Lecture 4: Atomic Spectra of Hydrogen

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Let's get started. A couple of announcements. A remember about the Periodic Table test which will be added next week on the 23rd. You can see the website if you want to download a copy of this to practice on it.

You are required only to put the one or two-letter chemical abbreviation, the symbol. Some people ask me if I want you to know the molecular weights and so on, and I thought that was bit excessive.

So I think if you just know the symbols for the ones that are up there, that would be just dandy. Here is the lady's scarf, very hot, black. See, trim lines, blue and gold, and it has elements on it.

Very hot. And there is a necktie, comparable fabric, comparable pattern. It is not geeky. This is very hot. You need this. Let's get on with the lesson. Last day we looked at Rutherford, Geiger, Marsden and the experiment that they conducted on the gold foil.

And out of that came the recognition that Thompson's plum pudding model is not sustainable in the light of this evidence. And in its stead Rutherford proposed something akin to a planetary system. And there is something I would like you to be tuned into here.

People to this day are looking for a theory of everything. And there is something very appealing about the notion that the rules that govern the motion of heavenly bodies might be applicable down at atomic dimensions.

This is very cosmic. There is a gestalt here. We ended the day with Bohr's model of the atom, which derived from the conceptualization of Rutherford. And this model is quantitative. And I think I put up the postulates.

And what I am going to do today is go in some detail through the Bohr model, so let's do that. I am going to put up the most primitive version. It is a one electron atom. It has a nucleus at the center charged positively.

This is where most of the mass resides and the positive charge resides. And orbiting around this is a lone electron out at some distance r. And I don't have to worry about these relative dimensions because we know from Marsden's calculations that, on the length scale that I have shown here, these would be just tiny little dots.

It is about 10,000 to one, the ratio of the nuclear dimension to the entire atomic dimension. And so we have a positive charge in the nucleus which is given by the number of protons. This is a one electron atom.

It could be hydrogen, it could be helium plus, it could be lithium double plus and so on. These are all one electron atoms, and they are gas, a single atom. This will be Z times the elementary charge.

Remember, e represents the elementary charge, not the charge on the electron. The charge on the electron is minus e. This is Z times e in the center. And out here we have the charge on the electron as simply equal to e.

I am going to highlight this by calling this q1, lower case q being the charge here on the nucleus. And lower case q sub 2 is the charge on the electron. And that is given by minus e. And so the other thing that we consider is the nucleus as being stationary.

We have an orbiting electron but a stationary nucleus. And Rutherford, who I have already described to you as being a little bit colorful in his language, backed up Bohr on this. He said when an elephant has fleas it is the fleas that do the jumping.

That is why we don't worry about the motion of the nucleus because the ratio of mass is about 2,000 to one. What I am going to do is use these postulates and go through the basis for the Bohr model.

The first thing we are going to do is describe the energy of the system. The total energy of the system, which we are going to get from postulate number four, which says the energy of the electron, which is the energy of the system, is the sum of the kinetic and the potential energy.

Let's write that. And, as I reminded you at the end of last day, the only reason I am doing this for you right now is I want to elucidate the principles. It is not because I favor derivations in class.

I've been teaching long enough to know that doing derivations in class is equivalent to pouring ice water on the proceedings, so I do this sparingly. It is a Newtonian system from the mechanical standpoint.

This is going to have one-half mv squared where this is the mass of the electron. And then the potential energy, the energy is stored here due to the coulombic force of attraction between the electron and the nucleus.

So, in the most general form, Coulomb's law is the product of the charge, q1q2 divided by their separation, which is r. And, in order to rationalize electrostatic units with mechanical units, we have to put a factor in here.

Otherwise, the units that come out of this calculation won't dovetail with the units that come out of this calculation. And that factor is 4 pi times epsilon zero. Epsilon zero is the permittivity of vacuum.

And now we can plug in q1 and q2. We know that we have one-half mv squared. And then q1 is plus qe, q2 is minus e. So then that will give us a net minus Z times the square of the elementary charge over 4 pi epsilon zero r.

And I am going to label this as equation one. And so this is really a mechanical term, if you like, and this is electrostatic or coulombic. I am going to use different words for that. And where do we find these things? You can find values of all of these if you look at your Table of Constants.

This is the table of constants in its entirety. If you look up really close, there is the elementary charge e. There is the mass of the electron. All of these are given here. This is 9.11 times 10 to the minus 31 kilograms.

