Fundamentals of Physics I

PHYS 200 - Lecture 21 - Thermodynamics

Chapter 1. Temperature as a Macroscopic Thermodynamic Property [00:00:00]

Professor Ramamurti Shankar: Alright class, welcome back. This is our last two weeks. We’re going to have a slightly different schedule for the problem sets. I’m going to assign something today which is due next Wednesday. I’m giving enough time so you can plan your moves. Then, I will probably give you one last problem set with two or three problems on whatever I do near the end. We’ll have to play it by ear.

Okay, so this is another new topic on thermodynamics, a fresh beginning for those who want a fresh beginning. And there’s also stuff you probably have seen in high school, some of it at least. So, the whole next four lectures are devoted to the study of heat, temperature, heat transfer, things like that.

So, we are going to start with the intuitive definition of temperature everybody has. So, hang on to that; that’s the right intuition. But as physicists, of course, we want to be more precise, more careful. So, let’s say you have the notion of hot and cold. Even that requires a little more precision. That introduces the notion of what is called thermodynamic equilibrium. Just like mechanical equilibrium, this is a very important concept.

So, I’ll tell you what equilibrium is with a concrete example. If you take a cup of hot water, and you take another cup of cold water, each cup, if you waited sufficiently long, is said to be in a state of equilibrium as long as the cups were isolated from the outside world and not allowed to cool down or heat up. We think they maintain a certain temperature. We say it’s in a state of thermal equilibrium because this temperature does not seem to change. Now, we have not defined what temperature it is precisely, but we can talk about whether whatever it is has changed or not changed. So, it will settle down to some temperature and it will maintain the temperature. Be very careful. If you leave a cup of coffee in this room, it will cool down because the room has got a different temperature. But I’m talking about a cup of coffee that’s been isolated from everything; it maintains the temperature. Here’s another cup of cold drink at what we feel is a lower temperature. They are both in a state of equilibrium.

Equilibrium is when the macroscopic properties of the system have stopped changing. If you now pour one of these cups into the other one, that’s going to be a period when the system is not in equilibrium in the sense that it doesn’t have a well-defined temperature. For example, if you just poured it from the top, the hot stuff is on the top, the cold stuff is on the bottom, there’s a period of transition when you really cannot even say what the temperature of the mixture is. Some parts are hot, some parts are cold; that system doesn’t have a temperature. But if you wait long enough until the two parts have gotten to know each other, they will turn into some undrinkable mess, but the nice thing is it will have a well-defined temperature. That’s, again, a system in equilibrium.

So, you’ve got to understand that temperature and thermal equilibrium represent gross macroscopic properties and they’re not always defined. At the microscopic level – it’s no secret – we all know everything is made of atoms and molecules. The atoms and molecules that form the liquid or the gas always have well-defined states. Each molecule has a certain location, certain velocity. But at a macroscopic level, when you don’t look into the fine details, focus on a few things like temperature, they don’t always have a well-defined value; that’s what you’ve got to understand. Things have a well-defined value when they have settled down.

How long does it take to settle down? That’s a matter of what system you’re studying. But generally, you can all tell when it has settled down. Here’s another example. Suppose you take a gas and you put it inside this piston here, put some gas inside, you put some weights, and everything is in equilibrium. We say that it’s in equilibrium because the macroscopic things, things you can see with your naked eye, nothing is changing. It’s going to just sit there. But if you suddenly now remove, say, a third of the weights, the piston’s going to rise up, shake around a little bit, maybe settle down in a new location. If you wait a few seconds, then the new location will again settle down, and you won’t see anything with the naked eye that looks like anything is happening.

In between you will see the pistons moving, the gas is turbulent, the pressure is high in some regions, low in some regions, then it settles down. This is the notion of systems in equilibrium, and in between, there are states of the system which are not in equilibrium. Now, whenever a system is in such equilibrium, we can assign to it a temperature that we call T. Right now, we don’t know anything about this temperature, so we’re going to build it up from scratch–other than your instinctive feeling for what temperature is.

One of the laws of thermodynamics is called a zeroth law –zeroth law because they wrote down the first law, then they went back and had an idea which was even more profound, and they said, “We’ll call it zeroth law.” Zeroth law says, “if a and b are at the same temperature, and b and c are at the same temperature, then a and c are at the same temperature.”

Now, I see disbelief in the audience today. Why do you call this a law? Look, I think that is the key to our being able to speak about temperature globally, is the assumption that if I take a thermometer and measure something there, and I come back and dip the thermometer here, and it reads the same number, then I may conclude these two entities, which never met each other directly, are also the same temperature. That’s not–That seems pretty obvious to you, but the whole notion of temperature is predicated on the fact that you can define an attribute called temperature that can be globally compared between two systems that never met directly, but met a third system.

Chapter 2. Calibrating Temperature Instruments [00:06:45]

Okay. So, once we have some idea of hot and cold, let us decide now to be more quantitative. It’s like saying, you know, somebody’s tall and short is not enough. We go into how tall, how many feet, how many inches, how many millimeters. So, we want to get quantitative. All we have right now is a notion of hot and cold. So, what we try to do is to find some way to be more precise about how hot and how cold. So, what people said is, “Let’s look at some things in the world that seem to depend on temperature.”

One thing that seems to depend on temperature is the following. You take this meter stick in the National Bureau Standards, kept in some glass case, at some temperature. You pull it out–or make a duplicate of it, you pull it outside and leave it in the room. What you may find is that if the room was hotter than the glass case, this rod then expands to a new length. So, one rod is outside the case, one rod is inside the case so the comparison is meaningful. Nothing has been done to this guy in the air-conditioned glass case, but this one is expanding. So, one way to define temperature is to simply ask how long is this rod, and somehow correlate the length of the rod with temperature by some fashion.

So, you can do that. So, what you need to do that–what you need to do first is to define, put some markings on it so that for each extra something it grows, we can say the temperature has gone up by some amount. So, there we need units for temperature, that’s completely arbitrary. And you need some standards, just like this meter stick, you know, it’s not–nothing intrinsic in nature about a meter, we just made it up and said “Let’s call that a meter.” In the case of the meter, the zeroth law is if you bring a meter stick next to mine and we agree, you can take the meter stick somewhere else and define that to be the meter because if this stick is as long as that one and as long as that one, then those two are equal in length. But temperature–You are similarly going to use this rod and say, “This rod is a certain length when kept on top of this bucket of some fluid and the same length when I keep it on that bucket, then the two buckets are the same temperature.” So, we can use markings on this rod compared to the unexpanded length as a measure of temperature.

