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0:00Professor Ramamurti Shankar: Alright class,
0:03welcome back. This is our last two weeks.
0:07We're going to have a slightly different schedule for the
0:10problem sets. I'm going to assign something
0:14today which is due next Wednesday.
0:17I'm giving enough time so you can plan your moves.
0:20Then, I will probably give you one last problem set with two or
0:25three problems on whatever I do near the end.
0:29We'll have to play it by ear. Okay, so this is another new
0:36topic on thermodynamics, a fresh beginning for those who
0:40want a fresh beginning. And there's also stuff you
0:45probably have seen in high school, some of it at least.
0:49So, the whole next four lectures are devoted to the
0:52study of heat, temperature,
0:54heat transfer, things like that.
1:01So, we are going to start with the intuitive definition of
1:04temperature everybody has. So, hang on to that;
1:07that's the right intuition. But as physicists,
1:11of course, we want to be more precise, more careful.
1:15So, let's say you have the notion of hot and cold.
1:21Even that requires a little more precision.
1:27That introduces the notion of what is called thermodynamic
1:32equilibrium. Just like mechanical
1:34equilibrium, this is a very important concept.
1:41So, I'll tell you what equilibrium is with a concrete
1:44example. If you take a cup of hot water,
1:49and you take another cup of cold water, each cup,
1:56if you waited sufficiently long, is said to be in a state
1:59of equilibrium as long as the cups were isolated from the
2:03outside world and not allowed to cool down or heat up.
2:06We think they maintain a certain temperature.
2:09We say it's in a state of thermal equilibrium because this
2:13temperature does not seem to change.
2:16Now, we have not defined what temperature it is precisely,
2:19but we can talk about whether whatever it is has changed or
2:22not changed. So, it will settle down to some
2:25temperature and it will maintain the temperature.
2:28Be very careful. If you leave a cup of coffee in
2:32this room, it will cool down because the room has got a
2:35different temperature. But I'm talking about a cup of
2:38coffee that's been isolated from everything;
2:40it maintains the temperature. Here's another cup of cold
2:43drink at what we feel is a lower temperature.
2:46They are both in a state of equilibrium.
2:49Equilibrium is when the macroscopic properties of the
2:54system have stopped changing. If you now pour one of these
2:58cups into the other one, that's going to be a period
3:01when the system is not in equilibrium in the sense that it
3:05doesn't have a well-defined temperature.
3:08For example, if you just poured it from the
3:09top, the hot stuff is on the top,
3:11the cold stuff is on the bottom, there's a period of
3:13transition when you really cannot even say what the
3:15temperature of the mixture is. Some parts are hot,
3:18some parts are cold; that system doesn't have a
3:21temperature. But if you wait long enough
3:23until the two parts have gotten to know each other,
3:26they will turn into some undrinkable mess,
3:28but the nice thing is it will have a well-defined temperature.
3:32That's, again, a system in equilibrium.
3:35So, you've got to understand that temperature and thermal
3:39equilibrium represent gross macroscopic properties and
3:44they're not always defined. At the microscopic level --
3:48it's no secret -- we all know everything is made of atoms and
3:51molecules. The atoms and molecules that
3:54form the liquid or the gas always have well-defined states.
3:58Each molecule has a certain location, certain velocity.
4:01But at a macroscopic level, when you don't look into the
4:05fine details, focus on a few things like
4:08temperature, they don't always have a
4:10well-defined value; that's what you've got to
4:12understand. Things have a well-defined
4:14value when they have settled down.
4:15How long does it take to settle down?
4:18That's a matter of what system you're studying.
4:21But generally, you can all tell when it has
4:23settled down. Here's another example.
4:25Suppose you take a gas and you put it inside this piston here,
4:30put some gas inside, you put some weights,
4:33and everything is in equilibrium.
4:36We say that it's in equilibrium because the macroscopic things,
4:39things you can see with your naked eye, nothing is changing.
4:43It's going to just sit there. But if you suddenly now remove,
4:47say, a third of the weights, the piston's going to rise up,
4:51shake around a little bit, maybe settle down in a new
4:55location. If you wait a few seconds,
4:57then the new location will again settle down,
5:00and you won't see anything with the naked eye that looks like
5:04anything is happening. In between you will see the
5:07pistons moving, the gas is turbulent,
5:09the pressure is high in some regions, low in some regions,
5:12then it settles down. This is the notion of systems
5:16in equilibrium, and in between,
5:18there are states of the system which are not in equilibrium.