It is 1.6 times 10 to minus 19 coulombs. There is the permittivity vacuum. It is all up there. So you have these at your disposal. Now let's do the next step which is a force balance. That comes out of postulate three.

Out of postulate three it says Newtonian mechanics applicable. Let's put a force balance. I am going to say if that electron is to stay in its orbit, that is to say it doesn't flee the atom, it doesn't collapse under the nucleus then the sum of the forces on the electron must be zero.

No net force. And so that will be the sum of a dynamic force plus an electrostatic. Or just to be pluralistic in our language, coulombic force. And so, you know from your Newtonian mechanics, as you were learning in 8.01, the dynamic force here is mv squared over r.

Again, m is the mass of the electron. And now the force, in its most general term is q1q2 over 4 pi epsilon zero, which is the conversion factor r squared. Force goes as one over r squared. Energy goes as one over r.

And, if you ever get confused, the way to remember these is that you know energy is the integral of a force moving through a distance. Energy is the integral of force moving through a distance. So the integral of one over r squared gives you one over r.

If you get them backwards, you will integrate one over r and will get log r. That makes no sense. So there we are. And we plug in our values and end up with mv squared over r minus Ze squared over 4 pi epsilon zero r squared.

And I am going to call this equation two. That is our force balance. And then, for the next piece of information, I am going to go to postulate number five. And postulate number five is some of Bohr's early genius.

Bohr says that the energy is quantized through its angular momentum. He is talking about the energy of an electron. No one, until this time, had suggested that a system would be subjected to quantization except for light.

Planck had already enunciated back in 1900 this e equals h nu. In other words, that the photon has a quantum of energy h nu. So quantization of radiation was already established as of about 1900. But no one dared say we have this little planetary model, I've got an electron orbiting a central nucleus and I am going to endow that electron with quantized states.

That was a big leap and that was Bohr in postulate five. Postulate five I am just going to reproduce. That is mvr, which is its angular momentum, does not take continuous values. It takes discrete values, multiples of some integer n, and the multiplication factor is the ratio of the Planck constant divided by 2 pi where n takes one, two, three and so on.

And I am going to call this one equation three. So I have three equations and three unknowns. And I am not going to take the time to solve it because I think that is a waste of class time, but those are the three unknowns.

The radius of the orbit, the energy of the system and the velocity of the electron, I am just going to present you the solutions. Let's look first of all at the radius of the orbit. If you solve for r, you will end up with this.

This is the permittivity of vacuum. Square of the Planck constant times pi mass of the electron. I am going to stop putting subscript e. It is the mass of the electron. Times the square of the elementary charge times n squared over Z.

Z is the proton number, number of protons in the nucleus. What do I see? I see that r takes on discrete values. R is a function of n and takes on discrete values. And, in fact, the functionality is that of the square of this number times a constant.

So this tells us that the radius of the electron orbit, first of all, there is more than one radius. There is a plurality of values. Secondly, that those values are discrete. They are quantized. And, third, it is nonlinear.

If I look at something that goes as n squared, if this is the edge of the nucleus here and if this is r1, it says when n goes to two the radius goes to four. Let's say this is r1, this is r2 and then this will be r3 and this will be r4.

And even that is not to scale. The higher the n number the greater the radius by far. And this applies to all one electron atoms. Let's look at the simplest case. Suppose we look at elemental hydrogen.

In the case of elemental hydrogen, Z will equal one. And then let's look at the lowest orbit, n equals one. That is the electron in its lowest orbit to the nucleus of atomic hydrogen. And this lowest orbit is termed the ground state.

First floor. If we put one here and one here, the value is simply given by this constant. And if you evaluate the constant you get that r1 is equal to 5.29 times 10 to the minus 11 meters or 0.529 angstroms.

I like the angstrom. It is 10 to the minus 10 meters. It is not an SI unit because it is not a multiple of 10 to the 3rd, but I like the angstrom. I don't like the nanometer. I like the angstrom because atomic dimensions are conveniently measured by angstroms.

And this is given a special symbol A nought. It is the value of the radius of the ground state electron orbit in atomic hydrogen. And this quantity is termed the Bohr radius. And you don't even have to look it up.

Because it, too, is given on your Table of Constants. There is the Planck Constant, number five. There is the Bohr radius down there, number 24. So you just look that up. And anything else is going to be some multiple of it.