So, what people do is to pick something a little easier than this rod. They notice the liquids expand when you heat them. That’s why in a summer day if you fill your gas tank, you have to leave some room at the top so the overflow can come out of the top; or you shouldn’t fill it completely, otherwise it’ll bust the tank. So, liquids expand. So, one way to measure temperature may be take some liquid, put it there, and then put it in hot rooms and maybe watch the liquid expand to the new height. And then draw some markings, and each marking can be a certain temperature.

But people had a better idea than this one. They had the following idea of a thermometer, where you have a lot of fluid in a reservoir, a very thin tube evacuated at the top, and the fluid, then, is here. So, what’s clever about this is that if this expands by one percent, your eyes should be good enough to see one percent increase in height. If this fluid expands by one percent in volume, that one percent in volume and it climbs up this narrow tube can climb to quite a bit [pointing to picture], because the extra volume you get by expansion will be the area of this tube times the extra Δx by which it expands. So, you’re magnifying the expansion by making all the expander fluid climb up this extremely narrow tube. In fact, the tube is so narrow, you cannot probably even see it well, which is why they have a little prism that magnifies the mercury or alcohol in the thermometer.

Okay, so we have some way of following temperature now. We can draw some lines, arbitrary lines, it doesn’t matter. That can be zero, that can be five, that can be 19; you’ve just got to make sure that it’s monotonic. Then whenever it’s on 21, we may argue that 21 is now hotter than 19. But you want a better scale than that. Even though that’s mathematically adequate in practice, what people decide is to do it as follows. They said, “We want to set up thermometers so that people all over the world, in different parts of the world, different countries, different labs can all agree. So, we will make it possible for everyone to make their own thermometer by the following recipe.” We will dip this guy in a bucket which has got some ice and some water. That’s called the melting point of water, so that–or the freezing point of water; melting point of ice, or freezing point of water, it doesn’t matter. We notice that as water cools down, in the world around us suddenly ice cubes begin to form. We go to the temperature at which that happens for the first time and we dip the thermometer there, and whatever reading we get we will postulate to be zero degrees centigrade. That is just a definition.

We believe that’s a good definition because people all over the world can do that. Of course, if you live in Kuwait, that’s not going to work for you; there’s no ice. But they figured out in parts of the world where you have ice, this is a very good definition. You get ice, you got zero degrees. Then they said, “Let’s find another universally accessible thing,” which, as you all know, is the boiling point of water. If you put water on the stove it heats up and heats up and heats up and suddenly it begins to bubble and boil and evaporate. That temperature is going to be called 100 degrees, 100 degrees centigrade. Then, you take this column between zero and 100, and you divide it into 100 equal parts. And that is postulated to be the temperature anywhere between zero and 100. If you have gone 79 percent of the way to the top, from here to here, the temperature is 79 degrees. That’s how the degrees were introduced, and that’s a centigrade scale, and you guys know there are different scales. You can have the Fahrenheit scale, you can have any other scale in which what you want to call the freezing point is different. Somebody thinks it’s zero, somebody thinks it’s 32. And you can again call this something else, and you can divide this interval into 100 parts, 180 parts, whatever you like. But the philosophy is the same. You have to find two points, which are reproducible, conveniently, and divide the region between them into some number of equal steps. If there’s 100 equal steps, you say it’s a centigrade scale, provided the lowest one is called zero. This is how you have thermometers.

Now, there are some problems with this. One problem is that the boiling point of water does not seem to be very reliable. Because if you boil water in Aspen, for example, you know it doesn’t seem to boil–it seems to boil more readily than in the plains. You can ask, “How do you know that?” maybe it is still doing the same thing. I know that because I tried to cook something, cook some rice and vegetables, I find, they don’t cook at all. In Denver, it boils before it cooks; that way we know it’s probably boiling earlier in the mountains than in the plains. So, who’s going to decide what the real temperature is? So, you have to be more careful when you say boiling point and freezing point, because things don’t seem to boil at a certain, predictable and fixed temperature.

This is a very deep argument I have never appreciated fully when I was learning the subject, is that it’s all cyclic definition. Because you may not know that the temperature is changing, because this thermometer by postulate, it’s going to be the temperature by definition. How can it be wrong? What’s wrong is that you know it’s not a reliable method because physical phenomena, like when your rice will cook, are not reproduced by the boiling point of water. It cooks in the plains, it doesn’t cook in the mountains, so we know the boiling point is to blame. Rice is the rice. That’s how we know that that’s not a good measure. So nowadays, people have much fancier measures, and I will tell you a little bit about that. But for a long time, this was a very good start. Don’t worry about the fact that water boils differently at different altitudes; you could go to sea level and that’s a good enough definition. Sea level is pretty much constant all over the world, and you can say the pressure of sea level is the pressure at sea level; just the ρgh of the atmosphere. Okay, so that’s the usual definition of temperature.

Now, the trouble started when people realized that if you make a thermometer with your favorite fluid, maybe mercury, and I make one with alcohol, they will agree at zero and they will agree at 100 because that’s how you fixed it. You rigged it so at zero everyone says zero; 100 everyone says 100.

But how about 74 degrees, or 75 degrees? I say it’s 75 if my fluid has climbed three-fourths of the way to the top. At that point, yours may not have climbed three-fourths of the way. In other words, you’ve got two things, two graphs, which have zero and 100 degrees; one graph may be like this, one may be like that. So that when I think it is 75, you may think it is 72. At 100, we will agree because we have cooked it up that way. In other words, it’s not true that all liquids expand at the same rate. So, you will have to then pick one liquid and say, “We swear by that liquid, and when that liquid’s gone halfway, we’ll say it’s 50 degrees.” So, you will have to pick a liquid, you’ll have to have an international convention, you know, there’s the alcohol lobby and there’s the no alcohol lobby; they argue. Finally, they found out a much better solution than these liquids.

They found out that if you use a gas–You can define temperature using gasses, which have some very, very nice properties. And this is the gas thermometer that I’m going to tell you now. So, here is how you build a gas thermometer. You take some gas in a container. A typical container for me in all–whenever I draw anything thermodynamics, it’s going to be gas inside some cylinder with some weights on it, and that defines the pressure of the gas. Of course, the pressure will be the mg of these weights divided by the area of the cylinder. That’s the pressure, plus atmospheric pressure. And the volume is this, whatever the volume is, base times height.

