Lecture 8: Born-Haber Cycle

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A couple of announcements. The contest ends today 5:00. Get the email to my address. Wednesday will be the test, Test 1, and I will say a few words about that, coverage through the end of lecture on Wednesday.

What I am talking about today, I will not be testing you, but maybe you might pay attention because about a month from now we will ask you a few skill testing questions about it. But it's one of those pedagogical things.

If I test right up until the last thing I say, I don't think that gives you enough dwell time. The flip side is there are going to be a couple of lectures now where it is not going to be on the test, so I think you will deal with it.

Monday I will be giving a seminar on engineering education in Halifax, so in my place will be Professor Ballinger who will competently substitute for me and keep the ball rolling. But I will be back Tuesday.

And you can contact me if you have some questions in connection with the material. First thing, I will begin with comments about the test. And that way we won't have to worry about running out of time at the end.

The test is Wednesday the 29th, and it will be written during the normal class time. So if you take the test at 11:00 come here, but there is a plurality of rooms because we want to have some vacancies here.

If your last name begins A through Ke, you will come to this room. If your last name begins Kh through Ya, you go to 50-340. That is the gymnasium on the third floor of Walker. There is a gym up on the third floor and will accommodate a couple hundred people there.

And for those of you at the very end of the alphabet, we have a special room for you, 1-190, so everybody has some elbow room. Please go to the assigned room. And, by all means, if you are scheduled to attend class at 11:00, please take the test at 11:00.

Don't all come crashing into 1:00 because we don't have the room or the number of exams. So please go to the assigned room. Here is what the mechanics are. You are going to just go to the room, as you normally do, come on in, sit down.

That's all you have to do, just sit down. What is the purpose of the test, by the way? Why do I give you a test? It is to give you a status report. I want to give you some feedback to tell you how you are doing soon.

There are two messages I can give you. Either keep up the good work or you better change your strategy because you are not mastering the material. That is all I am trying to do. I am not trying to run a cross-check on the admissions process.

As I told you, I believe that everybody that is in this room has the intellectual apparatus to pass this class, so there is no gamesmanship here. If you have been investing in the time that I have asked you to, you will pass handily.

There are no tricks. You can look at old exams, you can look at your homework, that's the kind of questions you can expect. Nothing to broadside you. On the other hand, if you haven't been keeping up, I hope that you won't pass so I can tell you, look, you have to do things differently.

That's all. Bring five things with you. You have five fingers, five things. Periodic Table, Table of Constants, and you are allowed an aid sheet, an 8-1/2 by 11 sheet of paper. You can write anything you want on it.

You can write on the front. You can write on the back. You can write on the edge. I don't care. You can photocopy. You can photo-reduce. You can take everything from the last ten years of 3.091 and make it microdots.

I don't care. Whatever you want. That is three, Periodic Table, Table of Constants, aid sheet. Something to calculate with. There will be some computations, so something to calculate with. And I don't care if it is a programmable calculator.

The only thing is no wireless devices. This has wireless Internet here. You are not going to be sending emails during and exam. No wireless devices. And Lord help you if a cell phone goes off. And last thing, something to write with.

And, believe it or not, I have students walk in and first thing a hand goes up and says do you have a pen? I cannot believe it. Make sure you have batteries that are in a good state of charge in your calculator.

We don't have 450 calculators for you. And you will write. We will give you the question paper. You just fill in on the paper, hand it in. You don't have to bring any paper. We will start you at 00:05 and we will stop you at 00:55.

Stop when we ask you to stop out of courtesy to your classmates. Everybody might like a little more time. But 00:55 you stop and we will have people collect them either down here or at the back. And we will get you out of here in an orderly fashion.

So that is the mechanics. Now, strategy. I want you to do well. I want you to be able to put your best foot forward. I don't want you to get all frustrated and fail to show what you know, so here are a couple of tips.

First thing, read the test. That comes as a big shock. A lot of people don't read the test. They just open it up, and they just start working. There's going to be, I don't know, four or five questions.

Well, I just put them down in the order that came out of my head. I don't put them in any strategic order. I don't put hardest first, easiest first. I don't even think they are in chronological order by topic.

I think it is just whatever. I said, well, let's do something on, I don't know, pick a topic. What are we asking you? I cannot remember. They are all made up and ready to go. But pick a topic. Pick something about the Periodic Table.