The other thing I would like to draw your attention to is this. I can simplify this in the following manner and say that r n, now that I have a Bohr radius for this bracketed quantity, I can write that as the product of the Bohr radius for the value of the orbit in any one electron atom.

Specify the quantum number n and divide by Z. That is this rewritten in a much more compact notation because you have the value of this. And note that as Z increases, as the proton number increases the radius decreases for a given n number.

If you look at the ground state in two different systems -- If you look at the ground state which is n equals one, think about this, if I increase the proton charge, the Coulombic force of attraction is greater.

And so, therefore, all other things being equal, the first orbit should be at a smaller distance. So this is making physical sense. Now let's look at energy. Let's quantify the energy value. If you go through and solve for energy, you will get this equation.

E is equal to minus mass of the electron times the elementary charge raised to the fourth power divided by e times the square of the permittivity of vacuum times the square of the Planck constant times Z squared over n squared.

Again, we see that e is a function of n. You see, the quantum condition, by putting quantization into the angular momentum it is propagated through the entire system. Orbit dimensions are quantized.

Energy is quantized. Velocity is quantized. Everything is quantized. Energy is quantized. It has a different n dependence. If I lump all of this together in some constant, I can then represent that formula as minus K times Z squared over n squared where n again chooses integer values one, two, three and so on.

And we can go through and calculate the value of this quantity in parenthesis. And, when we do so, we get the value 2.18 times 10 to the minus 18 joules. This is 10 to the minus 18 joules for this one atom.

I am going to draw attention to this. It is joules per atom. Or, if you multiply by Avogadro's number then you will get joules per mole. And if you do so, you will end up with 1.312 mega joules per mole for this quantity K.

And you can find that one, too. That is on your chart. If you multiply entry 23, which is 13.6 electron volts, I will show you what the electron volt is in a few minutes. We will just put it up here prematurely.

But that is given in your chart. And then the conversion of joules to electron volts is entry 42. If you multiply those two together you will end up with this quantity. And so let's take a look at what that does in terms of graphical representation.

I tried to give you a simple graphical representation. That was Cartesian space. When I plot r as a distance out from the nucleus that is sort of our simple-minded planetary model. Now let's look at energy.

I am going to put energy in the following manner. It is a conservative system so all the energies are negative. That quantity in parenthesis, I have a mass which is a positive number. Something raised to the fourth power has got to be positive.

Something raised to the second power has got to be positive. Everything inside the parenthesis is positive, so minus a positive number. All the energies are negative because it is a bound system. I start up here with n equals one.

And over here this is the value of minus K. And then, if we go to n equals two, what happens? We go to n equals two for a fixed value of Z. Let's make this Z equals one. We will do it for atomic hydrogen.

It is K over four. And then we go to n equals three. And this is not to scale. And so on. This is what the energy level diagram looks like. This is for atomic hydrogen gas. And I am going to put on here "not to scale."

There are some relative, the notion that the energy gap between n equals one and n equals two is greater than that for n equals two to n equals three. That is correctly represented. These are definitely not equally spaced, but they should be even more disproportionate than I have shown it.

And finally up here we have n equals infinity. When n equals infinity, r equals infinity. And so the energy is zero because the electron is no longer bound. It is free. The electron has been liberated.

This represents the free electron. It is no longer tethered to the nucleus so there is no energy stored in the system. And the closer it gets to the nucleus the greater the amount of energy, which means that you see this in the following manner.

This is called the ground state, n equals one. I already mentioned that. It's like the ground floor. I know some of you come from Europe and other parts of the world where the ground floor is rez-de-chaussÈe or some silly thing like this and you push a button on the elevator to one and you go up to the first floor.

You are in American now, and the International Chemical Community calls the ground state n equals one, not n equals zero. Please observe the traditions of the International Union of Pure and Applied Chemistry, not to mention Niels Bohr.

So this is ground state, these are the excited states and free electron at n equals infinity. And, at this point, we say that the atom has become ionized. That is to say an electron has been ejected.

And so now we have net charge. And we can calculate the ionization energy. The ionization energy must then be nothing more than, that is the energy to go from the ground state here to n equals infinity, so that would be the energy at state infinity minus the energy of the ground state.

Well, the energy at infinity is zero and the energy in the ground state is minus K. So minus minus K is plus K. K, in fact, is the ionization energy. And, in fact, if you go to the chart, to your Periodic Table indeed there it is.

This is a slightly different version from the one you have, but it is the same order. If you look on the Periodic Table this is atomic hydrogen. And, sure enough, there is 13.598, which is this number here in electron volts.