5:23Now, whenever a system is in such equilibrium,
5:27we can assign to it a temperature that we call
5:31T. Right now, we don't know
5:33anything about this temperature, so we're going to build it up
5:36from scratch--other than your instinctive feeling for what
5:39temperature is. One of the laws of
5:41thermodynamics is called a zeroth law --zeroth law because
5:45they wrote down the first law, then they went back and had an
5:48idea which was even more profound, and they said,
5:51"We'll call it zeroth law." Zeroth law says,
5:54"if a and b are at the same temperature,
5:57and b and c are at the same temperature,
6:01then a and c are at the same temperature."
6:04Now, I see disbelief in the audience today.
6:10Why do you call this a law? Look, I think that is the key
6:12to our being able to speak about temperature globally,
6:16is the assumption that if I take a thermometer and measure
6:18something there, and I come back and dip the
6:20thermometer here, and it reads the same number,
6:23then I may conclude these two entities, which never met each
6:26other directly, are also the same temperature.
6:28That's not--That seems pretty obvious to you,
6:31but the whole notion of temperature is predicated on the
6:34fact that you can define an attribute called temperature
6:38that can be globally compared between two systems that never
6:41met directly, but met a third system.
6:44Okay. So, once we have some idea of
6:49hot and cold, let us decide now to be more
6:52quantitative. It's like saying,
6:54you know, somebody's tall and short is not enough.
6:57We go into how tall, how many feet,
6:58how many inches, how many millimeters.
7:00So, we want to get quantitative. All we have right now is a
7:04notion of hot and cold. So, what we try to do is to
7:07find some way to be more precise about how hot and how cold.
7:12So, what people said is, "Let's look at some things in
7:16the world that seem to depend on temperature."
7:20One thing that seems to depend on temperature is the following.
7:23You take this meter stick in the National Bureau Standards,
7:27kept in some glass case, at some temperature.
7:30You pull it out--or make a duplicate of it,
7:32you pull it outside and leave it in the room.
7:35What you may find is that if the room was hotter than the
7:38glass case, this rod then expands to a new length.
7:42So, one rod is outside the case, one rod is inside the case
7:45so the comparison is meaningful. Nothing has been done to this
7:49guy in the air-conditioned glass case, but this one is expanding.
7:53So, one way to define temperature is to simply ask how
7:56long is this rod, and somehow correlate the
7:59length of the rod with temperature by some fashion.
8:03So, you can do that. So, what you need to do
8:07that--what you need to do first is to define,
8:10put some markings on it so that for each extra something it
8:14grows, we can say the temperature has
8:16gone up by some amount. So, there we need units for
8:19temperature, that's completely arbitrary.
8:22And you need some standards, just like this meter stick,
8:25you know, it's not--nothing intrinsic in nature about a
8:29meter, we just made it up and said
8:31"Let's call that a meter." In the case of the meter,
8:33the zeroth law is if you bring a meter stick next to mine and
8:37we agree, you can take the meter stick
8:39somewhere else and define that to be the meter because if this
8:43stick is as long as that one and as long as that one,
8:46then those two are equal in length.
8:48But temperature--You are similarly going to use this rod
8:51and say, "This rod is a certain length when kept on top of this
8:54bucket of some fluid and the same length when I keep it on
8:57that bucket, then the two buckets are the
9:00same temperature." So, we can use markings on this
9:04rod compared to the unexpanded length as a measure of
9:08temperature. So, what people do is to pick
9:11something a little easier than this rod.
9:14They notice the liquids expand when you heat them.
9:18That's why in a summer day if you fill your gas tank,
9:22you have to leave some room at the top so the overflow can come
9:25out of the top; or you shouldn't fill it
9:28completely, otherwise it'll bust the tank.
9:30So, liquids expand. So, one way to measure
9:34temperature may be take some liquid, put it there,
9:38and then put it in hot rooms and maybe watch the liquid
9:42expand to the new height. And then draw some markings,
9:47and each marking can be a certain temperature.
9:51But people had a better idea than this one.
