of the electron, to calculate the orbital radius of an electron that is in energy level 𝑛 equals two of a hydrogen atom. Use a value of 8.85 times 10 to the negative 12 farads per meter for the permittivity of free space, 1.05 times 10 to the negative 34 joule seconds for the reduced Planck constant, 9.11 times 10 to the negative 31 kilograms for the rest mass of an electron, and negative 1.60 times 10 to the negative 19 coulombs for the charge of an electron. Give your answer to three significant figures.

Okay, so this seems like a pretty long question. But actually, all we’re being asked to do is this bit: Calculate the orbital radius of an electron that is in energy level 𝑛 equals two of a hydrogen atom. The rest of the question just tells us how we can do this. So we’re told we can use this formula. And this part of the question defines what all the quantities in this formula are. And this last part of the question tells us the values of the constants in the equation.

We can recall that this formula which we’ve been given is derived from the Bohr model of the atom, which describes atoms as consisting of a positively charged nucleus with electrons making circular orbits around it. Now, the Bohr model has some limitations, but it’s still pretty accurate when describing systems with one electron such as the hydrogen atom in this question.

Now, in this question, we’re told that the electron occupies energy level 𝑛 equals two. We can recall that 𝑛 is the principal quantum number of an electron in an atom. 𝑛 takes whole number values, which describe the energy level that an electron has. The lowest value that 𝑛 can take is one, which would describe an electron in the lowest possible energy state of an atom. In the Bohr model, this would describe an electron in the innermost orbital around the nucleus.

However, in this question, we’re told that our electron is in energy level 𝑛 equals two, which means that the electron occupies the next orbital out. The orbital radius of an electron is simply the radius of the circular path that it follows around the nucleus. And as we’ve been told in the question, we can calculate the orbital radius of an electron using this equation.

One interesting thing to note about this equation is that it only actually contains two variables, the orbital radius and the principal quantum number. This means that according to the Bohr model, the orbital radius of an electron is proportional to the square of its principal quantum number. Now, we want to calculate the orbital radius of an electron in energy level 𝑛 equals two. In other words, we’re looking to find 𝑟 two. To find this, we simply substitute two in place of 𝑛 in this equation, which gives us four 𝜋𝜀 naught ℎ bar squared two squared over 𝑚 e 𝑞 e squared, where we’ve been told 𝜀 naught is the permittivity of free space. ℎ bar is the reduced Planck constant. 𝑚 e is the mass of an electron. And 𝑞 e is the charge of an electron.

Since we’re told the values of all of these quantities in the question, all we need to do now is substitute these values in and calculate the answer, which gives us this expression. And if we plug all of this into our calculators, it gives us an answer of 2.10 times 10 to the power of negative 10 meters, which is equivalent to 0.210 nanometers. And this is the final answer to our question. The orbital radius of an electron in energy level 𝑛 equals two of a hydrogen atom is 0.210 nanometers.

Okay, now we can summarize the key points that we’ve looked at in this video. We’ve seen that the Bohr model describes the atom as consisting of electrons moving in circular orbits around a small, dense nucleus. According to the Bohr model, the angular momentum of an electron orbiting an atom is quantized and proportional to its principal quantum number 𝑛, which is given by the formula 𝐿 equals 𝑛ℎ bar. The Bohr model also tells us that the orbital radius of an electron is proportional to the square of its principal quantum number 𝑛, which is given by this equation. And finally, we’ve seen that the Bohr model accurately describes single-electron systems such as the hydrogen atom. This is a summary of the Bohr model of the atom.