Here’s what we ask you to do. Take the product of pressure times volume for any sample of gas. Take some gas, put it in this tank, and now put it on different surfaces, like a hot plate, like a stove, like a tub of water, and measure the temperature using some standard method up to that point like a mercury thermometer. What you notice is that the temperature measured by some reasonable scheme shows that the product of P times V lies on a straight line [drawing diagonal line on board]. If you connect the dots, you find the product PV is linear in this temperature variable.

And this is zero degrees, and this is 100 degrees. Now, here is the beauty of the gas thermometer. If you take a different gas and you put a different amount of different gas in a different cylinder, you will get some other graph; it may look like this [drawing another straight line]. For you, that is zero and that’s 100. But the most important thing is that’s also a straight line. That it’s also a straight line, has the following implication you guys can prove at your own leisure, which is that, if I think that my gas has climbed 56 percent of the way of this height to the top, so the temperature is 56 degrees, I ask what’s your gas done, you will find yours also climbed 56 percent of the way. It’s the property of straight lines. You can show that if you took two straight lines, whatever be their slope, if they agree, if this is zero and this is 100, it has got a different slope, when you have climbed to the halfway point, draw a line at 50 degrees and ask what has any gas done, they will all have climbed to the halfway point from the zero point to the 100 point. In other words, gas thermometers will not only agree at the end points where they must, by construction, they seem to agree all the way in between.

But there is one requirement. This gas has to be very dilute. The more diluted it is, the better it comes out. So, take neon or Freon or whatever you like. Don’t pump it up with a lot of gas; put the least amount of gas you can get away with. Then, you find all gasses have the property that if you calibrate them at zero and 100, they agree in between. Is that clear to you?

Take the product P times V of your gas by putting it on different surfaces, measure the product, plot this graph. Whenever you’re on ice [freezing point of water] you call it zero; whenever you’re on boiling water you call it 100. You find they’re connected by a straight line, then every point in between, you’ve divided equally, leads to equal increase in the product P times V. P times V for a gas is better than the volume of mercury or volume of water because it doesn’t depend on the gas. So, everybody can use the gas thermometer. That’s why we prefer the gas thermometer.

So, this is the interesting issue about measurement or definitions and cyclic definitions–you’ve got to be careful. The laws of nature allow you to pick anything you like that varies with temperature and use that as a definition of temperature, as a thermometer. So, why are some thermometers preferred over the others? They’re preferred over the others if the laws of nature take the simplest form when described in terms of those thermometers. In other words, take a meter stick. What makes a good meter stick for a standard? You say the one that doesn’t expand, but we don’t know what that means. That meter stick is the standard; by definition it’s right. But then, you will soon find out that it’s not really that simple, because there are good and bad meter sticks. For example, the same meter stick at one time out of the year doesn’t match its own length at a different time of the year; then we know that it’s not a good meter stick. Similarly, there are good and bad thermometers, and people arrive on the gas thermometer this way.

Chapter 3. Absolute Zero, Triple Point of Water, The Kelvin [00:22:25]

If you have a gas thermometer, something very interesting came out of the gas thermometer. If you cool it below zero and you ask which way is it going, I don’t know how low you could go. In the old days, people couldn’t go far below zero, but now we can go to one-billionth of a degree above a certain point. I’ll tell you now, these thermometers indicate somehow the product PV vanishes at a temperature which is minus 273.16, suggesting that there is something very special about that temperature. Because if you took another gas–well, I’m going to do a little cheating here–that also extrapolates that same temperature. So, all gasses, all gas thermometers say there is something very special about this temperature because that’s when our pressures all vanish. So, as you cool a given amount of gas, even at a given volume, if you keep the volume constant and ask what pressure do I need, how many weights do I have to put on; that decreases and vanishes at this temperature. And this is called the absolute zero of temperature.

It’s called absolute zero for many reasons. One is that unlike the zero of the centigrade, which is by no means the absolute lowest possible temperature, the absolute zero is the lowest possible temperature. Why? Because the gas pressure can be reduced and reduced and reduced, but the worst that can happen is that it can go to zero. That’s it. It cannot go below having no pressure. We’ll find in other ways, also, this is the temperature at which you will see conceptually no further cooling is possible. That will require you to understand what hot and cold mean. But right now, this says all gas thermometers point at this temperature.

So, people decided, “You know what, calling this zero is kind of artificial.” That’s based on human obsession with water. But if you think laws of science describe the whole universe, what about planets where there’s no water? Right? You cannot describe–Suppose you’re talking to a different civilization; Planet of the Apes. You want to tell those guys, “We’re going to set up our temperatures; zero is when water freezes,” and they say, “What is this thing called water?” “You know, the stuff you drink.” You don’t know what these apes are drinking. Maybe they’re drinking methane or liquid hydrogen. We don’t know. On the other hand, you say, “Take any vapor and wait until the product of the pressure and volume go to zero, let’s call that zero,” that’s the universal standard. It’s not tied to something called water. It was fine for a while, but it is not fine as a universal aspiration for thermometers. So, zero of temperatures can be set from here.

Once they did that, they called that zero, they needed one other temperature. And they decided that if you’re starting the new temperature scale, you will put the zero not at the centigrade, but this is now called Kelvin. And everything will follow a straight line, but to define what one degree means, you’ve got to define one other temperature. That’s how we define the straight line; that temperature would be called 273.16. But this point is called the triple point of water. What’s the triple point of water? You know water and ice can coexist, and you know that water and steam can coexist at 100 degrees. But by varying the pressure and temperature and volume, you can actually find a certain magical point in which both ice, water and steam can coexist, simultaneously. It cannot pick between those three options.

Ice floating on water is when water has not decided whether to be ice or to be water. That’s the coexistence point of two things. And when the water starts boiling on your stove, that’s when water and steam coexist. But I’m saying that certain conditions of pressure and temperature and volume so that water, ice and steam will coexist. Now, that is a unique situation; you cannot get to that by any other means. And that temperature we will call plus 273.16 in these absolute units.

So, basically, what you have done by going to the absolute units is you’ve shifted the zero to a more natural point where all graphs meet; then, you define one degree Kelvin to be so that 273.16 of that Kelvin brings you to the triple point of water. So, if you found that confusing, I’m just saying the boiling point of water is not a fixed number. You go to the mountains, it changes. But only under one condition can water and ice and steam coexist. You cannot get that any other way. So, everybody will agree on that particular situation, that will be called 273.16 Kelvin.