9:55They had the following idea of a thermometer,
9:58where you have a lot of fluid in a reservoir,
10:01a very thin tube evacuated at the top, and the fluid,
10:07then, is here. So, what's clever about this is
10:11that if this expands by one percent, your eyes should be
10:15good enough to see one percent increase in height.
10:19If this fluid expands by one percent in volume,
10:21that one percent in volume and it climbs up this narrow tube
10:24can climb to quite a bit [pointing to picture],
10:27because the extra volume you get by expansion will be the
10:31area of this tube times the extra ∆x by which it expands.
10:36So, you're magnifying the expansion by making all the
10:38expander fluid climb up this extremely narrow tube.
10:42In fact, the tube is so narrow, you cannot probably even see it
10:45well, which is why they have a little prism that magnifies the
10:49mercury or alcohol in the thermometer.
10:52Okay, so we have some way of following temperature now.
10:56We can draw some lines, arbitrary lines,
10:58it doesn't matter. That can be zero,
11:00that can be five, that can be 19;
11:02you've just got to make sure that it's monotonic.
11:04Then whenever it's on 21, we may argue that 21 is now
11:09hotter than 19. But you want a better scale
11:13than that. Even though that's
11:15mathematically adequate in practice, what people decide is
11:18to do it as follows. They said, "We want to set up
11:21thermometers so that people all over the world,
11:23in different parts of the world, different countries,
11:26different labs can all agree. So, we will make it possible
11:29for everyone to make their own thermometer by the following
11:32recipe." We will dip this guy in a
11:35bucket which has got some ice and some water.
11:40That's called the melting point of water, so that--or the
11:44freezing point of water; melting point of ice,
11:47or freezing point of water, it doesn't matter.
11:50We notice that as water cools down, in the world around us
11:53suddenly ice cubes begin to form.
11:55We go to the temperature at which that happens for the first
11:58time and we dip the thermometer there,
12:00and whatever reading we get we will postulate to be zero
12:04degrees centigrade. That is just a definition.
12:08We believe that's a good definition because people all
12:12over the world can do that. Of course, if you live in
12:16Kuwait, that's not going to work for you;
12:18there's no ice. But they figured out in parts
12:20of the world where you have ice, this is a very good definition.
12:24You get ice, you got zero degrees.
12:27Then they said, "Let's find another universally
12:30accessible thing," which, as you all know,
12:32is the boiling point of water. If you put water on the stove
12:36it heats up and heats up and heats up and suddenly it begins
12:39to bubble and boil and evaporate.
12:41That temperature is going to be called 100 degrees,
12:45100 degrees centigrade.
12:50Then, you take this column between zero and 100,
12:53and you divide it into 100 equal parts.
12:59And that is postulated to be the temperature anywhere between
13:04zero and 100. If you have gone 79 percent of
13:07the way to the top, from here to here,
13:10the temperature is 79 degrees.
13:16That's how the degrees were introduced, and that's a
13:18centigrade scale, and you guys know there are
13:20different scales. You can have the Fahrenheit
13:22scale, you can have any other scale in which what you want to
13:25call the freezing point is different.
13:27Somebody thinks it's zero, somebody thinks it's 32.
13:29And you can again call this something else,
13:31and you can divide this interval into 100 parts,
13:34180 parts, whatever you like. But the philosophy is the same.
13:38You have to find two points, which are reproducible,
13:41conveniently, and divide the region between
13:43them into some number of equal steps.
13:46If there's 100 equal steps, you say it's a centigrade
13:49scale, provided the lowest one is called zero.
13:52This is how you have thermometers.
13:54Now, there are some problems with this.
13:58One problem is that the boiling point of water does not seem to
14:03be very reliable. Because if you boil water in
14:07Aspen, for example, you know it doesn't seem to
14:12boil--it seems to boil more readily than in the plains.
14:17You can ask, "How do you know that?"
14:18maybe it is still doing the same thing.
14:20I know that because I tried to cook something,
14:23cook some rice and vegetables, I find, they don't cook at all.
14:26In Denver, it boils before it cooks;
14:29that way we know it's probably boiling earlier in the mountains
14:32than in the plains. So, who's going to decide what
14:36the real temperature is? So, you have to be more careful
14:40when you say boiling point and freezing point,
14:42because things don't seem to boil at a certain,
14:46predictable and fixed temperature.
