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SPEAKER: So Bohr had this. And did it match reality, was the question.
What does it mean? I'm missing a slide.
So let's make a plot. We'll put zero here, and down here is minus
-2.18 x 10^-18 J. And for this drawing, I'll assume Z equals 1, hydrogen
atom. Just keep it simple.
So where are these energy states? Well, obviously when n equals 1, it's right
down here. When n equals 2, where is it?
n equals 2. Z is 1.
Well, it's 1/4 of the distance, approximately here.
When n equals 3, where is it? Up here.
It's like 1/9. When n equals infinity, where is it?
Zero, right? So what happens is, these states get--
we'll call them states already. These energy levels get closer and closer
and closer to zero. And so Bohr looked at this, and he said, now
I know where these lines come from.
Because he said, look, if in this gas discharge, I hit these gas atoms with
an electron with sufficient energy, I'm going to take an electron in its
ground state, and put it into one of these other states.
It can only go into one of those other states-- other level, sorry.
It can only go into one of those other levels. And then, it's going to fall back down.
When it does, it will emit a photon. So if it happens to be in n equals 2, you'll
get a photon of this energy difference, 2 to 1.
And if it happens to be an n equals 3, you'll get a photon
with energy 3 to 1. And I guess it's possible to get a 3 to 2.
But you'll get photons of very distinct energies. So he did the--
let me see where that page went-- I'll just wing it.
He said, well, if this is the final minus the initial, that'll be the
energy change, in going from the high state to the low state.
Of course, this will be a negative number, right, because it's going from
one state that's less negative in size to one that's going to have more
negative magnitude, I guess. So if I take this equation here--
I'll call this a constant k-- then I have that this is k times Z^2, 1 over
n final squared minus 1 over n initial squared.
Squared. That's the delta E. That's got to be the same
as the energy of the photon emitted.
You have to be careful about the sign, right? This is going to be a negative number, because
the final n is smaller than the initial n.
I should have had this minus sign, right? There's a minus sign here.
But if I say the absolute magnitude of this, then I have--
then I can multiply everything by -1. That's got to be the energy of the photon,
which is hc over lambda, because c equals lambda times wavelength.
In fact, if I solve this for 1 over lambda, what Plank found, amazingly,
was that he reproduced what was then known as the Rydberg formula.
When Z equals 1 for hydrogen-- this guy Rydberg, 40 years before, had been
studying emission from hydrogen gas, had these lines, and just found that
he could fit the position, one over the wavelength of the line, to this geometric
progression, where these were all integers.
In fact, Rydberg put them all together, because before that, there
was a guy named Lyman, who discovered a formula that-- when this was 1, then
n_i was 2, 3, 4, up to infinity. Those were all present in emission from hydrogen
atom. That's the Lyman formula.
In the Ballmer series, n_f final equals 2, where n_i is obviously 3, 4,
5, everything greater was called Ballmer. So this is Lyman.
This is Ballmer. So these formulas have been around.
Nobody understood them. And it was only through this quantization
model of what was happening in this complicated atom that you
could reproduce. So all these things were happening.
Nobody really knew what was going on. I mean, even Einstein didn't really know what
was going on. So we had quantization of light.
That could explain some of these measurements. We had quantization of angular momentum, that
now could explain some of these measurements.
It was all going crazy for them.