Now there is a rule, apparently. You can say, “degree centigrade,” you’re not supposed to say, “degree Kelvin.” There was a big deal made in a lot of books. I keep forgetting–In fact, I forgot again, and nothing terrible has happened to me. So, I don’t think you should pay too much attention to whether you can call something “degree Kelvin” or simply “Kelvin.” I think the purpose of language is to have no ambiguities. But when they say, “degree Kelvin” and I find that you guys don’t get confused, I don’t think that’s a big deal. But you’ll find if you’re a very erudite person, you will never write “degree Kelvin.” But having said that, don’t hold me to those standards–I just don’t feel any affiliation to this particular, completely artificial and empty convention. But you are supposed to remember, if you take the GRE or something, it’s not called “degree Kelvin.” Okay so, as far as we are concerned, the Kelvin scale is like the centigrade scale, except the zero has shifted to here. That’s it. That’s the temperature scale you will use. That’s the absolute temperature. Whenever I write T from now on, I’m talking about Kelvin, not centigrade. Now, that’s all about heat–I mean, all about temperature.

Chapter 4. Specific Heat and Other Thermal Properties of Materials [00:28:55]

Now, I’m going to talk about heat. So, heat is denoted by the symbol Q, and you’ve got to ask yourself, “What are we talking about when we talk about heat?” Again, let’s use your intuitive sense of what heat is. Say I have a bucket of water; I want to heat it up. And how do you do that? You put the bucket on top of something else which you think is hotter, and when the two are brought together, somehow the water begins to feel hotter and hotter. So, we say we’ve heated the water, and we say we have transferred heat.

Now, people were not sure what really was being transferred. What is it that’s going from the stove to the water? Why is it that the stove, if it’s not plugged in, is getting cooler and the water is getting hotter? They just decided to call it the caloric fluid. They imagined there was a certain fluid which is abundant in hot things, and not so abundant in cold things. When you put hot and cold together, this magical fluid flows from hot to cold, and in the process heats the cold thing. And they decided to measure it in calories. And so, you have to define what a calorie is. In other words, you want to ask, “How much heat does it take to heat this bucket of water?” And the rule they made up was, we’re going to define something called a calorie where the number of calories you need is equal to the mass of water times the change in temperature. That’s going to be calories.

In other words, if I had a container with 10 grams of water, and the temperature went up–I’m sorry, this is mass of water in grams. If you have 1 gram of water, and you did something to it and the temperature went up by seven degrees, you have, by definition, pumped in 7 calories. If this was a kilogram of water, this would be called a kilocalorie. Sometimes they use grams and calories; sometimes they use kilograms and kilocalories. But the definitions are consistent; if you put a kilo in the gram, put a kilo in the calories.

Okay. Now, suppose you say, “I don’t want to just talk about water, I want to talk about heating something else. Maybe I want to heat a gram of copper.” So, then you write down the following rule. The amount of heat it takes to heat up anything–pick your own favorite material–gold. Then, the amount of heat, I think we can all appreciate, must be proportionate to the amount of stuff you’re trying to heat up. That’s our intuitive notion. If you’ve got one chunk of gold that takes some number of calories, you have a second identical chunk; by definition, that should take the same number of calories. You put them together, it is clear that whatever this caloric fluid is, you need double that. So, it’s got to be proportional to the mass of the substance. And it’s got to be proportional to what you’re aiming for, namely, increase in temperature. But this is true for any substance, whether you’re heating copper or wood or gold; no matter what you’re heating, it is true the heat need is proportional to mass and to the [change in] temperature. So, what is it that distinguishes one material from another? We put a number here, and that number is called the specific heat. The specific heat is the property of that material.

You’ve got to understand certain formulas will depend on certain parameters in a genetic way, and some things that depend on the actual material. In fact, there’s a similar quantity. I mean, maybe I’ll take a second to tell you. If you go to liquids that I said were expanding, you can do the same thing. Take a rod and start heating it and ask, “How much will it expand if I heat it by some amount ΔT?” What will it be proportional to? Can anybody think of what it may be proportional to? Yes?

Student: Original length?

Professor Ramamurti Shankar: Depends on the original length of the rod. Now, why is that? Why do we think it’s got to be proportional to the length of the rod?

Student: Because it expanded based on what it had before.

Professor Ramamurti Shankar: Yeah, it’s based on what it had before. Yes?

Student: Well, each cycle the rod will expand by some amount, so [inaudible]

Professor Ramamurti Shankar: That’s correct. I think one way to say that is take a meter stick, it expands to some amount, put another meter stick next to it, that expands to the same amount by definition of identical things. For the two-meter stick it will expand by twice as much. So, we put the length of that. So, no matter what you’re heating – a block of wood, block of steel – this is true. But then, the fact that heat has different effects on copper versus wood, is indicated by putting a number here. That α is called the coefficient of linear expansion, and that depends on the material. These are true no matter what you are heating.

So, these specific numbers, these coefficients, these αs that come in are going to come in all the time, so you should get used to them. Here’s another one. Let’s play this game one more time. We can ask how much does the volume of a body change when I heat it. Well, the change in the volume, again, would be proportional to the starting volume times the increase in temperature. Then you put another number; that’s called the coefficient of volume expansion. And that depends on the material.

So, if you take copper, copper will have a certain α; iron will have a different α; wood will have a different α. Each material will have a different α. This is the property of the material. If you say, “Well, I had something and when I heated it up by one degree, it increased by nine inches; another one increased by two inches.” Is it clear that the first one expands more readily? It’s not, because the first one could have been a mile long, second one could have been a foot long. So, you have to take out certain factors that are universal, and the rest of it you put into a property of the material.

Similarly, when you come to specific heat, you ask how much heat does it take to heat some object, it depends on the mass. It doesn’t matter what you’re heating. Depends on the increase in temperature, because that’s the whole purpose of adding heat; it’s always going to be linear in the ΔT. This one is the property of the material, and by definition, c equal one calorie per gram, or one kilocalorie per kilogram for water. Once you’ve got–So remember, one calorie per gram for water is the definition. Once you define water to have a specific heat of one calorie per gram, you can define specific heat for other materials by the following process.

