So, in this slide, we're going to discuss the electrostatic energy of an ionic crystal. So, this will be more close to material science. We're going to apply the concept of less static energy in atomic physics, which is the field of solid state physics. Using that knowledge of total energy, we're going to see that is equal to the energy of a chemical change. In other words, the bonding energy in ionic crystals can be released, thought of, as mainly electrostatic energy between ions that are repeating in space. So, we will take a look at the cross section of a salt crystal, NaCl, where you have alternating plus and minus in a linear fashion. We are going to see how close the chemical energy is to the ones that we're going to calculate using the knowledge that we just discussed. All right, so how much energy will it take to pull all these ions apart? So in chemistry, there are ways to measure this, and in thermodynamics, we know that if you have two, well-defined states, then the energy between those two will not be different no matter which way you go. So, whatever kinetics you use, you will have the same results. So, one way to get to this state where every ions are separated is to first vaporize the salt. So, now you have phase transformation from solid state to gaseous phase. Still they are, in the NaCl, they are bonding each other and then you need to dissociate them in gas phase into one another. If I do the two-step experimental process, we can measure the energy to take apart fully and that's the bonding energy. That's happened to be 7.92 electron volts per molecule or you can change the units by changing it to joules per mole, then it becomes 7.64 times 10_5 joules per mole or that could be 183 kilocalorie per mole. We're going to stick to this electron volt unit. So, we will start from here. Now, let's see how we can calculate this number. First, let's think about the regular array of those NaCl ions in space, and we are just going to use first and second nearest neighbors or maybe third, and calculate how much energy we have in terms of electrostatic energy. So, we will pick out a particular ion and compute its potential energy with each other, which of the other ions and that's the twice the energy per ion, and that's the energy per molecule, to two ions per molecule. The energy of an ion with one of its nearest neighbors is, e square over a. Where e square is defined by q e squared over four pi epsilon naught, and this is because Na has one plus and Cl has one minus. They have the same amount of charge with different signs. So, that's this one. This is roughly 5.12 electron volt. So, even just thinking of their nearest neighbors, you got electrostatic energy of 5.12 electron volt per molecule, which is very close to 7.92, right? You can see that. Now, we are going to adding up all the contributions. So, here's the one line of alternating plus and minus ions. This is number one, the rule number one. For this linear combination, we're going to use this series of numbers to compute the total energy, along this line, and this is U1 equals two e square over a parenthesis minus one, plus one and a half minus one over three plus one fourth. Why? Because you have linear addition of distance between each other, right? If you calculate this infinite series, is log two, log two. So, you will have minus 1.386 times e squared over a. Now, with this knowledge, this is one row into this page, the number one, and we're going to think about the first neighboring linear combination of ions, which is number two, as you can see up and down. For up and down row, you can have U2 which is four e square over eight times minus one plus two over square root two minus two over square root of five, and so on and so forth. If you think about this, you can get the total energy, the total electrostatic energy, which is minus 1.747 times e square over a and this is minus 8.94 electron volt. Which is about 10 percent more than what we measured using the experiment that I described before. So already, with just summing up electrostatic energy, we can calculate the bonding energy between ions in salt crystal. So in essence, we can guess, "Oh, the energy of chemical bond is really coming from electrostatic energy." Okay? So, the coulomb force or forces, they are responsible for ionic bonding. So, our answer is about 10 percent above the experimentally observed energy minus the notation to show that you have attraction. It has a bonding energy. So, it shows that our idea that the whole lattice is held together by electric Coulomb forces fundamentally correct. It's only 10 percent off. The subject that tries to understand bulk matter behavior in terms of the laws of atomic behavior is called solid state physics as I just described, this is a huge field. Now, within this field, it starts from this electrostatic energy, and but you saw that there is an error of 10 percent. So, what about the error in our calculation? Where does it come from? Right? So, the first thing we can think of is, the repulsion between two ions. Melody, in our first lecture, we discussed about the repulsive force between two atoms when they get too close to each other or even the repulsive force between electrons and nucleus, when they are close to each other. Where does it come