That is the ground state energy of atomic hydrogen. The other thing to note is look at the dependence upon Z. Suppose instead of hydrogen we considered lithium 2plus. Lithium 2plus is a one electron atom.

Lithium has three electrons. If it loses two one remains. What is this formula telling us? It is telling us that the ground state in lithium would be Z squared. It would be nine times more intense.

It would be nine times less. That is to say minus nine is less than minus one. And that makes sense, too, because the positive three pulling on minus one has a tighter binding energy than positive one pulling on minus one.

So this drops by the square, whereas, you saw how the orbit goes. Just picking up on this ionization energy. The ionization energy we can define, this is the minimum energy to remove an electron from the ground state of an atom in the gas phase.

All the things we are talking about here are with reference to the gas phase. Gas phase single atom. So we don't have to deal with work function or any kind of energies associated with some condensed form of matter.

So we are really looking at this reaction here for ionization. It is H gas neutral goes to H plus in the gas phase plus the electron. And, furthermore, we can have multiple ionization energies. If you have multiple electrons we can lose them in sequence.

For example, I could look at the ionization of lithium. Again, lithium gas loses an electron to become lithium ion plus electron. I can then subsequently lose an electron from the lithium ion. Lithium ion loses an electron to become lithium 2plus.

And then a lithium 2plus still has an electron. Let's denude lithium 2plus. So lithium 2plus loses its electron. This is the analogy to the alpha particle. The alpha particle was the helium nucleus.

This is the lithium nucleus plus electron. We call these first, second, third ionization energies in sequence. This is the first ionization energy. This is the second ionization energy. That is to say the ionization energy of the second most electron.

And this is the third ionization energy. Now, where do we find these values? Well, if we look on the chart, the first ionization energy is what is reported in your Periodic Table. The first ionization energy of lithium is about 5.4 electron volts per atom.

We can get this from the Periodic Table, so reported is Periodic Table. And this is determined by measurement. This one here, I cannot do anything for you. You will have to look that one up. What about this one? This one is calculable by the Bohr model because lithium 2plus is a one electron system.

I just go through these equations, but be careful to put in Z equals three. And I can calculate this from Bohr model. Let's say calculable by Bohr model. And then remember Z equals three. I am showing some versatility here.

And, let's see, we've talked about radius, we've talked about energy. Let's talk about velocity just for completeness. So the velocity is given by this product of the quantum number n Planck constant 2 pi mass of the electron times the radius of the orbit, which itself is a function of n.

We can make some substitutions here using some of the derivation on the previous board which will give us the Planck constant divided by 2 pi mass of the electron times the Bohr radius. All of this times the ratio of Z over n where n equals one, two, three and so on **v=hZ/(2*pi*m*a0*n)**.

And we rarely look at velocity because, as you will learn very quickly in 3.091, we are going to abandon this strict adherence to a planetary model pretty soon. But it is interesting. Let's just, for an order of magnitude say what happens for ground state electron in atomic hydrogen? So we are going to put atomic hydrogen Z equals one, ground state n equals one.

That is just this quantity here. So let's find out. V1 in atomic hydrogen, is it fast? Is it slow? What is the number? Plug in the values. You get 2.18 times 10 to the plus 6 meters per second, which is about 1% of the value of the speed of light.

By the way, c, the value of speed of light, in case you cannot remember it, is the number one entry in your Table of Constants. So 1% the speed of light. I would say that is relatively fast. That is a pun.

I think about this as relatively fast. You people have no sense of humor, no sense of joy, no sense of joie de vivre. You are so young. You've only been here a week and are so jaded. You've got four years to go.

So where does c come from? How come the speed of light has the symbol c? Where is the symbol c? It comes from the Latin word celeritas which means swiftness. We get the English word accelerate from the same root word.

That is why we have one. That's why we have c. We are in pretty good shape here. We have a Bohr model, which is quantitative. It makes predictions that can be tested. We have energies. We have radii.

Is there any experimental evidence that can put this to the test? And the answer is yes. In the early 1850s, Angstrom, up at the University of Uppsala in Sweden was conducting experiments on atomic hydrogen.

He was doing experiments in a gas discharge tube. He had a tube with electrodes potted in it filled with atomic hydrogen. And by applying a voltage, he was able to get the gas to glow. And then he analyzed the components of that radiation and this is what he found.

