The next allowed energy level up, n equal two, has an energy of minus 5.44 times 10

to the minus 19th joules. And there's another level after that,

number three, and then number four, and you'll notice that spacing between these

levels is getting closer, and closer. So they're not equally spaced, and if we

look, at the physical properties hydrogen atoms, in these various energy levels.

We discover that's, what is happening is that the electron is moving further and

further away, in terms of the average distance from the proton.

And so for the ground state, remember that's the lowest quantum number, so n is

called a quantum number, it is the integer that indexes what level, what of

the allowed energy levels is occupied? The distance, the average distance, this

is actually an expectation value, is what we would call that in quantum mechanics.

So what is the most likely distance you would get after averaging over a large

number of experiments, is 5.3 times 10 to the minus 11th meters.

And so, if you remember an angstrom is about 10 to the minus 10th meters.

Its about half an angstrom. This unit of distance actually defines

the atomic unit of distance which is called the Bohr.

So one Bohr is defined by the hydrogen atom in its ground state and as I raise

the energy higher higher higher, ultimately I get the electron to the

infinitely separated from the proton. That is there is no interaction between

them and we define that to be zero there is no interaction so the energy is zero.

And that's why we see here at n equals infinity that zero, that would be an

ionized hydrogen atom. And relative to that zero, that allows us

to assign this number to the ground state.

Now if you focus on these energy levels, you would note that they have a certain

progression which can be well described by a simple equation.

It says that the energy of a given allowed level indexed by its quantum

number is equal to some constant divided by the square of the quantum number.

So, if n is one, then n squared is one and we would get the ground state energy,

which, sure enough, is right here, it just isn't expanded to as many digits, so

rounded to 2.18. We can also convert joules to a different

unity of energy. Here's the wave number.

So if you prefer reciprocal centimeters, it's 109,680 divided by n squared

reciprocal centimeters. So, this simple formula to compute the

energy levels would also let us compute the difference in energy between

different levels. And the energy required to ionize the

hydrogen atom, by letting n go to infinity at which point energy is zero.

So I'll let you take a moment here to actually do a little self assessment,

making use of this equation. And then we'll come back to continue to

look at atomic energy levels. Most of you will certainly have seen this

analysis of hydrogen before, and probably, the way you've seen it is in

the context of hydrogenic orbitals. So, what are the wave functions

associated with these allowed energy levels?

Well, the ground state energy level, there is one way to function with that

energy and we usually call that the 1S orbital.

So, the next level up n equals two, has four different wave functions.

All of which have the same energy. That's called degeneracy.

So when there are multiple solutions to the Shrodinger equation, all with a

common energy, we say that those solutions are degenerate.

So given that there are four possible solutions, the degeneracy is four.

And we often index degeneracy by g, and again a subscript to say which level.

If we go up to the third allowed energy level or the third quantum level

sometimes we would say now it turns out that there are nine orbitals or wave

functions that satisfy the Schrodinger equation that all have that energy.

So the degeneracy is now nine. And looking at these orbitals of course

you recognize here's the classic one s. Here's a two s, it has has a node

somewhere in it but we're drawing it solid so we can't see it.

And here are the two p orbitals. Here's a three s, two nodes hidden in

there somewhere. Here are the two p's and for some of

them, this one for instance, you can actually see the nodal structure.

And here you see it as well for the three p orbitals.

And these are the three d orbitals, there are five of those.

And these are the classic atomic orbits, more carefully, hydrogenic orbitals that

one is presented with very early in the study of chemistry.

Now, some of you may be experiencing a moment cognitive dissonance perhaps

because you've seen these orbitals arranged in energy ordering before and

someone has told you that the two s is lower in energy than the two p's.

Or the three s's is lower than the three p's is lower than the three d's.

That's actually not true, it's not true for this system.

As I've shown it, these degeneracies are correct.

All nine of the solutions for the third quantum level, the three s, the three p,

the three d, they all have the same energy.

In a one electron atom, normally we don't work with just one electron.

It's not just hydrogen in an excited state.

It is a more complicated atom. It turns out once you add additional

electrons to the system, that changes the energies of s and p and d solutions.

So everything that you've been taught hasn't been a lie.

But at least for the hydrogenic system, the degeneracy as shown here is what it

is, and you've probably noticed, it looks like it's n squared, right.

For n equal one, the degeneracy is one, for n equal two it's four, three goes to

nine, and sure enough that's correct. The degeneracy of hydrogenic solutions is

n squared where n indexes the quantum level.