So, what do you do? You take a container with some water in it. Let’s assume the container has zero mass, so I don’t have to worry about it. It’s an approximation. If you are worried about that, you know, take a huge container so that the volume of water dominates the surface area of the container. Anyway, container’s neglected; you’ve got some water. This water is of some initial temperature T₁, and I have some new material, lead, and I want to find its specific heat. So, I take the lead in the form of pellets and I heat the lead pellets to some temperature T₂, and I drop these guys into this water. That’d be an example where initially, the lead is in equilibrium, maybe on a furnace, at temperature T₂; water’s in equilibrium, maybe in the room, at temperature T₁. Then, I put the pellets into the water, and there will be a period when the temperature is not defined. Then, soon they’ll settle down to some common temperature called T_f.

We will now postulate–this is a postulate, or a law. The total change in Q is zero. In other words, if Q is lost by one body and gained by another body; the loss and the gain must equal. It’s a new law. You can make up all the new laws you want. You don’t know if they’re right, but this is the law you first make up. In that case, what can you say in this particular problem? In any of these heat problems, I urge you to draw the following picture. Here is one temperature, here is another temperature, here is the final one, which we don’t know, but we can measure with a thermometer and measure it. Then, you say the mass of the water, and specific heat of water, which is 1 times ΔT, which is the final temperature minus initial temperature. Ditto for the lead pellet; mass of the lead, lead has got a symbol Pb, times specific heat which I don’t know, times a change in temperature which is T_f- T₂ = 0. The sum of all the mc ΔTs is zero. This is the gain of heat, of the water. This, if you work it out, will be a negative number, because you can see T_f is below initial T. This will turn out to be negative, and the positive and negative will add up to zero. So, what is it you don’t know? Well, you know the mass of the water. Specifically, the water is 1 by definition; T_f and T₁ are measured by thermometers. Mass of lead is for you to measure; these are known; you can find c. So, this is a birthday present for you guys. If you ever see this in an exam, jump on this first because you’ve been doing this in high school, and I know kids love this kind of calorimeter problems. Yes?

Student: Looking at the volume in that equation it expands linearly but wasn’t the problem with the liquid, measuring liquid, changing volume, but it didn’t expand [inaudible]

Professor Ramamurti Shankar: Yes. That’s correct. So, the real point is, if everything expanded linearly, we wouldn’t have the disagreement between different thermometers. So, it turns out to an excellent approximation, the change of length is proportional to the length, but it’s not exactly proportional to the length. There will be terms involving higher powers of length. Not only that, specific heated materials is also not a constant. We said specific heated water is 1. Turns out at a certain temperature range it’ll be 1; at a different range in fact, it’s not quite 1. I told you long back. Everything I tell you is wrong. The question is, “How many decimal places do you have to go to before you honor my fallacies?” Specific heat of materials is not a constant, with the big industry calculating the specific materials starting from atoms and quantum mechanics. So, none of the things treated as constants are ever constant, including those alphas and betas.

I can always fudge it by saying α itself may depend on the temperature, and also the dependence on L may not be linear. But you should also look at dimensional considerations and say if it’s not L, if you want to put an L² as a correction to the formula to match the units, L² has to be divided by another length to keep the units. What other length do we have? It may turn out to be the inter-atomic spacing. So, once the atomic properties come into play, then you can find ways to calculate corrections.

So, all these laws are, in fact, very tentative and approximate. These are pretty ancient physics. I think the way I do the physics course here, sometimes I’m in the 1600s, sometimes in the 1400s, sometimes in the year 2000, but going back and forth. This is way back when people did not even know about atoms. So, they were trying to do the best they can, and what you found empirically is that once you found a specific heat for lead, right, you solve for it, then you can do another experiment using that value and you find if you use the right values, ΔQ does add up to zero. Again, when it adds up to zero, it adds up to zero to a very good approximation, during the epoch. Another epoch when people do more and more accurate experiments, everything is shot down. In fact, specific heats of all materials seem to go to zero when you approach absolute temperature. But you have to understand the laws of quantum physics to know why that happens.

So, this is in a period when people are probing temperature ranges which are around room temperature, or boiling or freezing point of water, which is a very narrow window in temperature. If you look at the history of the universe, you’ve got incredibly high temperatures near the Big Bang, and even now the rest of the universe is bathed at some temperature that happens to be very, very low, which is near three degrees; it’s called a blackbody radiation from the Big Bang. So, the temperature of the universe goes through huge ranges, and only when you probe different ranges you see different physics.

If you come to Sloan Lab, you can go to temperatures way below 1 degree Kelvin or hundredth of a Kelvin, and we heard a talk last year, physics at one billionth of a Kelvin. If you want to cool them and cool them and cool them, by zero degree Kelvin, see, there I go. Zero Kelvin is a barrier we’re not able to cross, just like the velocity of light is something we’re not able to cross.

These are all big surprises. The fact that velocity has an upper limit, not obvious even to Newton. Why not? Why not put rockets on top of rockets? Likewise, why not build better and better refrigerators? The reason you cannot go below zero is when you go to zero, all the mechanical attributes of pressure simply vanish, and they cannot have negative values. You will see more about this when you understand heat in greater depth.

Chapter 5. Phase Change [00:43:17]

Anyway, right now, ΔQ = 0 is the rule you use. I’m sure you guys know how to do these problems. Now, there’s a little twist that comes in, I just want to mention that to you. The twist is the following. So, I take some ice–ice, by the way, is not always at zero. You know, you can go below zero. Your refrigerator is several degrees below several tens below zero. So, let’s take ice, and let me measure–I take this container, I put some ice at, say, minus 30 degrees. I’ve gone to centigrade now so we can relate to ice. And I put it on some source of heat, and I watch how many calories are coming in. Let me arrange a device that will pump in a fixed number of calories every second. So, as a function of time, I’m expecting the temperature of this to go up. Do you understand that? In every second, I get some number of calories, and those number of calories are going to produce for me mc ΔT, m and c are constants, so ΔQ is proportional to ΔT. But if you divide both by the time elapsed, then the rate at which the temperature rises will be the rate at which the heat flows into the system. If heat is flowing at a steady rate, temperature should rise, and indeed it does. Temperature of the ice goes from minus 30 to minus 20 to minus 10 and so on. But once it hits zero, it gets stuck. I know heat is coming in, but it’s not getting hotter. But I notice that the ice is beginning to melt. There will be a period between here and here when I pump in calories, I don’t get any increase in temperature but I get conversion of ice into water. And there will be a period when this guy looks like some water with some chunks of ice floating on it. And until all the ice is converted to water, the whole system is stuck at that temperature.

