### Video Transcript

In this video, we’re going to be
looking at electron energy level transitions. Specifically, we’re going to be
taking a look at how we can represent the different amounts of energy that electrons
in atoms can have. And we’ll also see how we can
calculate the frequency of light that’s emitted or absorbed when an electron gains
or loses energy.

To start with, let’s remind
ourselves of some of the key principles behind electron behavior. We know that electrons are
negatively charged particles which surround the nuclei of atoms. Electrons can have different
amounts of energy. But crucially, the amounts of
energy that it’s possible for an electron to have are discrete. This means that the electrons in
atoms can only have certain specific amounts of energy.

We can represent this using an
energy level diagram, like this. Here, we have an energy axis
pointing upward. And these horizontal lines are
positioned at specific points on the energy axis corresponding to the amounts of
energy which is possible for electrons in a certain atom to have. So this diagram shows us that in a
specific atom, an electron’s energy could take any of these values marked on the
energy axis. But it couldn’t take any of the
values in between. The allowed amounts of energy
represented by the horizontal lines are often referred to as energy levels. And when an electron has a certain
amount of energy, we say that it occupies that energy level. We can show which energy levels are
occupied by electrons by drawing electrons into our diagram, like this.

So, right now, our energy level
diagram shows us that there’s one electron in this atom. And because this electron is on the
lowest possible line, this means that it has the lowest possible amount of energy
that an electron can have in this atom. We can denote different energy
levels using what’s known as the principal quantum number, which we represent with a
letter 𝑛. 𝑛 just takes a whole-number value
to describe which energy level an electron occupies from lowest to highest. So, for example, an energy in the
lowest energy level would have 𝑛 equals one, an electron in the second lowest
energy level would have 𝑛 equals two, and so on. So, although the value of 𝑛
doesn’t correspond to the exact amount of energy that an electron has, we can see
that larger values of 𝑛 correspond to larger amounts of energy. So, for example, an electron in the
𝑛 equals three energy level would have more energy than an electron in the 𝑛
equals two energy level.

Now, if an electron in an atom
gains energy, we can say that it transitions to a higher energy level. Similarly, if an electron in an
atom loses energy, we can say that it transitions to a lower energy level. We can represent these electron
energy level transitions by drawing arrows on our energy level diagram. For example, if we have an electron
in the 𝑛 equals one energy level and it transitioned to the 𝑛 equals three energy
level by gaining energy, then we could represent this transition by drawing an arrow
from the 𝑛 equals one energy level to the 𝑛 equals three energy level, like
this. If this electron then transitioned
into the 𝑛 equals two energy level, which it could do by losing energy, we can
represent this transition by drawing another arrow on our diagram like this.

Note that when an electron
transitions into a different energy level, it doesn’t necessarily transition into an
adjacent energy level. We’ve shown an example of this with
this arrow on our diagram. This shows an electron transition
from 𝑛 equals one straight to 𝑛 equals three. It doesn’t necessarily need to
transition from 𝑛 equals one to 𝑛 equals two and then from 𝑛 equals two to 𝑛
equals three. It’s also important to note that
it’s actually impossible for an electron in the 𝑛 equals one energy level to lose
energy because there aren’t any lower energy levels for it to transition into. Just like how if we drop an object
it can only fall until it reaches the ground, an electron in an atom can lose energy
only until it reaches the 𝑛 equals one energy level. For this reason, the 𝑛 equals one
energy level is commonly called the ground state of the atom.

