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- We've been talking about the Bohr model

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for the hydrogen atom,

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and we know the hydrogen atom has

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one positive charge in the nucleus,

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so here's our positively charged nucleus

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of the hydrogen atom and a
negatively charged electron.

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If you're going by the Bohr model,

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the negatively charged
electron is orbiting

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the nucleus at a certain distance.

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So, here I put the
negatively charged electron

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a distance of r1, and
so this electron is in

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the lowest energy level, the ground state.

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This is the first energy level, e1.

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We saw in the previous video
that if you apply the right

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amount of energy, you can
promote that electron.

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The electron can jump up
to a higher energy level.

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If we add the right amount of energy,

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this electron can jump up
to a higher energy level.

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So now this electron is a distance of r3,

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so we're talking about the
third energy level here.

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This is the process of absorption.

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The electron absorbs energy

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and jumps up to a higher energy level.

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This is only temporary though,

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the electron is not going
to stay there forever.

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It's eventually going to fall
back down to the ground state.

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Let's go ahead and put that
on the diagram on the right.

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Here's our electron, it's
at the third energy level.

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It's eventually going to fall
back down to the ground state,

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the first energy level.

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Here's the electron going back

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to the first energy level here.

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When it does that, it's
going to emit a photon.

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It's going to emit light.

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When the electron drops
from a higher energy level

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to a lower energy level, it emits light.

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This is the process of emission.

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I could represent that photon here.

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This is how you usually
see it in textbooks.

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We emit a photon, which is going to

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have a certain wavelength.

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Lambda is the symbol for wavelength.

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We need to figure out how to relate lambda

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to those different energy levels.

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The energy of the photon is,

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the energy of the emitted photon is

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equal to the difference in energy between

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those two energy levels.

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We have energy with the third energy level

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and the first energy level.

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The difference between those...

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So, the energy of the third energy level

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minus the energy of
the first energy level.

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That's equal to the energy of the photon.

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This is equal to the
energy of that photon here.

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We know the energy of a
photon is equal to h nu.

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Let me go ahead and write that over here.

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Energy of a photon is equal to h nu.

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H is Planck's constant,
this is Planck's constant.

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Nu is the frequency.

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We want to think about wavelength.

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We need to relate the
frequency to the wavelength.

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The equation that does that is of course,

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C is equal to lambda nu.

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So, C is the speed of light,
lambda is the wavelength,

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and nu is the frequency.

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So, if we solve

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for the frequency, the
frequency would be equal to

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the speed of light divided by lambda.

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Then, we're going to take all of that

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and plug this in to here.

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We get the energy of a photon is equal to

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Planck's constant, h,
I'll write that in here,

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times the frequency, and
the frequency is equal to

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c over lambda.

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Now we have the energy of the photon

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is equal to hc over lambda.

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Instead of using E3 and
E1, let's think about

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a generic high energy level.

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Let's call this Ej now,

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so this is just a higher energy level, Ej.

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The electron falls back down
to a lower energy level,

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which we'll call Ei.

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Instead of using E3 and E1,
let's make it more generic,

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let's do Ej and Ei.

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Let's go ahead and plug that in now.

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The energy of the photon would be equal to

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the higher energy level, Ej

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minus the lower energy which is Ei.

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So now we have this equation,

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let me go and highlight it here.

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We have hc over lambda
is equal to Ej minus Ei.

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Let's get some more room, and let's see

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if we can solve that a
little bit better here.

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So let me write this down here.

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We have hc over lambda
is equal to the energy

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of the higher energy
level, minus the energy

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of the lower energy level, like that.

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Alright, so in an earlier
video, I showed you

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how you can calculate the
energy at any energy level.

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We derived this equation.

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The energy at any energy
level, n, is equal to

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E1 divided by n squared.

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So, if we wanted to know the energy

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when n is equal to j, that
would be just E1 over j squared.

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We could take that and we
could plug it in to here.

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Alright, if I wanted to know the energy

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for the lower energy level, that was Ei,

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and that's equal to E1
divided by i squared.

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I could take all of this,

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I could take this and I
could plug it into here.

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Let's, once again, get some more room.

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Let's write what we have so far.

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We have hc over lambda is equal to

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Ej was E1 over j squared and
Ei was E1 over i squared.

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Okay, we could pull
out an E1 on the right.

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So we have hc over lambda is equal to E1,

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and so that would give
us one over j squared

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minus one over i squared, like that.

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We could divide both sides
by hc, so let's do that.

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So on the left, we could
have one over the wavelength

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is equal to E1 divided by hc,

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one over j squared minus
one over i squared.

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Again, from an earlier
video, we calculated what

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that E1 is equal to.

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So, one over lambda is equal
to E1 was negative 2.17

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times 10 to the negative 18 joules.

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So once again, you can
see that calculation

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in an earlier video.

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It took us a while to get there.

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We are going to divide by hc,
and this is one over j squared

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minus one over i squared.

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Well let's look a little
bit more closely at

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what we have right here.

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So, if I, for right now, not
worry about the negative sign,

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and just think about what we have.

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This is all equal to a constant.

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H is Planck's constant, and
c is the speed of light,

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so we have all these constants here.

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We could rewrite all of these as just R.

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I'm going to rewrite this as R,

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so this would be one
over lambda is equal to

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negative R times one over j squared

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minus one over i squared.

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So, R is called the Rydberg constant

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so let's see if we can solve for that.

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Over here R would be equal to 2.17,

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times 10 to the negative 18,

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over h is Planck's constant, that's 6.626

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times 10 to the negative 34,

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and then c is the speed of light.

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So we could use 2.9979
times 10 to the eighth

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meters per second as the speed of light.

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If you do all that math,

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I won't do it here just to save time,

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but if you do all that, you'll get 1.09

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times 10 to the seventh and
this is one over meters.

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I think I might have, it's possible

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I had a rounding issue in here,

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because if you use 2.18,
you get a better number.

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It's 1.097 times 10 to the seventh,

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which is the Rydberg constant.

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This is just, again I'm just
trying to show you the idea

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behind the Rydberg constant
here so we get this value

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for the Rydberg constant.

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So you could plug that in
for R if you needed to,

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and we're going to be doing
that in the next video.

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You could stop right there,

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and you have related the wavelength,

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right to your different energy levels.

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You could go a little bit further,

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let's just go a little bit further here.

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So, one over lambda is
equal to negative R.

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If we pull out a negative one,

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that would give us one over i squared

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minus one over j squared.

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Now we have these two
negative signs, right?

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These two negative signs out here

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which gives up a positive.

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This is now equal to one over lambda,

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one over the wavelength
is equal to positive R,

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the Rydberg constant,
times one over i squared

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minus one over j squared.

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Remember what i and j represented.

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I represented the lower energy level,

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and j represented the higher energy level.

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This is an extremely useful equation,

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so usually you see this called
the Balmer-Rydberg equation.

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We've derived this equation
using the assumptions

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of the Bohr model, and this equation

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is extremely useful because it explains

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the entire emission spectrum of hydrogen.

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This is again, this is
why we were exploring

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the Bohr model in the first place.

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We got this equation,
and in the next video

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we're going to see how this explains

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the emissions spectrum of hydrogen.

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Think about lambda or the wavelength,

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right as the light is
emitted when the electron

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falls back down to a lower energy state.