Maybe that's number one. But maybe that is not your strong suit, so don't start on number one. But you don't know that there is an easy one for you down at number four if you haven't bothered to look at number four.

So take a couple minutes and read the entire test. And zoom on the easiest question for you, that way two things happen. First of all, you build some confidence. Secondly, you settle your nerves and then move in ascending order.

The second thing is try every question because I give part marks. It's not all about the number. I only give, I don't know, about 10% or 20% for getting the right number in a computation. I tell my graders, if the student demonstrates minimal acceptable proficiency that student has to get a passing score.

If it is out of nine, it has to be five out of nine. If it is out of ten, it has to be five out of ten. If the student hasn't mastered the basic concept, we will make sure that the grade scheme gives a failing grade.

So if you demonstrate something by writing an intelligent answer just by outlining it and saying, well, what I would do is I would equate the energy and then solve for lambda, I can see that you know what is going on.

I can give five, seven out of ten for that. And then the person who actually plugs in all the numbers and gets the right number will get ten. But if you say, well, I don't know how to do this, I am not going to bother, don't grade the paper.

I will grade the paper. I have some people that say I am not doing this well, oh, forget it. No, keep going. Keep going. And leave tracks. Write your thoughts. You don't have to write everything in equations.

You can give me an outline how to solve. Work parametrically to the greatest extent possible. Let's say you have to go through three or four operations to get a final number, well, do it algebraically.

I have some people, as soon as they see it, oh, they have the frequency here. Oh, E equals h nu. I will plug in. E equals h nu, so I plug in. I look up h and plug in 6.6 times 10 to the minus 34.

They plug it in and make a keystroke error right off the bat. They got some number that looks like for the wavelength they got the distance from here to Chicago, and they are going to drag that through the whole calculation.

What are you doing that for? Conservation of keystrokes. Don't touch the calculator until the last line. Work parametrically, because I can grade easily. I can look at that and go she knows what is going on, eight out of ten.

Boom, we are done. You got the wrong number, forget it. She has a good grasp of the material. Instead, right off the bat you have some stupid number and you are dragging that through the calculations.

You have negative wavelengths. You have square root of minus one for something else. Focus. Answer the question. Don't play games with me. If I am talking something about the Bohr model and you start giving me some core dump about what you know about the Bohr model, it doesn't go by the pound.

I have read an entire page full of words and given it zero because it is not on topic. Just be surgical, give me the answer and move on. But don't start snowing me. My graders are smart, smarter than you.

They are. I have smart graders. Do your own work, I mean it. Do your own work. Here we are terrorists. Keep your eyes on your own paper. I don't want to see anybody like this. If I see that, I will just go up and reseat you.

And, by the way, if you ever feel that you are in a compromising position, someone is pressuring you for information, just raise your hand and ask to be reseated. We don't ask any questions. As far as we know, the person next to you has BO.

Just take yourself out of the situation. And, by the way, this business on working on every problem is important because I have seen some people who start working on number one, but that is not their strong suit.

Let's say you have four questions, you have 50 minutes, take out five minutes to read the paper, so that's 45 minutes. That's roughly 10, 11 minutes per question. There are some questions that might be a different length.

You spend 10 minutes on number one and haven't solved it yet. Well, another five more minutes. That's 15 minutes. Now they are getting steamed. It's 20 minutes into the exam and you are still working on number one.

And then they get really angry and say I am going to solve this if it is the last thing I do. Sure enough 11:55 comes, it is the last thing they are doing. Now, think about the math. Let's do the math here.

Suppose there is an epiphany at 11:54 and you get a perfect score on that question and it is worth 25% of the paper. You get 25 out of 25, terrific. You got zero on the rest because you didn't write anything.

You see what I'm saying? Try all the questions. And good luck. I mean. There is no attempt to deceive. Come prepared and you will do fine. And our goal is to grade those things and get them back to you the following day.

We give a rapid turnaround. And we will post the model solutions. And there will be a different exam in the afternoon, but I think it only makes sense for you not to discuss the paper with people that are taking it in the afternoon.

We're trying to run a school here that has some professional standards. And there will be plenty of information on the website and whatnot. All right. Well, let's get on with the lesson today. Just to refresh our memories, last day we looked at the average valence electron energy as a measure of reactivity.

