Video Transcript
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 1.05 times 10 to the negative 34 joule seconds for the reduced Planck constant.
We can start by recalling that the Bohr model is a simplified model of the atom that describes electrons as making circular orbits around atomic nuclei. The Bohr model is based on the idea that the angular momentum of electrons is quantized. This means that, according to the Bohr model, electrons can only have certain specific values of angular momentum. This quantization of angular momentum in the Bohr model leads to the prediction that electrons can only occupy certain specific orbits around the nucleus.
So each of these orbits represents a possible state that an electron can exist in within that atom. Each of these states is denoted by a specific value of a number 𝑛, which is also called the principal quantum number. By convention, the state corresponding to the orbit closest to the nucleus is given 𝑛 equals one. Then, the next orbit out is given 𝑛 equals two. Then, the next furthest out has 𝑛 equals three, and so on.
Each of these states corresponds to a specific value of angular momentum as well as a specific value of electron energy and a specific orbital radius, which describes how far the electron is from the nucleus. The 𝑛 equals one state is also known as the ground state. An electron in this state will have the lowest possible orbital radius, the lowest possible amount of energy, and the lowest possible angular momentum. And as we look at higher values of 𝑛, all of these quantities increase.
Note that this question specifically asks us about the angular momentum of an electron in a hydrogen atom. This is because the Bohr model only really makes accurate predictions for atoms with a single electron. This means it works well for predicting the behavior of hydrogen atoms, since these only have one electron.
Now, we’ve talked about the fact that in the Bohr model, angular momentum of electrons is quantized. But this question is asking us to actually calculate the magnitude of the angular momentum of an electron. In order to do this, we need to use this equation that comes from the Bohr model. 𝐿 equals 𝑛 times ℎ bar, where 𝐿 is the angular momentum of an electron, 𝑛 is the principal quantum number that describes the state of the electron, and ℎ bar is a physical constant known as the reduced Planck constant, which we’ve been given a value for in the question.
This equation enables us to calculate the angular momentum of an electron with any given principal quantum number. All we need to do is multiply its principal quantum number, that is, its value of 𝑛, by the reduced Planck constant.
Now, in this question, we’re being asked to calculate the magnitude of the angular momentum of an electron in the ground state of a hydrogen atom. Ground state just means 𝑛 equals one. So all we need to do is substitute 𝑛 equals one into this equation. Doing this, we have that the angular momentum 𝐿 is equal to one times ℎ bar, which of course we can simplify to just 𝐿 equals ℎ bar. And we’re told in the question that ℎ bar has a value of 1.05 times 10 to the power of negative 34 joule seconds. So this is the final answer to the question.
Because the Bohr model tells us that the angular momentum 𝐿 of an electron is just equal to its principal quantum number multiplied by ℎ bar, and an electron in the ground state has 𝑛 equal to one, then the angular momentum of an electron in the ground state is just equal to ℎ bar.