14:49This is a very deep argument I have never appreciated fully
14:53when I was learning the subject, is that it's all cyclic
14:57definition. Because you may not know that
14:59the temperature is changing, because this thermometer by
15:03postulate, it's going to be the temperature by definition.
15:07How can it be wrong? What's wrong is that you know
15:10it's not a reliable method because physical phenomena,
15:13like when your rice will cook, are not reproduced by the
15:17boiling point of water. It cooks in the plains,
15:19it doesn't cook in the mountains, so we know the
15:21boiling point is to blame. Rice is the rice.
15:24That's how we know that that's not a good measure.
15:26So nowadays, people have much fancier
15:28measures, and I will tell you a little bit about that.
15:30But for a long time, this was a very good start.
15:33Don't worry about the fact that water boils differently at
15:36different altitudes; you could go to sea level and
15:39that's a good enough definition. Sea level is pretty much
15:43constant all over the world, and you can say the pressure of
15:45sea level is the pressure at sea level;
15:47just the ρgh of the atmosphere.
15:49Okay, so that's the usual definition of temperature.
15:54Now, the trouble started when people realized that if you make
15:58a thermometer with your favorite fluid,
16:01maybe mercury, and I make one with alcohol,
16:04they will agree at zero and they will agree at 100 because
16:07that's how you fixed it. You rigged it so at zero
16:10everyone says zero; 100 everyone says 100.
16:12But how about 74 degrees, or 75 degrees?
16:17I say it's 75 if my fluid has climbed three-fourths of the way
16:21to the top. At that point,
16:22yours may not have climbed three-fourths of the way.
16:26In other words, you've got two things,
16:29two graphs, which have zero and 100 degrees;
16:32one graph may be like this, one may be like that.
16:35So that when I think it is 75, you may think it is 72.
16:38At 100, we will agree because we have cooked it up that way.
16:41In other words, it's not true that all liquids
16:44expand at the same rate. So, you will have to then pick
16:48one liquid and say, "We swear by that liquid,
16:51and when that liquid's gone halfway, we'll say it's 50
16:55degrees." So, you will have to pick a
16:57liquid, you'll have to have an international convention,
17:00you know, there's the alcohol lobby and there's the no alcohol
17:04lobby; they argue.
17:05Finally, they found out a much better solution than these
17:08liquids. They found out that if you use
17:10a gas--You can define temperature using gasses,
17:14which have some very, very nice properties.
17:17And this is the gas thermometer that I'm going to tell you now.
17:23So, here is how you build a gas thermometer.
17:29You take some gas in a container.
17:32A typical container for me in all--whenever I draw anything
17:36thermodynamics, it's going to be gas inside
17:38some cylinder with some weights on it,
17:41and that defines the pressure of the gas.
17:46Of course, the pressure will be the mg of these weights
17:49divided by the area of the cylinder.
17:52That's the pressure, plus atmospheric pressure.
17:55And the volume is this, whatever the volume is,
17:59base times height. Here's what we ask you to do.
18:05Take the product of pressure times volume for any sample of
18:08gas. Take some gas,
18:10put it in this tank, and now put it on different
18:13surfaces, like a hot plate, like a stove,
18:16like a tub of water, and measure the temperature
18:19using some standard method up to that point like a mercury
18:22thermometer. What you notice is that the
18:26temperature measured by some reasonable scheme shows that the
18:30product of P times V lies on a straight line
18:34[drawing diagonal line on board].
18:37If you connect the dots, you find the product PV
18:41is linear in this temperature variable.
18:49And this is zero degrees, and this is 100 degrees.
18:56Now, here is the beauty of the gas thermometer.
18:59If you take a different gas and you put a different amount of
19:03different gas in a different cylinder, you will get some
19:06other graph; it may look like this [drawing
19:08another straight line].
19:13For you, that is zero and that's 100.
19:15But the most important thing is that's also a straight line.
19:19That it's also a straight line, has the following implication
19:23you guys can prove at your own leisure,
19:26which is that, if I think that my gas has
19:30climbed 56 percent of the way of this height to the top,
19:36so the temperature is 56 degrees, I ask what's your gas
19:39done, you will find yours also climbed 56 percent of the way.
19:43It's the property of straight lines.