The amount of energy an electron
has relates to how strongly it’s bound to the nucleus of the atom. Electrons in the ground state are
the most strongly bound to the nucleus. When an electron gains energy and
transitions to a higher energy level, it becomes less strongly bound to the
nucleus. So, as an electron gains energy and
its value of 𝑛 increases, the force binding it to the nucleus actually becomes
weaker and weaker. We also find that as we look at
higher and higher energy levels, the gaps between adjacent energy levels get
smaller. For example, the gap between the 𝑛
equals four and the 𝑛 equals three energy levels is smaller than the gap between
the 𝑛 equals three and the 𝑛 equals two energy levels. This means that there’s a smaller
energy difference between the 𝑛 equals four and the 𝑛 equals three energy levels
than there is between the 𝑛 equals three and the 𝑛 equals two energy levels. In other words, transitioning from
𝑛 equals two to 𝑛 equals three would represent a larger change in energy than
transitioning from 𝑛 equals three to 𝑛 equals four. And transitioning from 𝑛 equals
four to 𝑛 equals five is an even smaller energy change than that.

Now, the energy level diagram that
we’ve drawn here doesn’t quite show the full story. This is because, in fact, all atoms
actually have an infinite number of energy levels. However, this doesn’t mean that
electrons in atoms can have an infinite amount of energy. In fact, because the energy levels
get closer and closer together as 𝑛 increases, we find that there is actually a
maximum amount of energy that an electron in an atom can have. This is the energy level for which
𝑛 equals ∞. And once an electron reaches this
level, it will no longer be bound to the nucleus.

Remember, the fact that 𝑛 equals ∞
at this point doesn’t mean that the electron has an infinite amount of energy. It just represents the fact that
there are an infinite number of energy levels between this point and the ground
state. In fact, the size of the energy gap
between the 𝑛 equals one and the 𝑛 equals ∞ energy levels is actually well known
for virtually all elements. And this amount of energy is known
as the ionization energy of an atom.

Okay, so now we’ve seen how to use
an energy level diagram, let’s quickly remind ourselves of the processes by which
electrons can gain or lose energy in atoms. So, in order to gain energy and
transition to a higher energy level, an electron needs to absorb a photon, which we
can represent with a wiggly arrow on our energy level diagram. Crucially, in order for the photon
to be absorbed, thus enabling an electron energy level transition, the energy of the
photon needs to be the same as the energy gap between the two energy levels. We can call the difference between
the two energy levels Δ𝐸, and we can say that this needs to be equal to the energy
of the photon 𝐸 p.

Just like a dropped ball naturally
falls to the ground, electrons that occupy an energy level higher than the ground
state will naturally tend to lose energy and fall back down to the ground state
after some time. And when an electron loses energy,
it does so by emitting a photon. And the energy of the emitted
photon, which we can also call 𝐸 p, will also be exactly equal to the energy
difference between the two energy levels.

If we wanted to cause an electron
to transition from the ground state to the 𝑛 equals three state, then we can do so
by directing a photon with energy equal to the size of this energy gap at the atom,
at which point it will be absorbed by the electron. Once this electron has transitioned
to the 𝑛 equals three state, it will naturally return to the ground state after
some time. This can happen in two different
ways.

We might find that the electron
transitions straight back to 𝑛 equals one from the 𝑛 equals three state, in which
case it will emit a photon with exactly the same amount of energy as the one it
absorbed. However, we might find that this
electron returns to the ground state in two separate transitions, going from the 𝑛
equals three to the 𝑛 equals two state and then from the 𝑛 equals two to the 𝑛
equals one state. And if this happens, then two
photons will be emitted, one for each transition that the electron makes. In this case, the photon emitted
when the electron transitions from 𝑛 equals three to 𝑛 equals two will have less
energy than the photon emitted when the electron transitions from 𝑛 equals two to
𝑛 equals one. This is because the energy gap
between the 𝑛 equals three and the 𝑛 equals two energy levels is smaller than the
energy gap between the 𝑛 equals two and the 𝑛 equals one energy levels. And in fact we find that the
energies of these two emitted photons add together exactly to the energy of the
absorbed photon.

Now, there’s one other important
thing to mention, and that is that the energy of a photon is closely related to its
frequency. The energy of a photon 𝐸 is equal
to the frequency of the photon 𝑓 multiplied by a constant. This is Planck’s constant,
represented by the lowercase letter ℎ. This equation tells us that the
energy a photon has is directly proportional to its frequency. Because atoms can only emit photons
with certain energies corresponding to the difference between energy levels, this
means they can only emit photons with specific frequencies. And we can use this equation to
calculate the frequencies of light that are emitted during certain electron energy
level transitions.

