Now that we understand something about the energy of radiation, we can use that

to analyze the energies of electrons and atoms.

Remember in the previous lecture I mentioned something about spectroscopy,

which is the study of matter by its interaction with radiation.

In particular, we can shine light on matter, atoms or molecules, and see what

frequencies of light are absorbed by them, by the atom or molecule.

Or, we can energize atoms and molecules, and see what frequencies of light that

they emit. We're going to focus on this particular

lecture on one substance, most notably, hydrogen, the simplest of all of the

elements. We're going to analyze something called

the Hydrogen Atom Spectrum. Here is the concept.

We're going to take a sample of hydrogen. We're going to put it in the electric

arc, that's going to energize the hydrogen in there.

And as a consequence of that, the hydrogen's actually going to emit a

number of frequencies of radiation. To see each of those frequencies, we're

going to pass them through a simple prism, that will separate the frequencies

of light spatially so that we can see them either with our own eyes or with

instruments. Here, we've illustrated what is called

the visible spectrum of Hydrogen. It's four different frequencies of light

which are emitted, or four different wavelengths of light which are emitted.

But these aren't the only ones, these are just the ones that our eyes will respond

to. In fact there are a huge number, of

frequencies of light which are omitted when hydrogen is excited in this way.

And, many of these have to be detected by instruments.

Here is a complete set well, a nearly complete set of the set of frequencies

which, and wavelengths. Which are omitted by excited hydrogen

atoms. And you can see, here, the relationship

that we, talked about before. Which is the inverse relationship between

the wavelength and the frequency. The shorter the wavelength, the higher

the frequency and vice versa. In this case, we've also identified the

region of the spectrum particularly there are those for frequencies that we saw in

the previous slide for the visible spectrum of hydrogen.

We look at the pattern of the data here, let's say we just sort of look down these

frequencies and say, anything there that looks like a pattern that we can observe.

and the answer turns out to be, not particularly, there just seem to be a set

of number there. We've put them in sequence.

But they don't necessarily seem to correspond to anything.

But it turns out there actually is a remarkably beautiful relationship amongst

these numbers. Won't go into the history of how that

relationship was developed, but it goes under the name of the Rydberg equation.

It turns out the Rydberg equation can predict every one of the frequencies

which is in this chart. And it is this simple formula that we've

illustrated back over here. And what is the idea here?

The idea is a pretty simple one. Pick a particular integer n.

And predict, say, one or two. We're taking another integer m.

Make it two or three. Don't make it the same number, and

calculate a frequency of radiation based upon that value of n and m.

And it turns out for every choice of n and m you want to make, there is a

frequency over here that will show up in the table.

Alternative for every frequency that's in that table there exists a choice of N and

M, which can predict it from the Rydberg equation.

Not it's not obvious why the Rydberg equation looks the way it does.

Why should there be a relationship between the frequencies and two different

integers, and what do those integers mean?

None of that is told to us immediately by this experimental data.

Because the experimental data is simply collected and then analyzed to determine

to have this particular form. Let's see how we can now interpret the

fact there are only a select set of frequencies which were emitted by the

hydrogen atom. We're going to walk through a line of

reasoning here, that tells us that individual atoms are, of hydrogen, are

only emitting very specific frequencies of radiation.

That's the experimental observation. Now we're going to combine that

information with what we learned in the previous Concept of Elements study.

Which is that the frequency of radiation is related to the energy of the photons

which are being emitted since only certain frequencies are emitted.

Then only certain photons with certain energies are emitted.

And as a consequence, what we can say is that the atom itself is only capable of

losing rather specific energies. That is, if I'm emitting, if I'm an atom,

and I emit a photon. I lose the energy associated with that

photon. If only certain energies of photons are

permitted then I can only lose certain energies.

But I can only lose certain energies, it must be true, that I can only have

certain kinds of energy transitions that can take place.

In particular, we think back over here, let's imagine first.

I've got an atom and it is emitting radiation, and that radiation corresponds

to an energy loss of H nu. And that must correspond to the energy

loss of the individual electrons in the atoms and as a consequence, if I try to

plot What the energies must for the electrons.

