Video Transcript
An electron in a hydrogen atom has an angular momentum of 3.15 times 10 to the negative 34 joule-seconds. Under the Bohr model of the atom, what energy level is the electron in? Use a value of 1.05 times 10 to the negative 34 joule-seconds for the reduced Planck constant.
Let’s start by recalling that the Bohr model is a simplified model of the atom in which negatively charged electrons occupy circular orbits around the positively charged nucleus. An important feature of the Bohr model is that electron angular momentum is quantized. This means that according to the Bohr model, electrons in atoms can only have certain specific values of angular momentum. This in turn means that electrons can only occupy certain specific orbits around the nucleus. The particular orbit that a particular electron occupies is denoted by that electron’s principal quantum number 𝑛. An electron in the closest possible orbit to the nucleus has 𝑛 equals one. An electron in the next furthest out orbit to the nucleus has 𝑛 equals two. The next furthest out has 𝑛 equals three and so on for greater values of 𝑛.
Now, as well as corresponding to specific orbits and specific amounts of angular momentum, these values of 𝑛 also correspond to specific amounts of energy that the electron can have. For this reason, the values of 𝑛 are often referred to as energy levels. So, where this question asks us to find what energy level an electron is in, it’s really just asking us to find the value of its principal quantum number 𝑛. Now, the Bohr model proposes a very simple relationship between the energy level, or principal quantum number, of an electron and the amount of angular momentum that it has. This is given by the equation 𝐿 equals 𝑛 ℎ bar, where 𝐿 is the angular momentum of an electron, 𝑛 is its principal quantum number, and ℎ bar is a physical constant called the reduced Planck constant.
As an aside, it’s worth noting that the reduced Planck constant ℎ bar is equal to the Planck constant ℎ divided by two 𝜋. This equation makes it easy to work out the angular momentum of any electron given its principal quantum number 𝑛. What we have to do is multiply 𝑛 by the reduced Planck constant, which we’re given a value for in the question.
In this question though, we haven’t been told the principal quantum number of the electron. Instead, we’ve been told its angular momentum, and we need to work out what it’s principal quantum number, or energy level, is. To do this, we just need to rearrange this equation to make 𝑛 the subject. So, dividing both sides of the equation by ℎ bar, we have 𝑛 equals 𝐿 divided by ℎ bar. All we need to do now is substitute in the value of angular momentum and the value of the reduced Planck constant, both of which we’re given in the question. This gives us 𝑛 equals 3.15 times 10 to the power of negative 34 joule-seconds divided by 1.05 times 10 to the power of negative 34 joule-seconds.
At this point, we can see that we have units of joule-seconds in both the numerator and the denominator. This means that these units will cancel out, and we’ll get a dimensionless number as a result. Next, we can see that we have a factor of 10 to the power of negative 34 in both the numerator and the denominator. Again, these cancel out, leaving us with just 𝑛 equals 3.15 over 1.05, and the value of this is exactly three. And this is the final answer to our question.
According to the Bohr model of the atom, an electron with an angular momentum of 3.15 times 10 to the power of negative 34 joule-seconds must be in the third energy level of the atom. In other words, it’s in the energy level denoted by 𝑛 equals three.
One final thing to note is that this question specifies that we’re talking about a hydrogen atom. Now, this doesn’t actually change the way we calculate the answer to the question, but it is worth noting that the Bohr model is generally only accurate for atoms that have a single electron. This means that we commonly see the Bohr model applied to problems involving hydrogen atoms because they only have one electron each.