In order to do this, there’s just
one more thing we need to know. And that’s the actual energy values
of these energy levels. Now, each element actually has its
own unique energy levels. So let’s take a look at the energy
levels for the simplest element, hydrogen. These values tell us how much
energy an electron would have in each energy level of a hydrogen atom. Because these values relate to how
strongly an electron is bound to the nucleus, they’re also referred to as the
binding energy of an electron in each energy level. So, for example, we can say that an
electron in the ground state of a hydrogen atom has negative 2.2 times 10 to the
negative 18 joules of energy. Or we can say that it has a binding
energy of negative 2.2 times 10 to the negative 18 joules.

Now, one of the first things we
might notice about these values is that they’re all negative, except for this one,
which is zero. Now, it might seem a bit weird to
describe something as having a negative amount of energy. However, this is really just a
convention to show that you need to give extra energy to an electron in order to
free it from an atom. So we say that an electron that’s
not bound to the nucleus of the atom has zero joules of energy. And all the other energy levels are
relative to that.

Since we know that electrons in
other energy levels are bound to the atom and have less energy than a free electron,
we therefore say that electrons in these energy levels have negative amounts of
energy. This convention makes it easy to
work out the amount of energy required to free any electron from an atom. We just take the magnitude of the
energy level value. For example, consider an electron
in the 𝑛 equals two energy level. If this electron has an energy of
negative 5.4 times 10 to the negative 19 joules and a free electron has an energy of
zero joules, then it becomes clear that in order to free this electron from the
atom, we would need to give it 5.4 times 10 to the negative 19 joules of energy in
order to increase its energy to zero. In other words, the energy
difference Δ𝐸 between these two energy levels is 5.4 times 10 to the negative 19
joules.

We can find the energy involved in
other electron transitions by calculating the difference between the two energy
levels involved. For example, let’s consider an
electron in the 𝑛 equals one energy level transitioning to the 𝑛 equals three
energy level. The difference in energy Δ𝐸
between these energy levels can be found by subtracting the energy of an electron
with 𝑛 equals one, we’ll call that 𝐸 one, from the energy of an electron with 𝑛
equals three, 𝐸 three. So, for a hydrogen atom, that’s
negative 2.4 times 10 to the negative 19 joules minus negative 2.2 times 10 to the
negative 18 joules. This gives us a positive result of
1.96 times 10 to the negative 18 joules. This is the amount of energy which
would need to be given to an electron in the 𝑛 equals one energy level in order to
cause it to transition to the 𝑛 equals three energy level. This means it’s the amount of
energy a photon would need to have in order to cause this electron transition to
happen. And this is exactly the same as the
amount of energy that will be released in the form of a photon by an electron
transitioning the other way from 𝑛 equals three to 𝑛 equals one.

Now, because the energies involved
in electron transitions are very small, it’s common to express them in units of
electron volts rather than joules. Recall that one electron volt is
equal to 1.60 times 10 to the negative 19 joules. This means that to convert a value
from joules into electron volts, we just need to divide it by 1.60 times 10 to the
negative 19. So the energy levels in a hydrogen
atom take these values when expressed in electron volts. And the energy difference between
the 𝑛 equals one and the 𝑛 equals three energy levels is 12.1 electron volts.

So, now that we’ve calculated the
amount of energy that’s associated with an electron transition between two specific
energy levels, we can use this equation 𝐸 equals ℎ𝑓 to calculate the frequency of
a photon that would either be absorbed or emitted during such a process. Let’s clear some space on the left
of the screen and see how this works.