And en route we discovered that we can divide the Periodic Table into metals, which is about 75% of the Periodic Table. These have low average valence electron energies. Which means that these elements are very good electron donors.

And up in the other extreme are elements with very high average valence electron energy. These are electron acceptors. They cling to their electrons quite tenaciously. And, subsequently, we looked at photoelectron spectroscopy which is a technique that allows us to determine binding energies, ionization energies being just one example.

But we will see later on that PES can give us other important information that we cannot calculate. And then, towards the end of the lecture, we started looking at reactivity. And we put up this hypothesis that our observation is that octet stability seems to be an attractive electron configuration.

So we saw the Noble gases. And then we said you can also achieve octet stability and electron transfer if you look at elements that are just a little bit rich of electrons versus octet stability or a little bit lean of electrons.

And so we got to ionic bonding. And I think we just got to the point where we were able to start looking at the energetics of that. And we ran out of time so I want to resume that. I think I had managed to get to this point here where I show what happens when the cation, here sodium, is in contact with the anion here, chlorine.

And we reasoned that these two eventually reach some kind of an equilibrium separation which we are using lowercase r to represent. This is the separation measured from the center of the sodium nucleus.

And there is this separation which is a balance of attractive forces because the chloride is net negative and the sodium is net positive, but both of them, regardless of net charge, have electrons. And so when they become very, very closely spaced there is mutual repulsion of those electrons.

And so there is a repulsive term here. And so the balance of the attractive term and the repulsive term eventually leads us to this situation where we have the equilibrium spacing. And this equilibrium spacing is denoted r sub zero or r naught.

And r naught really is nothing more than the sum of the ionic radius of sodium and the ionic radius of chloride. So we could just say r plus plus r minus gives you r naught. And right off the bat you notice that I am making a huge assumption here.

I am modeling the ions as hard spheres. I am modeling them as billiard balls of a finite dimension. And so they eventually reach this equilibrium position. And we saw last day that we have a Coulombic force of attraction which you have seen before.

Cation is plus one, anion is minus one. But it doesn't have to be. This could be magnesium. Magnesium would be 2 plus, so in here you would put 2e. Or I could have an oxide here, so I would put minus 2e.

In this case we just have one and one. And then the repulsive term, some positive coefficient and r to a very high number. N lies between 6 and 12, and this is known as the Born exponent. And this is a very high number because it depends upon electron-electron repulsion.

These electrons, first of all, there are many of them. And they are in motion and moving at very, very high speed. And so the result is that this repulsive term is very, very intense. But very short order.

Beyond a modest distance you don't feel the electron-electron repulsion. So you need the mathematics to imitate reality. And that is why this is such a high number. And down here, what do we have? We have the sum of these two bottoming out.

See, at very, very high values of r, one over r dominates. At very, very low values of r, one over r to the 10th dominates. And in between the two cancel, and here we are at the minimum. And we can find r naught by looking for the minimum.

How do we get the minimum? We differentiate dE by dr. We take dE by dr equals zero at r equals r naught. And, coincidently, what is the partial of energy with respect to distance? That is force.

So what am I saying? The net force is zero. Attractive and repulsive forces are balanced. So everything makes sense here. And we can solve this. And, if we do so, we can actually get a handle on that Born exponent by solving the equation.

And I am not going to do it. I am just going to put it up here without taking time. And so what I get is the energy of the system at the equilibrium separation is given by minus z plus. This is the most general form.

This is the valence, the charge on the cation times -- I will common factor this. The charge on the cation times e. The charge on the anion times minus e, so there is the minus e squared, and divided by 4 pi epsilon zero r naught, because now I am evaluating this function at r naught, one minus one over n where n is the Born exponent.

And this applies only at, obviously, r equals r naught. But that is good because that is the equilibrium separation. So that tells us the energy of what? What have I formed here? I formed a bond, an ionic bond.

So this is the energy of a single ionic bond. And what do we notice? We notice that the value of E at r naught is negative, as it should be. It's a negative number. And what are the characteristics of this bond.

Well, there are two things that I want to highlight. First of all, it is omnidirectional. What do I mean by that? There is no preferred direction. It could form in any direction because the Coulombic field is radial in all directions.