First of all, let's take a look at his experiment. There it is. He used the prism spectrograph to analyze his results. So what do we have here? Here is the gas discharge tube. This is the diagram taken right from your text, there are the two electrodes coming in and this is atomic hydrogen in the gas tube.

And so if he gets the voltage high enough, this gas discharge tube will glow. And the light will be going in all directions, obviously. It is a glass tube. He wants to narrow the light so he puts a couple of slits here.

And then he just takes that narrow column of light and he admits it to a prism. And what the prism does is takes a tiny, tiny difference in wavelength and refracts it through a different angle. And if the prism is of sufficient dimension that angle can be magnified.

And then further you put some recording device, which in those days was a photographic plate, far across a room. So you take a tiny, tiny angle and you go far enough across the room. And now you get separation on the order of centimeters.

And we record here. And this is the blowup of what is on that photographic plate. And this was painstaking work. What you have here is the photographic plate demonstrating a spread. First of all, the glow is made up of constituents.

There are distinct lines here. What appears to the naked eye to be just glowing is actually superposition of different lines of distinct frequency. So, you see, this is quantized. And so he published his results around 1853.

And so they lay. 1853. Let's get Angstrom up on the board here since he did the work. He was Uppsala, in Sweden. He measured the line spectra of atomic hydrogen. By the way, your text has a glaring error.

The label above this figure refers to H2. Take the fattest marking pen you have, scratch that out and replace it with atomic hydrogen. You do not get this line from molecular hydrogen. We have the line spectra lying out there in the literature, and people read the literature.

They read. And one of the people that was reading this literature was a high school teacher who was teaching mathematics in Basil, Switzerland. This man also had the initials J.J. as in J.J. Thompson.

This is J.J. Balmer. And J.J. Balmer in 1885 playing with the numbers associated with the lines there, you notice how the lines are colored? Your book isn't colored. See, I colored them for you. I care about you.

I wanted to illustrate that this is representing different wavelengths in the visible. And the doctor told me it is good for me to express my creative urges. So I got out my pen and I did that. Where was I? What Balmer did, all he has is those four numbers.

He has the wavelengths of those four numbers. And he is playing with those numbers. He is determined to make sense of those four numbers in some kind of a sequence, and this is what he came up with finally.

He came up with the relationship that nu bar, which is called wave number -- He found the reciprocal. What he did was coined this. But he found that by taking the reciprocal of the wavelengths, you have four numbers, he took the reciprocals of those wavelengths and found that they fit a numerical sequence.

One over two squared minus one over n squared where n takes values three, four, five, six. And there is a constant out here. And the constant, if we redo it in modern SI units, would take on the value of 1.1 times 10 to the 7th reciprocal meters.

How does this support the Bohr model? Well, in order to answer that question, I have to give you a little more background information. The first thing I have to do is give you the understanding of the physics of the gas discharge tube.

Let's take a look inside that tube. Let me redraw it. I am going to redraw it horizontally. This is a glass tube. There are some feedthroughs here to allow electrodes to be potted. These are gas-tight feedthroughs.

And these electrodes are then connected to a variable voltage power supply. And it is possible to change the voltage on the plates. And the way I have this configured, the left electrode is going to be charged negatively and the right electrode is going to be charged positively.

So the left electrode will be the cathode as I have configured it, and the right electrode will be anode. And what is inside the gas discharge tube is gas at low pressure. As we increase the voltage, we get to a critical value at which it is possible to actually draw electrons across the gap.

Electrons will actually boil off the cathode and accelerate across. You have a positive electrode here and the electron is charged negatively. It will be drawn to the positive electrode. So it will accelerate from rest and crash into the anode.

If we have nothing here, we see nothing. If we have too much gas in here, in other words, if the gas is at too high a pressure, the collisions with the gas molecules will consume the energy of the electrons and, again, we will see nothing.

If we get the pressure in there just right, we will allow for gas electron collisions and then we get the glow that ultimately was sent through the slits. That is the operation of the gas discharge tube.

Now let's look at the physics. What is the energy of the electron, the energy of this electron accelerating from rest and crashing into the anode here? Well, it is going to be mechanical. One-half mv squared.

And how about potential energy? It is not bound. It is a free electron so there is no potential energy to be accounted for. So it is simply one-half mv squared. But I can equate the amount of mechanical energy in the electron to the amount of electrical energy that was imparted through the electrodes.