19:45You can show that if you took two straight lines,
19:48whatever be their slope, if they agree,
19:51if this is zero and this is 100, it has got a different
19:54slope, when you have climbed to the halfway point,
19:58draw a line at 50 degrees and ask what has any gas done,
20:01they will all have climbed to the halfway point from the zero
20:05point to the 100 point. In other words,
20:08gas thermometers will not only agree at the end points where
20:12they must, by construction, they seem to agree all the way
20:16in between. But there is one requirement.
20:20This gas has to be very dilute.
20:26The more diluted it is, the better it comes out.
20:30So, take neon or Freon or whatever you like.
20:33Don't pump it up with a lot of gas;
20:35put the least amount of gas you can get away with.
20:37Then, you find all gasses have the property that if you
20:41calibrate them at zero and 100, they agree in between.
20:45Is that clear to you? Take the product P times
20:48V of your gas by putting it on different surfaces,
20:52measure the product, plot this graph.
20:55Whenever you're on ice [freezing point of water]
20:58you call it zero; whenever you're on boiling
21:01water you call it 100. You find they're connected by a
21:05straight line, then every point in between,
21:08you've divided equally, leads to equal increase in the
21:11product P times V. P times V for a
21:15gas is better than the volume of mercury or volume of water
21:18because it doesn't depend on the gas.
21:20So, everybody can use the gas thermometer.
21:25That's why we prefer the gas thermometer.
21:27So, this is the interesting issue about measurement or
21:31definitions and cyclic definitions--you've got to be
21:35careful. The laws of nature allow you to
21:37pick anything you like that varies with temperature and use
21:41that as a definition of temperature, as a thermometer.
21:44So, why are some thermometers preferred over the others?
21:47They're preferred over the others if the laws of nature
21:50take the simplest form when described in terms of those
21:54thermometers. In other words,
21:56take a meter stick. What makes a good meter stick
22:00for a standard? You say the one that doesn't
22:03expand, but we don't know what that means.
22:05That meter stick is the standard;
22:06by definition it's right. But then, you will soon find
22:09out that it's not really that simple, because there are good
22:12and bad meter sticks. For example,
22:14the same meter stick at one time out of the year doesn't
22:16match its own length at a different time of the year;
22:19then we know that it's not a good meter stick.
22:21Similarly, there are good and bad thermometers,
22:23and people arrive on the gas thermometer this way.
22:25If you have a gas thermometer, something very interesting came
22:29out of the gas thermometer. If you cool it below zero and
22:34you ask which way is it going, I don't know how low you could
22:39go. In the old days,
22:40people couldn't go far below zero, but now we can go to
22:44one-billionth of a degree above a certain point.
22:47I'll tell you now, these thermometers indicate
22:50somehow the product PV vanishes at a temperature which
22:55is minus 273.16, suggesting that there is
22:59something very special about that temperature.
23:04Because if you took another gas--well, I'm going to do a
23:08little cheating here--that also extrapolates that same
23:11temperature. So, all gasses,
23:13all gas thermometers say there is something very special about
23:17this temperature because that's when our pressures all vanish.
23:21So, as you cool a given amount of gas, even at a given volume,
23:25if you keep the volume constant and ask what pressure do I need,
23:29how many weights do I have to put on;
23:31that decreases and vanishes at this temperature.
23:34And this is called the absolute zero of temperature.
23:42It's called absolute zero for many reasons.
23:45One is that unlike the zero of the centigrade,
23:48which is by no means the absolute lowest possible
23:51temperature, the absolute zero is the lowest
23:55possible temperature. Why?
23:58Because the gas pressure can be reduced and reduced and reduced,
24:00but the worst that can happen is that it can go to zero.
24:03That's it. It cannot go below having no
24:06pressure. We'll find in other ways,
24:09also, this is the temperature at which you will see
24:12conceptually no further cooling is possible.
24:14That will require you to understand what hot and cold
24:17mean. But right now,
24:18this says all gas thermometers point at this temperature.
24:21So, people decided, "You know what,
24:24calling this zero is kind of artificial."
24:27That's based on human obsession with water.
24:31But if you think laws of science describe the whole
24:34universe, what about planets where there's no water?
24:38Right? You cannot describe--Suppose
24:42you're talking to a different civilization;
24:45Planet of the Apes. You want to tell those guys,
24:49"We're going to set up our temperatures;
24:50zero is when water freezes," and they say,
24:53"What is this thing called water?"