So the first thing we’ll do is
rearrange this equation to make 𝑓 the subject. This gives us 𝑓 equals 𝐸 over
ℎ. Recall that 𝐸 is the energy of the
photon and ℎ is a constant known as Planck’s constant. The value of Planck’s constant is
6.63 times 10 to the power of negative 34 joule-seconds. It’s also commonly expressed in
units of electron volt seconds, in which case it takes a value of 4.14 times 10 to
the power of negative 15.

The reason that we sometimes
express Planck’s constant using different units is that we commonly express the
energies of things like photons and electrons in joules or electron volts. And the version of Planck’s
constant that we use depends on which of these units we’re using to express
energy. So, in this equation, if our energy
is expressed in joules, then we should also use the version of Planck’s constant
that’s expressed in joule-seconds. Conversely, if 𝐸 is expressed in
electron volts, then we’ll use the version of Planck’s constant that’s expressed in
electron volt seconds.

So, to calculate the frequency of
the photon which is emitted when an electron transitions from the 𝑛 equals three to
the 𝑛 equals one energy levels of a hydrogen atom, we first need to use the fact
that the energy of the photon will be exactly equal to the energy difference between
those two energy levels. And expressed in joules, that’s
1.96 times 10 to the power of negative 18. Substituting this into our
equation, we have 𝑓 equals 1.96 times 10 to the power of negative 18 joules divided
by whatever Planck’s constant is in the appropriate units. In this case, that’s 6.63 times 10
to the power of negative 34 joule-seconds. Typing this into our calculator, we
get a value of 2.956 times 10 to the power of 15 hertz. Or rounded to the nearest
significant figure, that’s three times 10 to the power of 15 hertz.

Now, as long as we did everything
correctly, the answer we get isn’t dependent on whether we use joules or electron
volts for the photons’ energy. So let’s try repeating this
calculation, but this time we’ll use the energy as expressed in electron volts. So, this time, the energy is 12.1
electron volts. And since we’re using electron
volts, we use the version of Planck’s constant expressed in electron volt
seconds. And if we type this into our
calculators, we should find that we get the same answer. The frequency of the photon that’s
emitted when an electron transitions from the 𝑛 equals three to the 𝑛 equals one
energy levels of a hydrogen atom is three times 10 to the power of 15 hertz. This is also the frequency of the
photon that’s absorbed when an electron transitions from the 𝑛 equals one to the 𝑛
equals three energy level.

Okay, so now we’ve done some
calculations, there’s one more thing to talk about that relates specifically to
hydrogen atoms. When we look at an energy level
diagram like this, we can see that because there are an infinite number of energy
levels, then there are, at least in theory, an infinite number of electron energy
level transitions that can take place. We actually find that electron
transitions are most common within the lower energy levels of an atom. Now, because these energy levels
are unique to specific elements and each of these transitions produces a specific
frequency of light, this means we can actually identify different elements by
looking at the frequencies of light that they emit. The set of frequencies of photon
emitted by a specific element is known as its emission spectrum.

The emission spectrum of hydrogen
in particular has been very well studied by physicists. And names have been given to the
different groups of frequencies which are emitted by hydrogen. Those frequencies, which are
emitted when an electron falls into the ground state of a hydrogen atom, are known
as the Lyman series. Frequencies produced when an
electron falls into the 𝑛 equals two energy level of a hydrogen atom are known as
the Balmer series. And frequencies produced when an
electron falls into the 𝑛 equals three energy level of a hydrogen atom are known as
the Paschen series.

Now, let’s summarize the key points
that we’ve learned about electron energy level transitions. Firstly, we saw that the energy
level of an electron which is bound to a nucleus is denoted by the principal quantum
number 𝑛. The lowest possible energy level
has 𝑛 equals one, which is known as the ground state. And higher values of 𝑛 correspond
to greater electron energy and weaker binding to the nucleus. We’ve also seen that electrons gain
energy by absorbing photons and lose energy by emitting photons. And the frequencies of these
photons correspond to the difference between the energy levels. And finally, we saw that certain
hydrogen transitions have names, such as the Lyman, Balmer, and Paschen series.