Video Transcript
Use the formula 𝑟 𝑛 equals 𝑎 naught times 𝑛 squared, where 𝑟 𝑛 is the orbital radius of an electron in energy level 𝑛 of a hydrogen atom and 𝑎 naught is the Bohr radius, to calculate the orbital radius of an electron that is in energy level 𝑛 equals three of a hydrogen atom. Use a value of 5.29 times 10 to the power of negative 11 meters for the Bohr radius. Give your answer to three decimal places.
So in this question, we’ve been asked to calculate the orbital radius of an electron that’s in energy level 𝑛 equals three of a hydrogen atom. And the question tells us that we can do this by using this formula, 𝑟 𝑛 equals 𝑎 naught times 𝑛 squared. Now, this formula comes from the Bohr model, a simplified model of the atom that describes negatively charged electrons as making circular orbits around a positively charged nucleus. Now, although the Bohr model gives a simplified description of how electrons in atoms behave, it still makes fairly accurate predictions for atoms that only have one electron. This means that the Bohr model is useful for handling problems involving hydrogen atoms like the one in this question because hydrogen atoms only have one electron.
One of the main features of the Bohr model is that it tells us electrons can only occupy certain orbits around the nucleus. The orbit that’s occupied by a certain electron depends on its energy level, which we represent with the symbol 𝑛. This is also known as the electron’s principal quantum number. Now, the energy level 𝑛 can only take positive whole number values. This means that the lowest energy level is 𝑛 equals one. An electron in the 𝑛 equals one energy level will have the lowest amount of energy that it’s possible for an electron in that atom to have. And as a result, it will occupy the closest possible orbit to the nucleus.
An electron in the 𝑛 equals two energy level will have the second-lowest possible amount of energy and occupy the second-closest orbit to the nucleus. And an electron in the 𝑛 equals three energy level will have the third-lowest possible amount of energy and occupy the third-closest orbit to the nucleus. And this pattern continues for higher values of 𝑛. So, we can see that as an electron’s energy level increases, the distance of its orbit from the nucleus also increases. Now, the distance between an electron’s orbit and the center of the nucleus is known as the electron’s orbital radius.
In the question, we’re given an equation that enables us to calculate the orbital radius of an electron: 𝑟 𝑛 equals 𝑎 naught times 𝑛 squared. In this equation, 𝑟 𝑛 represents the orbital radius of an electron in a given energy level of a hydrogen atom. 𝑎 naught is a constant known as the Bohr radius. And as we’ve seen, 𝑛 is the principal quantum number or energy level of the electron. This equation enables us to calculate the orbital radius of any electron in a hydrogen atom as long as we’re given its principal quantum number 𝑛. All we need to do is square 𝑛 then multiply it by this constant 𝑎 naught, and we’ll have the orbital radius.
As an example, let’s try doing this for an electron in the 𝑛 equals one energy level. To do this, we substitute 𝑛 equals one into this equation. This means that on the right-hand side, we have 𝑎 naught times one squared. And on the left-hand side, we now have 𝑟 one. And the subscript one is just a way of notating that we’re talking about the orbital radius of an electron with energy level one. Now, in this case, we have one squared, which is just equal to one. And since we now have 𝑎 naught times one, we can just write 𝑎 naught.
So, we can see that the orbital radius of an electron in the 𝑛 equals one energy level of a hydrogen atom is actually just equal to 𝑎 naught, the Bohr radius. And in fact, this is the definition of the Bohr radius. This is an important physical constant, not only because it’s the smallest possible orbital radius in the simplest possible atom, but also because all other orbital radii are calculated using this constant.
Now, in this question, we’re being asked to calculate the orbital radius of the electron in energy level 𝑛 equals three. So, in our diagram, that’s this distance here. To find this, we just need to substitute 𝑛 equals three into our equation. So this time, we have 𝑟 subscript three is equal to 𝑎 naught times three squared. Now, three squared is equal to nine. So we can write 𝑟 three equals nine 𝑎 naught.
Now, we just need to substitute in the value of 𝑎 naught, the Bohr radius, which is given in the question. That’s 5.29 times 10 to the power of negative 11 meters. So, in total we have that the orbital radius of an electron in the 𝑛 equals three energy level of a hydrogen atom is equal to nine times 5.29 times 10 to the power of negative 11 meters. And we can enter this multiplication into our calculator to get a value of 4.761 times 10 to the power of negative 10 meters. Now, this is the correct answer. However, because it’s such a small length, it’s conventional to express it in units of nanometers.
We can recall that one nanometer is equal to 10 to the power of negative nine meters. In order to convert our result from meters into nanometers, it’s useful to first express one meter in units of nanometers. We can do this by multiplying both sides of this expression by 10 to the power of nine. On the right-hand side, we have 10 to the power of negative nine times 10 to the power of nine, which is simply equal to one. And on the left-hand side, we have 10 to the power of nine times one, which is equal to 10 to the power of nine. So, we’ve now shown that one meter is equal to 10 to the power of nine nanometers. This means that in our answer, instead of writing meters, we can write 10 to the power of nine nanometers.
So, we now have 4.761 times 10 to the power of negative 10 times 10 to the power of nine nanometers, which we can also express using a multiplication sign like this. Notice that here we’ve got 10 to the power of negative 10 times 10 to the power of nine. This simplifies to 10 to the power of negative one. So, we’ve now expressed 𝑟 three in units of nanometers. One final simplifying step is to note that multiplying by 10 to the negative one is equivalent to moving this decimal place one space to the left, enabling us to express our answer as 0.4761 nanometers. The final thing to do is just round our answer to three decimal places as specified in the question. 0.4761 to three decimal places is 0.476. So, this is the final answer.
According to the Bohr model, the orbital radius of an electron in the 𝑛 equals three energy level of a hydrogen atom is 0.476 nanometers.
