Video Transcript
An electron in a hydrogen atom has
an angular momentum of 6.30 times 10 to the power of 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
power of negative 34 joule-seconds for the reduced Planck constant.
First things first, let’s recall
that the Bohr model is a simplified model of the atom that describes negatively
charged electrons as occupying circular orbits around a positively charged
nucleus. An important feature of the Bohr
model is that in the model electron angular momentum is quantized. What this means is that according
to the Bohr model, electrons in atoms can only have certain specific values of
angular momentum. This quantization of angular
momentum means that electrons can only occupy certain orbits around the nucleus.
The particular orbit that a certain
electron occupies is denoted by that electron’s principal quantum number 𝑛. If an electron occupies the closest
possible orbit to the nucleus, we say that it has 𝑛 equals one. An electron occupying the second
closest orbit to the nucleus is said to have 𝑛 equals two. An electron in the third closest
orbit to the nucleus has 𝑛 equals thre, and so on for higher values of 𝑛 as we get
further away from the nucleus.
So, each value of 𝑛 corresponds to
a specific orbit as well as a specific amount of angular momentum. It also corresponds to a specific
amount of energy that the electron has. For this reason, values of 𝑛 are
also referred to as energy levels. So, while this question asks us to
find out what energy level an electron is in, it’s really just asking us to find the
principal quantum number or value of 𝑛 for that electron.
Now, the Bohr model proposes a very
simple relationship between the energy level of an electron and the amount of
angular momentum that it has. This is given by the equation 𝐿
equals 𝑛 times ℎ bar, where 𝐿 is the angular momentum of the electron, 𝑛 is its
principal quantum number, and ℎ bar is a physical constant known as the reduced
Planck constant.
At this point, it’s useful to
remember that the reduced Planck constant ℎ bar is equal to the Planck constant ℎ
divided by two 𝜋. Now, ordinarily, this equation
gives us an easy way of calculating the angular momentum of an electron given its
principal quantum number 𝑛. All we need to do in that case is
multiply 𝑛 by the reduced Planck constant, and that gives us the electron’s angular
momentum.
In this question though, we’ve been
given the angular momentum of the electron and we need to work out the principal
quantum number. To do this, we just need to
rearrange this equation to make 𝑛 the subject. We can do this by dividing both
sides of the equation by ℎ bar, giving us 𝑛 equals 𝐿 over ℎ bar. We can now substitute in the
angular momentum of the electron, and the reduced Planck constant, which are both
given in the question. This gives us 𝑛 equals 6.30 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, giving us a dimensionless number as a result. We can then notice that we have a
factor of 10 to the power of negative 34 in both the numerator and the
denominator. Again, these cancel out,
simplifying our expression to just 6.30 divided by 1.05. And if we type this into our
calculator, we find that it’s equal to exactly six. And this is the final answer to our
question.
If an electron in an atom has an
angular momentum of 6.30 times 10 to the power of negative 34 joule-seconds, then
the Bohr model tells us that it must be in the sixth energy level of that atom. In other words, we can say that its
principal quantum number 𝑛, also known as its energy level, is equal to six.
One final thing to note is that
this question specifies that we’re talking about an electron in a hydrogen atom. Now this doesn’t actually change
the way we calculate the answer to our question, but it is worth noting that the
Bohr model is only really accurate for atoms that just have one electron. This means that we often come
across the Bohr model in a context of hydrogen atoms because hydrogen atoms are the
simplest atom with only one electron.