24:55"You know, the stuff you drink." You don't know what these apes
24:57are drinking. Maybe they're drinking methane
25:01or liquid hydrogen. We don't know.
25:03On the other hand, you say, "Take any vapor and
25:07wait until the product of the pressure and volume go to zero,
25:11let's call that zero," that's the universal standard.
25:15It's not tied to something called water.
25:18It was fine for a while, but it is not fine as a
25:20universal aspiration for thermometers.
25:23So, zero of temperatures can be set from here.
25:27Once they did that, they called that zero,
25:30they needed one other temperature.
25:33And they decided that if you're starting the new temperature
25:37scale, you will put the zero not at the centigrade,
25:40but this is now called Kelvin. And everything will follow a
25:44straight line, but to define what one degree
25:47means, you've got to define one other temperature.
25:50That's how we define the straight line;
25:51that temperature would be called 273.16.
25:57But this point is called the triple point of water.
26:01What's the triple point of water?
26:03You know water and ice can coexist, and you know that water
26:08and steam can coexist at 100 degrees.
26:10But by varying the pressure and temperature and volume,
26:13you can actually find a certain magical point in which both ice,
26:17water and steam can coexist, simultaneously.
26:22It cannot pick between those three options.
26:27Ice floating on water is when water has not decided whether to
26:30be ice or to be water. That's the coexistence point of
26:33two things. And when the water starts
26:35boiling on your stove, that's when water and steam
26:37coexist. But I'm saying that certain
26:40conditions of pressure and temperature and volume so that
26:44water, ice and steam will coexist.
26:47Now, that is a unique situation; you cannot get to that by any
26:51other means. And that temperature we will
26:55call plus 273.16 in these absolute units.
26:59So, basically, what you have done by going to
27:02the absolute units is you've shifted the zero to a more
27:05natural point where all graphs meet;
27:08then, you define one degree Kelvin to be so that 273.16 of
27:12that Kelvin brings you to the triple point of water.
27:16So, if you found that confusing, I'm just saying the
27:19boiling point of water is not a fixed number.
27:21You go to the mountains, it changes.
27:23But only under one condition can water and ice and steam
27:28coexist. You cannot get that any other
27:32way. So, everybody will agree on
27:34that particular situation, that will be called 273.16
27:38Kelvin. Now there is a rule, apparently.
27:42You can say, "degree centigrade," you're not
27:45supposed to say, "degree Kelvin."
27:47There was a big deal made in a lot of books.
27:50I keep forgetting--In fact, I forgot again,
27:53and nothing terrible has happened to me.
27:56So, I don't think you should pay too much attention to
27:59whether you can call something "degree Kelvin" or simply
28:01"Kelvin." I think the purpose of language
28:04is to have no ambiguities. But when they say,
28:06"degree Kelvin" and I find that you guys don't get confused,
28:09I don't think that's a big deal.
28:10But you'll find if you're a very erudite person,
28:13you will never write "degree Kelvin."
28:16But having said that, don't hold me to those
28:18standards--I just don't feel any affiliation to this particular,
28:22completely artificial and empty convention.
28:26But you are supposed to remember, if you take the GRE or
28:29something, it's not called "degree Kelvin."
28:32Okay so, as far as we are concerned, the Kelvin scale is
28:36like the centigrade scale, except the zero has shifted to
28:39here. That's it.
28:41That's the temperature scale you will use.
28:45That's the absolute temperature. Whenever I write T from
28:48now on, I'm talking about Kelvin, not centigrade.
28:51Now, that's all about heat--I mean, all about temperature.
28:57Now, I'm going to talk about heat.
28:59So, heat is denoted by the symbol Q,
29:02and you've got to ask yourself, "What are we talking about when
29:05we talk about heat?" Again, let's use your intuitive
29:09sense of what heat is. Say I have a bucket of water;
29:12I want to heat it up. And how do you do that?
29:15You put the bucket on top of something else which you think
29:17is hotter, and when the two are brought together,
29:20somehow the water begins to feel hotter and hotter.
29:23So, we say we've heated the water, and we say we have
29:26transferred heat. Now, people were not sure what
29:30really was being transferred. What is it that's going from
29:34the stove to the water? Why is it that the stove,
29:37if it's not plugged in, is getting cooler and the water
29:40is getting hotter? They just decided to call it
29:44the caloric fluid.