There is no preferred orientation. The field is radial. This is the electric field. So there is no preferred direction. And the second thing is it is unsaturated. What do I mean by that? Well, I show you one sodium and one chloride.

There is nothing saying that that sodium, its positive charge has not been neutralized. It still has positive charge, and there is nothing saying another chloride could not stick to it. We get another chloride and another sodium sticking to the chloride.

This means it is not a strict bond one-to-one. It is not monogamous is another way of putting it. It allows for plurality of bonds from any ion. Plurality of bonds can form from any ion. And, if you take all of these ideas, omnidirectional and unsaturated, it means that ions can keep glomming on.

And that result is we get large arrays of ions. Well, what does give us? Well, if I have a large number of species glommed together that means it is going to be solid at room temperature because gases exist as discrete particles.

We have either single atoms or molecules consisting of small numbers of atoms. You don't have gas molecules consisting of thousands of atoms. When it gets to thousands of atoms things start to condense because there is a capacity to form other bonds secondarily, which we will get to in a little bit.

So just take it on faith. If you have a large number of atoms in an aggregate, it is going to require that the substance turn solid and condense at room temperature. And word array is loaded. These don't just sit down anywhere.

They sit down in a special way. And so what you're going to end up with is, well, let's go over that. Here is sodium and chloride. They keep attaching and attaching and attaching, but we have two sizes.

We are modeling them as hard spheres. And so they are going to fit together in a regular array. And so this means that they will form a crystal. So we are going to have solids and we are going to have crystal.

And what do we mean by crystal? Crystal is an ordered solid, as opposed to a disordered solid. And we will talk about both in a little bit. Now I want to show you that this comes necessarily from an energetic standpoint.

It is a really cool derivation. I want to compare two situations. Here is an ion gas. This is my humble drawing to indicate discrete sodium chlorine pairs. This is one of these pairs that I have shown up here.

It has a negative energy. We see that we get a negative energy by attaching the chloride to the sodium. And I am saying suppose these form, as they have to initially, in the gas phase. And they remain two atoms per ion pair discretely bombing around in the gas phase.

That is situation number one. Situation number two is I take the same number, and I make a row. They are all in one row. So what happens? I have only drawn three, but now let's pump up the volume.

Here, let's get N Avogadro. If I have Avogadro number of these things in a long line, you can bet it is going to be a solid. It's going to be a one-dimensional crystal. The question is can I show you, from an energetic standpoint, that the crystal is favored over having the discrete set of pairs? And you know the answer is yes because I wouldn't set myself up for failure like this, would I? So let's take a look.

First of all, situation number one, well, you know how to do that. Because you know the energy of one pair. So let's just multiply that by Avogadro's number. E1 will simply equal N Avogadro times the energy of the ion pair at r equals r naught, which we can then substitute N Avogadro.

I am doing this for plus one minus one like sodium chloride. We can make it more complex later. And you have to get N from experimental evidence. You cannot pull that out of theory. The Born exponent is determined experimentally, and there are tables of that.

Now let's get the energy for the line. For that I am going to draw a line of these. All right. So here is the sodium and the chloride on each side, and then we will go for another sodium and another chloride on each side.

And so now what I want to do is start here at the center. And I am going to count the energetics. And what I am going to do is say start with this ion, add up the energy associated with the interactions between that ion and everybody else in the row and then multiply it by Avogadro's number, because that is the number of atoms there are in a row.

Sauce for the goose is sauce for the gander. You say, yeah, what about edge effects? I got Avogadro's number of these things. What do you think the error is going to be by failing to recognize that at the end of 6 times 10 to the 23rd I have over-counted some energy? It is vanishingly small.

So don't get preoccupied by the end. Work the big middle. And the middle is everything here. So we can start. This distance here is r naught. I know the energy in this first pair would equal -- That is just going to equal minus e squared over 4 pi epsilon zero r naught.

That is just what we have seen before. But, in addition, the field is continuing to radiate so it is unsaturated and it is omnidirectional. This cation feels the repulsive force of the cation over here.

This cation is attracting the chloride next to it and it is repelling the sodium as the next nearest neighbor. So that is a repulsive term. I have to add that up. I have got to put that in because that is a repulsive energy and that is going to decrease the energy of the system.