from? The interatomic forces. Exactly. Interatomic forces or interparticle forces when they become so close together, then there's uncertainty rising. Which is uncertainty in momentum. So, quantum mechanical effect plays here to account for the repulsive force that we see, and it is because of repulsion of between the ions at close distances, that they are not perfectly rigid sphere. So, when they're close together, they are partly squashed, and some energy is used in deforming them. So, when ions are pulled apart, the energy is released. This energy is released, and we can find some analogy between human interaction as well. If two persons like each other and they form a bond, but when they dissociate, the dissociation energy will be decreased by the amount of the things they don't like about each other, right? The same thing happens here. So, the bonding energy is discounted by the fact that they had already repulsion when they were binding to each other, right? So, from a measurement of the compressibility of the whole crystal, it is possible to obtain a quantitative idea of the law repulsion between the ions and therefore of its contribution to the energy. So, if we subtract this contribution, the repulsive part, then we obtain 7.99 electron volt for dissociation energy per molecule. This is very close,7.99 versus 7.92. I guess it is less than three percent, which is usually the experimental error allowable in most cases. But we want to do more, so we do have one more correction here. If we take into account that we are not working at absolute zero temperature. At absolute zero temperature, Melody, do we have kinetic energy? No. We don't have kinetic energy. Why? Kinetic energy is proportional to kT, the Boltzmann constant temperature, the absolute temperature, and if we are at zero absolute temperature, it's zero. However, we are at room temperature. So, we have certain amount of kinetic energy that vibrates those ions, and this vibrational energy will also discount the bonding energy. If we take that into account, we end up with 7.92 electron volt, which is right on the spot and which is in no good agreement with the measured value of 7.92. So, with one, two, three with attractive forces between ions, Coulombic forces, of course, there's repulsive forces as well between like ions. We had roughly 10 percent overshoot, and then I'm taking into account the quantum mechanical effect or compressibility, we drag that down to less than three percent or even less. Then finally, by taking into account the kinetic energy, we could predict the bonding energy with almost 100 percent accuracy, okay? So, from here we're going to discuss a rather difficult and very deep question that is related to the location of energy. Where can we locate our energy that we discussed? Where does electrostatic energy come from? So, energy in the electrostatic field is sometimes difficult to visualize. So, we will go through that question step-by-step and convince you that we can locate the energy where electric field is located. In other words, we can locate energy at any arbitrary point in space. So, we now consider other methods of calculating electrostatic energy. They can all be derived from basic equation, the sum over all pairs of charge that we just discussed of the mutual energies of each charge pair. First, we wish to write an expression for the energy of a charge distribution. So, take a look at the equation that is depicted here. So, the total energy U is equal to one over two, integrate all spaces, row one, row two over four pi epsilon naught r12 dV1 and dV2. So, the main difference you see here is, we change the discrete quantity, the charge, into continuous quantity. So, we smear the charge into a smushy distribution, and we put one over two in front of the equation. So, I want to ask Melody. Why do we have one over two all of a sudden here? So, basically if you're integrating over all positions, then some of them are going to be double counted, for example when one is counted to two and matched to two, and then on the second time when two is at one and matched with two. Yeah, excellent. Yes, something like that. That is correct. So, unlike the summation where we said just all pairs, when you do integration, it's hard to do it because the math doesn't allow you to do that. So, you have to count those two times, one being charged and the other being the field, the one being field, the other being the charge, so on so forth. So, you have to put one or two for the double counting in the integration. So, and we can see also that we noticed that the integral over dV2 inside this one is just the potential at location one. So this one, integration of rho at the position two over four pi epsilon naught r12 dV2 is really the potential felt by the cluster of charge distribution at 0.2 on one, right? So, you can replace part of this integration by the electrostatic potential at position one. Knowing that, you will be able to understand that we can replace the equation u by this one, where you have one over two integration of rho times phi times dV. This equation can be interpreted as follows. The potential energy of the charge rho dV, that's the potential energy of the charge rho dV, is the product of this charge and the potential at the same