And the electrical energy is the product of the charge on the electron times the voltage which is the potential difference through which the electron was accelerated. And so this allows me to, by increasing the voltage, increase the energy on the electron.

You can see low voltage, low energy, high voltage, high energy. It is a linear function. And how about order of magnitude? Let's see. Let's do everything unit. That is a good way. What is the unit charge? The smallest charge I could put here would be 1.6 times 10 to the minus 19 coulombs.

And what if I took a unit charge and I multiplied it times unit voltage, one volt? Unit charge times unit voltage then would give me what? If I multiply these two, what is the coulomb times a volt? Here is the value of using SI units.

SI units means everything goes in, in SI, everything comes out in SI. What is the energy unit in SI? The joule. What is the unit of charge in SI? Coulomb. What is the unit of potential difference? Volt.

With impunity. If I use coulombs here and volts here, I don't even think. I write 1.0 times 1.6. Even that I can do in my head. That is 1.6 times 10 to the minus 19 and I write joules. That is the power of having a rationalized system of units.

That is the good news. The bad news is I hate this number. Look it. 1.6 times 10 to the minus 18, excuse me, we have 1.6 times 10 to the minus 19, 2.18 times 10 to the minus 18, etc. Cannot we get simple numbers like three and seven? Whatever happened to good-old numbers? I have an idea.

Why don't we define a new unit of energy that is nearer to what we have to measure? And that was done. What people did is they said let's define a unit of energy that represents a unit charge accelerated across a unit potential difference, and let's call that the electron volt.

That is one electron volt. The electron volt is not a unit of potential difference. It is a unit of energy. Now, this is 1.0 eV. Now you can see 2.18 times 10 to the minus 18 joules can be 13.6 eV.

Now, I don't have to fill my head with 2.18 times 10 to the minus 18. 13.6, that's a cool number. I can remember that. That is how that works. By the way, this is the cathode. And we have a beam.

It is an electron beam. In the 1890s, the latter part of the 19th century a very fashionable word was "ray." Because it came from optics, you know, rays of sunlight and so on. And, in fact, even when we first started pulling synthetic fiber, the first synthetic fiber cellulose was referred to as rayon because it looked like a ray that was being pulled.

So rather than calling this an electron beam, this was called a cathode ray. And this was a tube at low pressure. And if I evacuate it, I have a cathode ray tube. And, in fact, instead of having just an electrode here, what I could do is I could have an electrode that is hollow.

And then I could put a flat end here and I could put phosphorus here. And then I could put a couple of plates and I could raster the beam so that when I am looking here the beam goes back and forth about 80 times a second.

And the phosphorus glow. And I can be watching a science documentary. All right. The other thing I have to teach you is matter-energy interaction so that we can see how the Balmer series validates the Bohr model.

So there is one more thing you need to know. And I am going to put up this energy level diagram again. This is infinity. Let's call this 3, 2, 1. This is E1 ground state, E2, E3 and so on. And this is zero.

Let's look at the energetics of one of those electrons crashing into a hydrogen atom inside the gas tube. Here is an electron. And I am going to let this arrow somehow indicate the value of the incident energy.

And I am saying electron, but more generally it applies to any incident particle. Later on we could do this with a proton. We could do it with a neutron. Rutherford used alpha particles. We could use argon ion.

There are all kinds of different ways of bombarding, in this case, hydrogen gas. Now, if this incident energy is great enough it will take an electron out of the ground state and promote it. But the electron cannot reside anywhere except in one of these quantized states.

Suppose the E incident is greater than the energy in the transition going from ground state to n equals two. What will happen? That energy will be absorbed by the hydrogen atom, the electron will rise from n equals one to n equals two.

And that amount of energy will be subtracted. And then over here this electron will continue. And I have purposely made the arrow shorter to indicate that it has been slowed because we are going to argue its mass didn't change, so the only way to change its energy is to change its velocity.

This will we call scattered now. It's interacted. This is the scattered particle. But it doesn't end there. So how much energy is there? Let this represent the total energy of the incident electron.

And if that is the energy to go from n equals one to n equals two then this is the amount of energy that has to be left as kinetic energy of the electron. It is like an elastic collision between billiard balls.

You've seen these things. But it doesn't end there because the electron is sitting up here at n equals two very nervous because there is a lower energy state and there is a coulombic attraction to the nucleus.