So I have a distance here of 2 r naught. And at a distance of 2 r naught, I have a positive repulsive term. I just plug into the Coulomb's Law. And so that is going to be plus e squared over 4 pi epsilon zero 2 r naught.

Now what happens? Let's keep counting. Now the next one is the chloride. It is of opposite charge and it exerts an influence, only that is minus to plus so that is going to give me a minus term here.

That's 4 pi epsilon zero 3 r naught and so on and so on. I think you can see the way this is going. Well, what about the Born term? Forget the Born term. The Born term is not going to be felt, except in this first one.

I am going to forget it now, and then I will patch it in at the end. You don't need Born repulsion at 2 r naught. That is a waste of memory. Then what are we going to do? I am only counting in one direction, so I have to multiply this whole thing by two because I have to count both directions.

Now I am going to common factor this and will get minus e squared over 4 pi epsilon zero r naught times two, times this series, one minus one-half plus one-third minus one-quarter, etc. So this series is broken into two terms that are conveniently representing two physical ideas.

This is electrostatics right here. The pre-factor embodies all the electrostatics. And what is all this stuff? Y is at one, one-half, one-third? This is the instant geometry. That is to say the instant atom arrangement.

In this case it is a line. Now what I am going to do, this is off of just that one central cation. So now let's do it for all of the ions. What I am going to do now is I am going to multiply by N Avogadro and then add Born repulsion.

And we put all that together and here is what it looks like. It looks like. E equal to minus N Avogadro e squared over 4 pi epsilon zero r naught, one minus one over n. And I am going to call this whole geometric factor here, I am going to put it into one number, and I am going to denote that the Madelung constant.

The Madelung constant is simply the geometric factor. That is a number. Let's put that there. And so now the question is, what I am trying to prove to you is that E2 is more negative than E1. So, ultimately, what I want is what is the relative magnitude of E2 over E1? What is it? Is it greater than one? That's the question.

If I can prove that it is greater than one that means that the line is more stable than the set of ion pairs. And if you look at this series, the series log of one, what is the series expansion? Log of one plus x, what is it? It is x minus x squared over two plus x cubed over three, etc.

If you plug in x equals one that is our series. One minus one-half plus one-third, it is all there. If you go and work that out, log of two and you have two log two, by math you can show that for the line r series m is equal to two natural log of two, which is 1.386 which is greater than one.

So what have I shown? I have shown that by taking Avogadro's number of individual ion pairs and putting them all together in a line, the system's energy became more negative. So that is to say that it is favored.

We could say an energy level diagram -- This would be energy. Here is zero. If this is the energy of the pair, let's make this E over E of the ion pair, so this is one, and we have just shown that for the line, so this is Avogadro's number of pairs, this is Avogadro's number of ion pairs in a line, that is 1.386.

And then, if you go the real sodium chloride crystal and you do this same calculation but in three-dimensions. So that you take every sodium and you count all of its nearest neighbors, next nearest neighbors, next next nearest neighbors, not only on a line but in the plane and above and below in all three dimensions and go through that calculation, it is even more negative.

It is 1.7475. Conclusion: if I start to react sodium with chlorine, the result is a crystal, a three-dimensional array. Because at every point the first thing happens is electron transfer. Why? To get a stable octet.

The result, I've got ion pairs. Pluses attract to minuses, minuses attract to pluses. They start glomming together, but they don't stop with one plus one. One plus one and then an ion pair grabs another one, another one.

And, before you know it, you have crystal. So this is simply the energetics of crystal formation. This is a three-dimensional crystal. You can consider this line a one-dimensional crystal because it is ordered.

So I would call it a one-dimensional crystal. This says that when it comes to ions line-dancing trumps ballroom dancing. That is what it is saying. They would rather be in a line than dance two-by-two.

That doesn't apply to all things, but it does in this case. Now let's look at properties. Let's talk about properties. See, we've gone a long way. We started with this hypothesis of octet stability and I'm talking about properties of ionic crystals.

They are solid at room temperature. Why? What are the bonds? These are huge energies, Coulombic forces, high melting points and boiling points. Why do they have a high melting point and boiling point? Because, when you ask yourself that question, you should consider what is the thermal energy versus the bonding energy? If the bonding energy is very strongly negative, thermal energy isn't great enough to disrupt those bonds and allow those bonds to be broken and then have fluidity.