point. So, you have a product of the charge and the potential at the same point. So, total energy is therefore the integral over this one, right? We already mentioned why we have one over two in front of this integration. We have a double counting in our integration. Then with this knowledge, now we're going to convince you that electrostatic energy can be located at the point where we have the electrostatic field, right? So, if we can locate the energy like in the case of heat energy, we know in the campfire, we can locate heat at each position, and we can also think of a heat flow in any arbitrary position using the diffusion equation, right? So, we might then extend our principle of the conservation of energy with the idea that if the energy in a given volume change, we should be able to account for the change by the flow of energy into or out of that volume as we discussed before. We realized that, our early statement of the principle of conservation of energy is still perfectly all right, even for the case if some energy disappears at one place and appears somewhere else, far away without passing in the space between. Mathematically, that's okay. Intuitively, it's hard to understand, but at least on the principal level that is okay. So, principal of the local conservation of energy, the energy in any given volume change only by the amount that flows into or out of the volume will be the local conservation energy, and in nature energy is conserved locally. So, we can find formula for where the other energy is located and how it travels from place to place. In that case, you need to have a pathway to move one energy to the other place. So, that's a local energy conservation. If it is global energy conservation, you don't need that. It can appear somewhere out of nowhere, right? Disappear out of nowhere as well. So, you may wonder, why are we so obsessed with this question, to locate the energy, where it is. Because you will now know that we are able to locate the energy in case of gravitational energy, based on some of the physics that we already learn. So, here's the thing. There's also physical reason why it is imperative that we're be able to say where energy is located. Now, according to the theory of gravitation. Thanks to Isaac Newton. All mass is a source of gravitational attraction. We know the gravitational force is F equals G, the gravitational constants are m1, m2 over r-squared. So here, you see the mass is a source of gravitational attraction. From Einstein's revelation, we know energy is equivalent to mass or mass is equivalent to energy through this E equals mc-squared, a very popular equation, so that mass and energy are equivalent. Meaning, if all mass is source of gravitational attraction, where the mass is, the energy is there. So, energy can be located and all energy is therefore, a source of gravitational force. So, that's one thing you should take into account. If we couldn't locate the energy, we couldn't locate all the mass, and we wouldn't be able to say where the sources of the gravitational field are located, and the gravitational theory would be incomplete. So, with this kind of thinking, you can now understand we're able to locate where the energy is in classical physics, and classical mechanics. Now, we're going to extend that idea to see if we can locate the electrostatic energy in the space. So, if we restrict ourselves to electrostatics, there's really no way to tell where the energy is located. It's really hard, it's a hard question to answer. The complete Maxwell equations of electrodynamics gives us much more information. We will start from there and we will end up being able to locate the energy in the space. So, we will end up with the conclusion that energy is located in space where the electric field is. In order to prove this, we need to have an equation, where on the left side we have energy, on the right side, we only have electric field. All right. We thus described the energy, not in terms of charges, but in terms of the fields they produce which is written here. The energy is equal to Epsilon naught over 2, integration of electric field dot electric field in the space. So, if we can prove this equation, then we can prove the fact that energy is located in space where the electric field is. If we can do that, if we can prove this, then we can then interpret this formula as saying that when an electric field is present, there is located in space an energy whose density or energy per unit volume is u equals Epsilon naught over 2 times E dot E which is Epsilon naught E squared over 2. Okay. So, this is a big if. So, let's see how we can solve this. We're going to derive this from the equation that we just discussed shortly before. So, U, the internal energy or electrostatic energy is equal to 1 over 2 integration of Rho times Pi over the space, and we have a relationship here between the density of charge, and electrostatic potential. So, I'm going to ask my teaching assistant, Melody. Where did we get this from? I think it was a couple slides ago we talked about it, but it's Poisson's equation. Exactly. It's Poisson's equation where the Laplacian of electrostatic potential is equal to the charge density divided by permittivity and put minus sign in front of it. So, if we rearrange