That’s a very interesting property. Now, if you really took a real pot and you put a chunk of ice on it, you know what will happen, right? The bottom of the ice will melt; it may even evaporate. That’s not what I’m talking about, because that’s not a system where there’s a globally defined temperature. I want you to heat the ice so slowly, the minute you put a little bit of calories, give it enough time for all these guys to share that heat, so that the whole system has one single common temperature. Let’s watch the temperature rise. I’m saying it gets stuck at zero, but your calories are getting you something; they’re converting ice into water. Then you can ask, okay, what penalty do I have to pay, that’s called a latent heat of melting, and again, I know only in calories per gram, it’s 80 calories per gram for water. Some of your ΔQ now goes not to raise the temperature, but to melt that amount of stuff at the latent heat of melting. That’s how much Q you need to melt that amount of stuff and the L varies from substance to substance, but water is 80 calories per gram. If you want to melt mercury from solid mercury to liquid mercury, it will have a different number. Then, once everybody has become water, then that uniform system of water starts growing. And this is called a phase change. A phase change is when it changes its atomic arrangement from a regular array; for example, that forms a solid into a liquid. In a solid, everybody has its place; you can shake around where you are, but liquid you can run around.

The specific heat of ice is not the same as the specific heat of water, so you’ve got to be careful. Even though it’s still made up of water molecules, the calories needed to heat one gram of ice is roughly half what it takes to heat one gram of water. So, in these problems, don’t make the mistake.

Okay then, you go along and I guess you know what the next stopping point is. When you come to 100 degrees, again, it gets stuck until everybody vaporizes, and then you get steam. Then, you can have super-heated steam, which is at even higher than 100 degrees. So, that’s the latent heat of vaporization. I really don’t know what–you want to write something, I think it’s 500 and something calories per gram. That’s information I don’t carry in my head. So, if I tell you I took some ice at minus 30 and I dumped in 5,000 calories, where will it end up? You’ve got to first spend a few calories going from here to here, you got some more money left you can start melting this, maybe you’ll run out of stuff there, and that’s what you will have. Some amount of water and some amount of ice. If you have even more calories at your disposal, you can melt it all and start heating it. You may come this way and you may be running out of calories; if not, keep going here and there and there, and you may end up there if you got enough calories.

Or one can ask a question, “How many calories does it take to convert ice at minus 30 to, say, water at 100?” You’ll have to do the mc ΔT for that, m times latent heat for this, mc ΔT for that, and m times latent heat of vaporization for that.

So, the kind of problems you can get are fairly simple most of the time. Only kind of problem where you can really get in trouble is the following. I will mention that to you. Suppose I take some water and some ice, so this is zero. The ice is at, say, minus 40, the water is at plus 80. In fact, let me make that water plus 40. I bring them together and I ask you what will happen.

Now, this is a subtle problem. If you had two–If you had water at 40 and you had water at 20, you can easily guess that it’ll end up somewhere in between; you can calculate it. Now it’s more subtle. You’ve got water at 40, you’ve got ice at minus 40, you bring them together and ask what happens. Well, the answer will depend on how much of the stuff you have. If by water at 40 you mean the Atlantic Ocean, and by ice you mean a couple of ice cubes, we know what’s going to happen. These guys are going to get clobbered; they’re going to melt; you will end up somewhere here. Then, you can easily calculate the final temperature by saying mc times this ΔT for water, in magnitude, is going to be the heat given to this. Heat given to this is the mc ΔT to come here; then, the heat to melt this amount of ice, then the heat to raise this amount of water to that final temperature. Then, you can solve for the final temperature.

So, if you want to solve this problem, and I give you some mass for this ice, of water, and I give you some mass for the ice, you can first make the optimistic assumption that you will end up as water, but at an unknown temperature. We call the unknown temperature T; this is the T₁, this is the T₂. Write your equations, except you’ll have one more term there. That’s the heat it takes to melt the ice. You solve for T. If you get a positive answer you can use it, because the assumption that you ended up on water meant you heated up the ice, you melted the ice into water, then heated up the water from zero to the final water. But if you did the calculation and got a negative value of T, that answer cannot be blindly used, because the assumption that you are on the other side of ice is wrong. Then, you can try something else; you can assume you’re down here. If you think you’re down here, then you’ve simply heated the ice from here to here. This water you brought down to zero, sucked out mc ΔT from that, then you’ve taken out now the latent heat of melting. You take out heat when you freeze, and then you’ve taken even more to come down here. Then, all those losses of the original water is equal to the gain of this ice. You can assume it here, you can solve for this T. When you solve for this T, if you’ve got a negative number, then you’re okay. That will be a good assumption if I say I sprinkled two drops of water on a big iceberg; we know it’s going to end up as ice and that’s a good starting point.

But if I give you numbers which are kind of wishy-washy, where I don’t know whether this will win or that will win, there’s a third possibility. The third possibility is at the end of the day, you end up here with some amount of water and some amount of ice at zero degrees. So, that’s a third option you may have to consider, if neither of them works. Then, the question is not what is the final temperature. But what’s the question then? What do you want to know in that case? How much is ice and how much is water? That’s the question.

And there are several ways to figure that out. Let me just say in words, I don’t want to do this algebra because for you guys it would be fairly easy. If it’s a question of–Suppose both of the things I try fail. I took a positive T, assumed I’m up here, and I assume the ice melted, and I get a negative answer; that’s shot down. I take a negative T and assume everybody froze and that doesn’t work. Then, I’m down to this option, which is some amount of water and some amount of ice. And the question is, “How much is left?” You solve that by doing the following. You say all this ice went from here to there. It does that by absorbing that mc ΔT; mass of the ice times specific heat of ice times ΔT. Maybe it was minus 40, the ΔT is plus 40. You give that heat to this guy; that heat you suck, out of this guy. When you suck that out of this guy, first you bring this to zero, then you still have some more heat you can extract from him, you will use that to convert water into ice at the price of 80 calories per gram. Maybe you can freeze 5 grams or 5 kilograms of water; that will be the extra ice, the rest will be the water you started with. The total mass will be the same, but if you got 60 grams of water, you bring the 60 grams to zero and you still have some more heat to be extracted; maybe you’ll convert 10 grams to ice and 50 will remain as water. So, the final answer will be 50 grams of water, 10 grams of ice plus whatever grams of ice you started with. That’s about the most complex heat-exchange problem. If you guys want me to tell you some more I will, or I can move on. I don’t know what your view on this is. Do you understand what you have to do in each problem?