So strong bonds means high melting points and boiling points. Transparent to visible light. A sodium chloride crystal is transparent to visible light. I know table salt is white, but that is because you have power and you have multiple surfaces scattering, but a large crystal of sodium chloride is clear and colorless.

And it is faceted. You can see that it has the shape. If you take one of those individual grains of sodium chloride, look at it carefully, you will see the edges look like this. Why is it transparent to visible light? Here is how you ask that question.

When you want to know if something is transparent to visible light or not, here is the mystery material and here is visible light. And what is going to allow for absorption or some interaction with visible light? Inside here you have electronic states.

And you have to ask yourself is the energy difference here, the delta E in the electronic states, how does that compare with the E of visible light? What is the E of visible light? You know it is about two to three electron volts.

This has what? Sodium plus. It is isoelectronic with neon. Chloride minus. Isoelectronic with argon. What do we know about the average valence electron energies? We just saw it in the first slide.

It is up around 13, 14, 15 electron volts. This thing is not going to touch it. It won't lay a glove on it. So what happens? It goes through. Nothing. All that glitters is not gold, but it must have free electrons.

This has no free electrons. They are all bound. That is how you determine if something is transparent. It is going to be electrical insulators. There are no electrons to move around. They are all bound, hard and brittle.

Why? Because they have strong bonds. Strong bonds means the material will resist and applied force. You apply a force and the material will resist it until you have a force great enough to break the bonds.

So you don't have ductility, which is what you have in metal. That is what makes metal so fascinating, because metallic bonding allows the atoms to glide over one another without resulting in catastrophic failure.

Whereas, ceramics, ionic crystals will resist until they shatter. There are only two categories. It is either elastic with full resistance or destruction. Again, the bonds soluble in water. Other polar solids, we will have to come back to that later.

They melt to form ionic liquids. Ionic solids form ionic liquids. I showed you last day what happens when they form and we were able to electrolyze and make fantastic metals like magnesium. Where do we find these ionic crystals? Where do we look? Well, you need electron transfer.

You need a donor and you need an acceptor. So where are the good donors? Donors are over here, the metals on the left end of the Periodic Table. And where are the acceptors? On the right side of the Periodic Table.

If you take some from column one or two and you mix it with some from column eight and nine, bingo, you have reaction. Here is an example that shows some very, very metallic elements, groups one and two, and some non-metallic elements over here, five, six and seven.

And so you can imagine that if you mix these, if you take sodium and you mix it with chlorine you get sodium chloride. But there is nothing saying you cannot take magnesium and mix it with oxygen and make magnesium oxide.

And you would expect that that would be a much stabler compound because, instead of plus one attracting plus one, now you have plus two attracting minus two. So the Coulombic forces are much, much greater.

And that will be reflected in what? Higher melting point. Higher boiling point. Greater resistance to reactivity. Does it strike you odd that maybe magnesium oxide, one of its applications would be refractories? That is to say brick work in high temperature furnaces, maybe tiles on the Shuttle, to resist high temperatures because of the high internal bonding.

You see, there is a system here. Chemistry isn't a litany of facts. We started out with a few simple assumptions and look at what we are doing now. We are designing systems, building engineering systems, making materials choices.

It is fantastic. We can do this one more time. Let's do it one more time. I want to show you something else. I want to show you another way to get a sense of these energies. And for that I want to look at the energy starting from real material.

See, up until now I have been talking about gas phase, single atoms, all this stuff. Let's look at this reaction. Sodium solid plus chlorine gas. It is a diatomic molecule. And I want that to react to give me sodium chloride as a solid and crystal.

**Na (s) + 1/2 Cl2 (g) --> NaCl (s, xtal)** I want an ordered solid. For this I need to invoke a couple of bits of science. First one is Hess' Law. Hess' Law states that for any chemical reaction, the energy change is path independent.

That is to say I can now reroute that sodium plus chlorine reaction and go a different way, but I still end up with the same change in energy. Why? It is because it is state function. It is the same thing as gravitational energy.

Whether you take the elevator to the top of the Hancock Tower or whether you walk up the stairs, the change in gravitational potential is identical. You might be winded doing it the second way, but when you get to the top and you are at the 60th story your gravitational potential is independent of how you got there.