They can't just be anywhere. Because if they could be anywhere I could

lose any amount of energy. Since I can only lose specific amounts of

energy, then there must only be specific energy levels that can exist for the

electrons. Such that I could lose that amount of

energy, or I could lose that amount of energy.

But I couldn't lose just some sort of random amount of energy out here

corresponding to, say that energy loss. Because that wouldn't correspond to the

transition between any two energy levels here.

I might have that one. But I could not have just any random

number energy loss. Therefore since only certain energy

transitions are possible, I can conclude that only certain energies are possible

within the atom. There are only certain energies that the

electron can have in the Hydrogen atom. As a consequence, looking at the spectrum

of hydrogen atom, we can clearly conclude, that the hydrogen atom

electrons must be in one, of a number of quantized energy levels.

Let's see if we can figure out what those energy levels are, from the Rydberg

equation. Let's back our way up, here's the Rydberg

equation again. Remember, this is the frequency of light

which is being emitted over here in the diagram.

What that corresponds to, a certain photon frequency, h nu, that's being

emitted. But that photon energy must correspond to

a certain amount of energy lost by the electron.

So in the next line in the equation here, what we have specified Is that the energy

of the photon is now the negatives of the energy change of the electrons.

So I've inserted a minus sign in there. So the amount of the energy that the

electron can lose is minus h times the frequency of the photon emitted and the

photon emitted must fit the rydberg equation.

Well if that's the case then we can also just algabraically rewrite the change in

the energy of the electrons. All we've really done here is to divide

the previous equation into two pieces corresponding to the n squared and the m

squared, so here's the n squared piece. And here's the n squared piece.

But if I now analyze those, what I clearly see, is that this looks like the

difference, which between two different terms, that look very similar.

Both of those terms have a minus hR, divided by an integer squared And I

subtract by a minus HR divided by a different integer square.

So the energy difference is the difference between two very similar terms

that is strongly suggestive that the energy the electron is simply given by a

very simple formula. Minus H a proportionality constant called

Plank's Constant times R. Proportionality constant called Rydberg's

constant, divided by n squared, where n is just some integer.

One, this actually tells us what the energies are, associated with an electron

in an hydrogen atom. And this formula exactly predicts the

spectrum from the Rydberg equation because we derived it from the Rydberg

equation. But it also tells us something funny

here, it says that the energy of a hydrogen electron depends upon an

integer, n, and that integer is a quantum number.

A quantum number n that has just shown up rather naturally, by examining the

experimental data. So we have our first observation, then,

of quantized energy levels corresponding to individual quantum numbers.

Now that's all true for Hydrogen, what about for other atoms?

Turns out each atom has it's own characteristic spectrum, you can look

these up on the internet, on any place that you want to.

I actually recommend a particular site from the University of Oregon.

Which I have pulled up here for you. That actually allows you to see the

frequencies of light which are emitted, by whatever element you might be

interested in. Lets click on hydrogen here.

Here are those four frequencies of light that we've seen before.

Lets click on Helium. There are quite a few frequencies of

light here. Notice again that these are only the

visible spectrum that we are looking at. There are a number of frequencies which

are outside the visible range for each of these elements.

Pick on your favorite element. How about phosphorus?

We can click on this. There are a significant number of

different frequencies in the visible range.

The important point of this conversation is that each Atom from each element has

it's own characteristic set of frequencies.

And since it has it's own characteristic set of frequencies, they don't apply the

Rydberg equation doesn't apply. So this formula that we created does not

apply to other atoms only to the Hydrogen atom.

However, we can conclude that since each atom has its own characteristic set of

frequencies, then each atom must have its own characteristic set of energy levels.

And those energy levels can actually be determined experimentally and measured

simply by measuring the spectrum that is omitted, the spectrum of frequencies that

is omitted. By each individual atom.

Now it's a different question to ask why can you only have certain energy levels.

Or for that matter, why are those energy levels characterized by a quantum number.

We're going to pick that up by digging into quantum mechanics in the next

lecture.