that equation in this form, we can plot this into our equation here, and we end up with U equals minus Epsilon naught over 2 integration of Pi Laplacian of Pi over the space. So, already here, we can see some smell that energy is related to electric field. So, in this slide, we are going to derive it using mathematics. So, writing the components of the integrand, we see that Pi times Laplacian of Pi is equal to Pi times parentheses round squared Pi over round x-squared plus round squared Pi over round y-squared plus round squared Pi or round z squared. if you do some mathematical trick, we can make this equal to the following equations. So, you're adding something extra and subtracting that extra later. By doing this trick, you can rearrange this to be equal to the divergence of Pi del Pi, which is electrostatic potential times electric field minus electric field, minus the dot product of del Pi del Pi, and del Pi is really minus electric fields, so it's dot product between electric field. So then, our energy integral can be written as two terms instead of one and that will be Epsilon naught over 2 integral of del Pi dot del Phi dV minus Epsilon naught over 2 integral del dot Pi del Pi dV. We are going to prove the latter term will be zero. Because this will be true, no matter what the size you are thinking of. We're going to go extreme. We're going to include all of the charges including those at infinity. By doing so, we're going to use Gauss theorem to prove this should be zero. So, from the Gauss theorem, we can change the second derivative which is the volume integral of divergence, to our surface integral of flux. So, volume integral of divergence of Pi del Pi will be the flux of Pi del Pi over the surface. So, in the second term, if we increase the Gauss surface to include every charge of interest, then the integrand of the second term which is Pi del Pi, this one will be proportional to 1 over R times 1 over R-squared, and the da, which is a surface integral portion, will be proportional to R-squared. Therefore, in a nutshell, the whole part will be proportional to 1 over R and if 1 over R goes to infinity, the integral will be zero. So, in that way, we can understand, we can omit the second part of the integral. So, as you can see, the total electrostatic energy is equal to Epsilon naught over 2 integral of del Pi dot del Pi dV, where you can change del Pi 2 minus E which is electric field, and you end up with Epsilon naught over 2 integral of E dot E dV. We see that it is possible to represent the energy of any charge distribution as being the integral over an energy density located in the field. So, if we can locate the field in space, we can locate the energy in the space. So, we will think about a much more difficult question and the one that will be the source of incompleteness of electrodynamics, which is the energy of a point charge. So, our new relation says that even a single point charge q will have some electrostatic energy because from a single point q, we know it emanates electric field in a spherically symmetric way, and the magnitude will be q over 4 Pi Epsilon dot r squared. So, from our previous slide, we were able to locate energy where the electric field is. Therefore, we are able to calculate the energy density at a distance r from the charge which is Epsilon naught E squared over 2 and replace E by this one, and we end up with q squared over 32 Pi squared Epsilon naught r to the power of 4. So, the total energy, as you can see, of this system where you only have one point charge, should be from zero to infinity. Yes. But you see instantly there is a problem. What problem do have, Melody? It goes from infinity to zero? So, it goes from infinity to zero. If I put infinity to r, there is no problem, it's zero. But if I put zero to the equation, what happens? It becomes infinity. It becomes infinity, so it diverges. So, the energy diverges, which is not right, it is not physically right. We don't have infinite energy. So, here is the dilemma. So, energy of a point charge is a dilemma in electrodynamics and it is a dilemma when we think about how to locate the energy created by a point charge. Okay. So, that is still unsolved. So, unlimited potential energy of a point charge, the equation says that there's an infinite amount of energy in the field of a point charge, although we began with the idea that there was energy only between point charges. In the original energy formula for a collection of point charges, we didn't include any interaction energy of a charge with itself. We didn't, we excluded that. So, we must conclude that the idea of locating the energy in the field is inconsistent with the assumption of the existence of point charges. Alternatively, we could say that there's something wrong in our theory of electricity at very small distances, or with the idea of the local conservation of energy. So, there are difficulties with either point of view. So, these are the remaining challenges. Sometimes later, when we have discussed some additional ideas such as the momentum in the electromagnetic field, we will give a more complete account of these fundamental difficulties in our understanding of nature. Okay. With that, we'll wrap up our lecture here, and we will come back soon. Bye-bye.