And that is what this is saying. It is analogy. Energy change is path independent. Now I am going to break this into elementary steps and look at the relative contributions. What did we do so far? We talked about this reaction here where we had chloride ion in the gas phase plus sodium ion in the gas phase.

We started with gaseous sodium to make gaseous sodium ion and gaseous atomic chlorine to make gaseous chloride ion through electron transfer. All the gas phase. Very idealistic, but that's not the way the world works.

I want to get to there starting from real sodium and real chlorine. The first thing I am going to do is I am going to convert the sodium into vapor. This is sodium going to sodium gas. And then I am going to take sodium gas and I am going to make this into sodium gas plus electron.

Oh, by the way. This is called sublimation. This process is sublimation, solid to vapor. This is called ionization, gas phase species loses an electron. That is the strict definition of ionization.

So I am already there with sodium. Now, chlorine I have to do a little bit more heavy lifting here because chlorine starts as a diatomic molecule. The first thing I have to do is bust that in half.

This is called dissociation. I need to dissociate the chlorine. And now I have to convert atomic chlorine into ionic chlorine. I am going to do that by adding an electron. And this reaction here is the inverse of ionization, isn't it? Its ionization would be I take one of the electrons on chlorine and throw it away.

Here I am actually throwing an electron onto chlorine. So this reaction is called electron affinity. You can think of electron affinity as sort of the ionization in reverse of the ion. It is the minus.

It loses an ion. And this reaction is what we call crystallization. Those are the five steps. Let's see if we can find information. I am going to call this one. Let's see. I think I labeled this one two, ionization I labeled three, electron affinity I labeled four and crystallization I labeled five.

And thanks to Professor Hess, I can now say that the total energy to form sodium chloride should equal -- In other words, delta H formation of sodium chloride should be independent of paths. I should be able to take the sum of all of those delta Hs.

Oh, I'm using H. Forgive me. This is the term enthalpy. It is essentially equal to internal energy for condensed systems, but when you look in the books sometimes they will use this term. For a first pass, let's just say for condensed systems we can go with it.

We are going to take the sum of the delta H's or the deltas H for all of these little reactions. Let's get some numbers here. Let's see. This is called the Born-Haber Cycle, what I am showing you right now.

And this was put up by two people, Max Born and Fritz Haber. Actually, I put this up -- If you are interested in reading about these people, there is a Nobel prize website here you can learn about all these guys.

This is Haber. He got the Nobel in 1918. He invented ammonia synthesis with catalysis by iron. A brilliant chemist. And Born worked on a number of things, including the Born exponent, and taught himself crystallography sort of as something to do during World War I.

And that is how come he got involved in this, because obviously this is talking about crystal formation. So that is why this is called the Born-Haber Cycle. Anyway, if you go to your Periodic Table you will find that on the one side you can get what is called the enthalpy of atomization.

And that is this first term. Delta H1 is enthalpy of atomization. And if you look it up, for sodium it is about 108 kilojoules per mole. Actually, what I did was used a different source so I have 107.

But that is OK. What's a kilojoule between friends? Number two is dissociation energy. And dissociation energy is 122 kilojoules per mole. And number three is ionization. If you flip the Periodic Table over you will have the ionization energy of sodium, first ionization energy, and it is about 5 eV which, when you convert, is a whopping 496 kilojoules per mole.

This is looking bleak, people. Look at this. I have three out of the five energies, and they are all positive. This thing better start turning negative soon or we are going to change the way the world works here.

And I know that sodium chloride forms because, if I look on the webpage at 3.091, you see this down here, this is sodium in kerosene. That is metallic sodium that would keep in kerosene so it doesn't react with the oxygen in the air.

And that is an Erlenmeyer flask that has been filled with chlorine and sodium has been added to it. And there is the sodium chloride crystal that forms between the two of them. And, because I looked at the website, I know this reaction has a negative number.

There is an interesting observation about electronic structure, by the way. Metallic sodium is toxic to humans. Chlorine gas, toxic to humans. Electron transfer, we cannot live, we die without sodium chloride.

So this tells you that it's electron structure that governs. Let's find some good news here. Well, I have some good news. Number four, electron affinity. Chlorine has a high valence electron energy.

It gobbles up electrons. And that is a negative number. That turns out to be negative 349. That is for part four. And then the last one, huge. If you go to the Madelung calculation, it is basically this one over here.