Okay. So, it’s the conservation of heat that’s applied. So, the most tricky part is phase change, when you’ve got a phase change, you’ve got to remember that the formula mc ΔT–ΔQ has one more term, the one more term is this. Okay, so next question we ask is, “What’s the manner in which heat manages to flow?” We say you got these calories, I mean, how does it flow, what’s the rate at–what makes it flow. So, it turns out there are three popular ways of heat transfer; one is called radiation.

Chapter 6. Heat Transfer by Radiation, Convection and Conduction [00:55:06]

Radiation is when the heat energy leaves some hot body and comes to you without the benefit of any medium, like heat from the Sun. So, that’s really electromagnetic radiation that comes from hot, glowing objects, and directly comes to you. Electromagnetic radiation doesn’t need air, doesn’t need anything. In fact, if it needed air, we would not get any heat from the Sun because there is no medium between the Earth and the Sun. Most of it is just vacuum. So, if you took one of these space heaters, you know, with glowing red coils, and you feel warm. If I start pumping the air out of that room, of course, you will be dying very rapidly, but your last thoughts will be, “I am still warm” [laughter] because the radiation will keep coming to you. Okay? That’s radiation heat. There are lots of laws for radiation; I don’t want to give them to you because there are formulas you memorize, and you don’t understand too much of the physics right now. Other than to say it’s electromagnetic radiation, whatever that means–we haven’t gotten to that yet. That’s what comes from there to here and can come in vacuum. It doesn’t need a medium, is the key.

Then, the second way of heat transfer is called convection. So, convection is explained by the following example. You’ve got water; you put it on a hot plate. Then, in the lower part of it, the water gets hot. When it gets hot it expands, and when it expands the density goes down; therefore, by loss of buoyancy it will start raising up. Remember, a chunk of water belongs in water. A chunk of something else with lower density will float to the top. But the point is, water doesn’t have a fixed density. If you heat it up, the density goes down, so the water guys downstairs have lower density– they’re like a piece of cork, they will rise to the top. When they rise to the top, the cold water with the higher density will fall down. So, you set up a current. Hot rises to the top and cold comes down. And this also happens in the atmosphere. On a hot day, the air next to the ground gets really heated up and it rises, and the cold air comes down and you set up these thermal currents. So, here you’re trying to equalize the temperature between a region which is cold and a region which is hot by the actual motion of some material. In radiation, you don’t have the medium transferring heat because a medium is not even present in radiation. In convection, the medium actually moves. The hot guys physically move to the other place and the cold guys come here, and by that process, the heat is transferred.

The heat transfer I want to focus on a little more quantitatively, is conduction. So, heat conduction is something you’ve all experienced. I mean, if you have a skillet, why does it have a wooden handle? Simple reason; if you had a steel handle, you put it on a hot stove and you put your hand here, the fact that your body is at whatever, 98 degrees, and this one is God knows, 200 degrees, you’re going to have heat flow from here to here. So, we want to understand what’s the rate at which heat flows from the hot end to the cold end.

So, you can imagine a rod of some cross-section A, one end of the rod is in some reservoir at some temperature T₁, other end is at temperature T₂. By the way, I’m now introducing a new term called reservoir. Reservoir is another body like you and me, except it’s not like you and me. It’s enormous. It is so big that its temperature cannot be changed. You can sit on it, you will fry and you’ll evaporate, but its temperature will not change. No body is really a reservoir. If you drop an ice cube in the Atlantic, you’ll lower the temperature of the Atlantic but by a negligible amount. So, take the limit of Atlantic goes to infinity, then you have a reservoir. Reservoirs have one label, namely, what’s our temperature. So, something big enough can be–this room is like a reservoir. You put a cup of coffee here, you say it will come to room temperature. Actually, the room temperature meets the coffee, not halfway but slightly up. But the room is large enough so that we can attribute to the room temperature quite independent of bodies that go in and out of it. So, this is connected on the left to an enormous tank of maybe a water-ice mixture at zero degrees; this is a water-steam mixture at maybe 100 degrees. You put a rod there. We know heat is going to flow from the hot body, from the hot end to the cold end. And we want to write a formula for how much heat flows per second. Again, I’m going to write these formulas over and over again. So, you’ve got to ask yourself, what will it depend on? What are the properties it will depend on, in general, independent of what the rod is made of? Can you think of one? Yes?

Student: [inaudible]

Professor Ramamurti Shankar: You said the cross-section. Now, why do we say–what reason can you give for cross–

Student: If you just want to consider a rod with twice the cross-section area, you’re going to come up with [inaudible] two rods and twice [inaudible]

Professor Ramamurti Shankar: Yes, okay let me look at this argument. You take one rod, and for convenience let’s just take it to be a rectangular rod. Take another rod, rectangular rod; they will both transfer the same amount of heat for a given amount of time. Just glue them together and say here is my new rod. We know it’s going to transmit twice the amount of heat. So, it’s going to be proportional to the area. And why is the heat flowing? It’s flowing because of a temperature difference. So, that’s always there; that’s the underlying force for heat transfer. That’s the dynamics in thermodynamics; that’s what makes the heat flow. But then, we find as an empirical fact, that if these two reservoirs are separated by that distance, then the heat flow is a lot less than when they are closer. It seems to depend on how much temperature difference is packed in spatially. So, you want to divide by a Δx is not infinitesimal; it’s the length of the rod separating the hot and cold ends. In other words, if you dilute the temperature difference over one mile, the heat flow will be correspondingly reduced, whereas if there’s huge temperature difference between a very small spatial separation, there will be very robust flow of heat; that’s what we’re saying. These happen to be true, you realize, independent of what material I’m talking about. When I said one rod plus one rod is two rods, it doesn’t matter what it’s made of. Again, having put all these factors which you can argue on general grounds, you have to now ask, “What happens when this is a copper rod versus silver rod versus wooden rod?”

So, you’ve got to put one more number which is kappa [κ] here; not k you guys, it’s κ, and it’s called the thermal conductivity of that material. Sometimes you put a minus sign; minus sign just means it flows from hot to cold. I don’t care whether you put the plus sign or don’t put the minus sign; anybody knows that the heat is going to flow from hot to cold. So, just remember that direction of flow, and that’s all I care about, this sign here. This κ is the property of the material.