So what happens almost instantaneously after the collision, the electron up here falls back down. And when it falls back down that energy is given off. And it is given off in the form of a photon. And what is the energy of that photon? Well, the energy of the photon, we know from Planck, is h nu, which is hc over lambda.

Now I have lambdas associated with transitions between energy levels in atomic hydrogen. Gee, I wonder if I could come up with a set of transitions occurring inside atomic hydrogen that just might match what Angstrom measured back in 1853.

That is going to be the challenge. The interesting thing here is that I can change the incident velocity continuously. This is continuously variable because I can vary voltage continuously. Change the voltage, I change the incident energy.

What about the energy of the emitted photons? Are they continuously variable? No, they are discrete. They are discrete. And, furthermore, they are characteristic of the target gas element. Let's say I got up late and I am racing to get to class.

I just burst in now, I look up there and go that's atomic hydrogen. Because only atomic hydrogen has that set of lines which means I could then take the spectra of gas phase species and use that information to identify.

This is now a technique of analysis. I could take an unknown gas and put it in and measure it and then say, wow, those are the lines that are characteristic of sodium. Gee, have you ever wondered how they know the composition of stars that are light-years away? Do they get sample bottles coming in? How do they get that information? What is a star, if not just a giant gas bottle, containerless? It is glowing.

And why is it glowing? It is glowing because electrons are excited. And they are excited and they are jumping up and down and they are emitting. But they are just not emitting anything. I mean, why are some stars this color and some stars that color? Did somebody paint them that way? They are different because they have different compositions.

So you could use this in order to identify the contents of stars. All of this. What did Bohr do? I want to tell you a couple more Bohr stories. The guy was great. Here is what happens. Here is a story.

Capstone of Bohr's work on the Balmer formula. Let me simply tell you that in time I am going to show you that this formula is inconsistent with the Bohr model. You will see that at the beginning of next lecture.

In 1896, Charles Pickering from Harvard found a series of lines in starlight which he attributed to hydrogen, even though they did not fit Balmer. They were off by a factor of four. Now, what do you know about energy? Already from this class, if I told you that I gave you energies in some spectrum but they were off by a factor of four, what would you think? Maybe the Z is wrong.

So, even though the lines were wrong, he says, no, that is hydrogen. Bohr says, no, that is not hydrogen. It is helium. That star has ionized helium, helium plus. Z equals two. Energy will be four times more intense.

Pickering was a nasty SOB and said you are wrong. Because, in fact, it is 4.0016. And the 0.0016 is greater than what I would have as experimental error, so your theory is wrong. Bohr says oh, yeah? He goes back and redoes the calculation, only in this case he says what I am going to do is I am going to redo the calculation for helium, but I am going to consider not just the mass of the electron but the reduced mass of the system.

So I will take into account that there is some contribution of both the nucleus and the electron. And he gets, you are absolutely right, it is not 4.0, it is 4.00163. Take that. What you have is with a simple calculation five significant figures.

That shut him up. Now, just to show it doesn't end, more nonsense at Harvard. This is Cecilia Payne. First woman graduate student in astronomy. First PhD in astronomy. First woman to receive tenure at Harvard.

She was awarded tenure in 1938, denied a professorship for 18 years, and when she presented her Ph.D thesis she was the one that determined that the sun is made dominantly of hydrogen, not ion, the way most of the old goats thought.

Because the earth is made of iron, heaven knows everything is made of the same stuff, all heavenly bodies. Even though the sun is glowing and the earth isn't, but they must be made of the same thing.

And, after all, we are Harvard so we know. Here is what she did. She was fastidious. Look at this. What do you see there? Well, you see iron. But this is misspelled. You would make an excuse and say the detector isn't working.

This is a noisy system. That is what everybody else is doing. But what Cecilia is doing is saying I have got to keep looking. I need a consistent picture. This is what spectroscopy is. Spectroscopy is how you go and look at patterns, not just individual lines.

When she defended her thesis, she was forced to put this phrase in her thesis. "The enormous abundance of hydrogen is almost certainly not real." This is the way science welcomes newcomers. This is the way science on some occasions welcomes new findings.

Just up the street. And I am not talking about the 1600s. This was in the 1930s. This is a painting of her. Dudley Herschbach, a Harvard chemistry professor who has a Nobel prize had this painting hung at one of the halls.

And it is obviously referential to Vermeer's "The Astronomer." We have come a long way from the gas discharge tube to the heavens. But it shows you that with a little bit of understanding of quantization you can go a long way.

OK, we will see you Friday.

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