Multiply by 1.7475 and you end up with minus 787 kilojoules per mole. When you add this whole thing up, it gives you a net of minus 411 kilojoules per mole or about 4 eV per ion pair. Good. I was worried there.

That is the Born-Haber Cycle. And the Born-Haber Cycle allows you to see what the relative values of the different energy components are. Again, what's the utility of the Born-Haber Cycle? It deconvolves that reaction of sodium plus chlorine into elementary steps that are related to electron transfer and acquisition.

That is the value of Born-Haber. Here is a chart that just shows the positive energies, and you can see the various components, and then the negative energies and the net. But look at the lattice energy, the Madelung energy component is huge which tells you that when ions form they really want to continue to glom onto one another and form that giant crystal.

And so this is now a chart of just a component number five, the lattice energies. Keep this number in mind, 787. Let's check. Here is sodium plus chlorine. It's 787. So that is one plus against one minus.

But look at this one. There is magnesium with oxygen. That is 3,791. It is almost 4,000 kilojoules per mole. And magnesium oxide has a melting point of 2,800 degrees C. Sodium chloride is about 801 degrees C.

Aluminum oxide, aluminum plus oxygen. Look at this. Aluminum oxide almost 1,600 kilojoules per mole. Alumina is really stable, which is why it takes so much energy to make aluminum. That is why aluminum is so expensive because the oxide is so stable.

And why is it so stable? Because aluminum ions have charge of plus three and oxygen ions have charge of minus two. When you go into the Coulombic term here, instead of one times one, it's three times two.

That is it. So you want high temperature bricks, what are you going to make them out of? It's pretty obvious. It's all here. It all falls out. It is fantastic. And this is an interesting thing.

This is ionic radii with a noble gas configuration. All of these have engaged in electron transfer. So, just to get your bearings, there is our pal sodium plus. Sodium plus is isoelectronic with neon.

It only has 10 electrons, not 11. Magnesium 2 plus, it is also isoelectronic with neon. What do you see? You start with O double minus, so that is oxygen plus two electrons. It is 1.45. Ten electrons again, 1.33.

Ten electrons again. Ten electrons again. Ten electrons again. What is happening? The same number of electrons, but the number of protons keeps going up, up, up. The positive force of attraction of the electrons on the outside is greater and greater, and that pulls everything in.

And so not only is aluminum highly charged, but it is physically small. It is tight, buff, a lean, mean fighting machine. It goes after oxygen because it is small. It is a high charged density. And likewise down here.

You see all of these. Look at beryllium 2 plus, very tiny, charge plus two. Why do we need to know all of this? I can take you and immediately start designing batteries, because we need ions in motion, we need small ions in motion.

All of this stuff is related, fantastic stuff. What you see is that the radius changes with atomic number for constant electron number. Well, we are getting close. I talked a lot about the exam so I don't have a lot of time left to talk about the structural metals.

I will just give you this one little slide so you get a sense of what we are up against. Number one structural metal is steel. Number two is aluminum. Number three is magnesium. Aluminum and magnesium are both made by ionic liquid electrolysis, just as I showed you last day.

But look at the tonnage differences. Aluminum is 25 million tons per year for the whole planet. Magnesium was 60,000 tons. Iron or steel, 800 million tons, almost a billion tons of steel. That is virgin metal.

That is not recycled. And steel is the most recycled material. It sells for about 20 cents a pound, so that represents about one-third of a trillion dollars. 20 cents a pound. You build a steel plant with a capital cost of several billion dollars.

You bring iron ore from one part of the planet, carbon, limestone, you put it into this reactor and you ship this all over the world. Go to the grocery store tonight. Tell me what you find on the shelf that sells for 20 cents a pound.

This is amazing. This is modern industrial chemistry showing its strength. Before you go, it is the weekend, I will send you off with a joke. We studied the Heisenberg's uncertainty principle. The story goes Heisenberg is giving a seminar.

He has a rental car and is zooming to the airport. He is doing 100 miles an hour down the Mass Pike and gets pulled over by a State Trooper. The State Troopers says I want title and registration. He says, hey, buddy, where is the fire? Do you know how fast you were going? And Heisenberg says, no, but I know where I am.

Get out of here.

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