Once again, let me tell you–You can say, “Well, I have two reservoirs, hot and cold. I connected them with two different rods. This rod carried twice the amount of heat per second as the other rod. Is it necessarily a better conductor?” No. Maybe it had 10,000 times the cross-section. So, what you want to do is to make the playing field level, and compare rods of the same cross-section, same temperature difference, same length, then ask who conducts more heat. That depends on the material and that’s the thing you pulled out specific to the material. That is the property of wood or copper of steel; that’s the heat conductivity.

Chapter 7. Heat as Atomic Kinetic Energy and Its Measurement [01:03:27]

Okay. Now, the final topic is just going to be more hand-waving now. I don’t want to get into too many details. It really has to do with what is heat. In the old days, people just said that it was a fluid, and they postulated the conservation law for the fluid. You can postulate what you want, you’ve got to make sure it works, and it seems to work, in the sense that all the ΔQs in any reaction add up to zero. But then, people are getting hints that maybe this thing that we call heat is not entirely independent of other things we have learned.

So, where do you get the clue? One clue is, long back when we studied mechanics, we talked about two cars that come and collide; they slam into one big lump. Now, you’ve got no kinetic energy, no potential energy. Potential energy is always zero, they’re moving on the same height, kinetic energy was ½ mv² for this, ½ mv² for that; at the end there’s nothing. No kinetic, no potential, we just gave up and said, “Look, conservation of energy does not apply to this problem.” We just say it’s inelastic. On the other hand, we find whenever that happens, we find the bodies become hot. Here’s another thing you can do, you can take a cannonball, drop it from a big tower. This is how some people in the French army, I think, first detected this feature; you dropped cannonballs from a big height. When they hit the sand, they start heating up. Or you drill a hole in a cannon, that’s what Count-somebody did, and he also noticed that you need to constantly pour water to keep the drill bit from heating up. You’ll find very often, mechanical energy is lost and things heat up. So, you get a suspicion whatever the underlying mechanism, maybe there’s a rule that says if you lose so much mechanical energy that you cannot account for, then it translates into a fixed number of calories. If that is the case, then we at least get a dictionary on–between calories and joules.

So, joules is energy you can see, calories is energy you cannot see. That was going to be the premise. But first, you’ve got to prove that every time you lose some number of joules, you get a fixed amount of calories. And that experiment is due to Joule.

Here is the Joule experiment. It’s very, very simple and tells you the whole story. You have a little container in which there is a paddle. This is a shaft with a pulley, and there is a weight here. So look, try to imagine this guys. You got rope wrapped around the top pulley, and when you let this weight go down, it’s going to go down like this; it’s going to spin the shaft. And put some water here, and I have some fins that are sticking out, so they churn up the water. So, it’s like this thing, the egg-beater, right? In fact, I tried to do the experiment with an egg-beater this summer to a bunch of high school kids, and I got thoroughly humiliated because nothing happened as planned. But the idea is the same. You agitate the water in some fashion. But this guy did it in a particularly simple way. My egg beating was not good enough; you will see maybe in a while why that’s not good. What he did was to put these paddles, let the weight go down from there to here. Now, we can keep track of how much mechanical energy is lost, right? Because if this mass was at rest, and a drop to height mg drop to height h, it’s supposed to have mgh kinetic energy. Let’s say it’s got some kinetic energy, which is not equal to mgh. So, mgh minus kinetic energy is missing. So, some number of joules are gone. So, the water gets hot.

When the water gets hot, you can immediately ask how many calories were supplied to the water. Because that water heats up the same way whether or not you put it on a hotplate, or whether or not you churn it. It doesn’t seem to depend on how it got hot. This has the same effect. This water is hot in every real sense. So, you must have put some calories. You can find out how many calories you put in by looking at the mass of the water; specific heat of the water is 1; looking at the increase in temperature. So, some joules are missing, some calories have been pumped into the water. Then you ask, “Is there a proportionality between joules and calories?” And you find that it is. And that happens to be 4.2 joules per calorie.

In other words, if you can expend 4.2 joules of mechanical energy, you got yourself one calorie to be used for whatever heating purposes. So, in the example of the colliding cars, this had some energy, that had some energy, all measured in joules; they slammed together, they come to rest. That means you can take those many joules, divide it by 4.2 and get some number of calories. Imagine the whole car is made out of copper. Then those calories will produce an increase in temperature, right, equal to ΔQ is mc ΔT. That will be the rise in temperature of the car.

In practice, there will be other losses, because you heard the sound, well, that’s some energy gone; you won’t get it back. Maybe some sparks are flying, that’s light energy; that’s gone. You subtract all that out, you find that in the end, the calories explain the missing joules.

So, that made people think that this is just another form of energy. Because if you add this to your energy balance, there is no reason to go on apologizing for the Law of Conservation of Energy. Law of Conservation of Energy is not in fact violated, even at the inelastic collision, if you include heat as a form of energy. And the conversion factor is 4.2 joules per calorie. But the question is, “What right do you have to call it energy?” Energy, we think–primarily, when you say somebody’s energetic, you mean that someone’s running around mindlessly, back and forth. Energy is associated with motion. These two cars were moving, and we have every right to say they have energy. How about potential energy? Well, if the car starts climbing up a hill and slows down, we think it’s got potential. If you let it go, it’ll come back and give you the kinetic energy. So, most people’s idea of energy is just kinetic energy. That is lost. And yet, you get calories in return, so you ask yourself, “What can it be?”

Well, the correct answer to that came only when we understood that everything is made up of atoms. Once you grant that everything is made up of atoms, then it turns out that the kinetic energy of atoms is what we call heat. But you’ve got to be very careful. Take a tank full of gas. I throw it at you. That whole tank is moving, that’s not what I call heat. Okay? That motion you can see. I’m talking about a tank of gas that doesn’t seem to be going anywhere; yet, it got motional energy because the little guys are going back and forth. So, what we will find is what I’m going to show you next time, is that if you kept track of the kinetic energy of every single molecule in this car, every single molecule in that car, before and after, and you added them up, you would get exactly the same number. The only difference will be originally the car has got global common velocity; macroscopic velocity you can see. On top of it, it’s got random motion of the molecules that make up the car. So does the other car. When they slam together, the macroscopic motion is completely gone, and all the motion is thermal motion. But it’s still kinetic energy, and that’s what we will see the next time.

[end of transcript]

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