29:49They imagined there was a certain fluid which is abundant
29:53in hot things, and not so abundant in cold
29:56things. When you put hot and cold
29:58together, this magical fluid flows from hot to cold,
30:01and in the process heats the cold thing.
30:04And they decided to measure it in calories.
30:07And so, you have to define what a calorie is.
30:10In other words, you want to ask,
30:12"How much heat does it take to heat this bucket of water?"
30:16And the rule they made up was, we're going to define something
30:22called a calorie where the number of calories you need is
30:27equal to the mass of water times the change in temperature.
30:32That's going to be calories. In other words,
30:37if I had a container with 10 grams of water,
30:40and the temperature went up--I'm sorry,
30:43this is mass of water in grams.
30:51If you have 1 gram of water, and you did something to it and
30:54the temperature went up by seven degrees, you have,
30:56by definition, pumped in 7 calories.
31:04If this was a kilogram of water, this would be called a
31:09kilocalorie. Sometimes they use grams and
31:12calories; sometimes they use kilograms
31:14and kilocalories. But the definitions are
31:16consistent; if you put a kilo in the gram,
31:18put a kilo in the calories. Okay.
31:23Now, suppose you say, "I don't want to just talk
31:26about water, I want to talk about heating something else.
31:31Maybe I want to heat a gram of copper."
31:35So, then you write down the following rule.
31:39The amount of heat it takes to heat up anything--pick your own
31:44favorite material--gold. Then, the amount of heat,
31:48I think we can all appreciate, must be proportionate to the
31:51amount of stuff you're trying to heat up.
31:53That's our intuitive notion. If you've got one chunk of gold
31:56that takes some number of calories, you have a second
31:59identical chunk; by definition,
32:00that should take the same number of calories.
32:02You put them together, it is clear that whatever this
32:06caloric fluid is, you need double that.
32:08So, it's got to be proportional to the mass of the substance.
32:12And it's got to be proportional to what you're aiming for,
32:16namely, increase in temperature.
32:18But this is true for any substance, whether you're
32:22heating copper or wood or gold; no matter what you're heating,
32:25it is true the heat need is proportional to mass and to the
32:28[change in] temperature.
32:29So, what is it that distinguishes one material from
32:31another? We put a number here,
32:33and that number is called the specific heat.
32:37The specific heat is the property of that material.
32:47You've got to understand certain formulas will depend on
32:50certain parameters in a genetic way, and some things that depend
32:53on the actual material.
32:57In fact, there's a similar quantity.
32:59I mean, maybe I'll take a second to tell you.
33:01If you go to liquids that I said were expanding,
33:03you can do the same thing. Take a rod and start heating it
33:08and ask, "How much will it expand if I heat it by some
33:12amount ∆T?"
33:18What will it be proportional to? Can anybody think of what it
33:20may be proportional to? Yes?
33:23Student: Original length? Professor Ramamurti
33:24Shankar: Depends on the original length of the rod.
33:25Now, why is that? Why do we think it's got to be
33:27proportional to the length of the rod?
33:29Student: Because it expanded based on what it had
33:34before. Professor Ramamurti
33:36Shankar: Yeah, it's based on what it had
33:37before. Yes?
33:38Student: Well, each cycle the rod will expand
33:40by some amount, so [inaudible]
33:41Professor Ramamurti Shankar: That's correct.
33:43I think one way to say that is take a meter stick,
33:46it expands to some amount, put another meter stick next to
33:50it, that expands to the same amount by definition of
33:53identical things. For the two-meter stick it will
33:56expand by twice as much. So, we put the length of that.
34:00So, no matter what you're heating -- a block of wood,
34:03block of steel -- this is true. But then, the fact that heat
34:07has different effects on copper versus wood, is indicated by
34:12putting a number here. That α is called the
34:16coefficient of linear expansion, and that depends on the
34:22material. These are true no matter what
34:24you are heating.
34:29So, these specific numbers, these coefficients,
34:30these αs that come in are going to come in all the
34:33time, so you should get used to them.
34:34Here's another one. Let's play this game one more
34:37time. We can ask how much does the
34:39volume of a body change when I heat it.
34:40Well, the change in the volume, again, would be proportional to
34:44the starting volume times the increase in temperature.
34:47Then you put another number; that's called the coefficient
34:50of volume expansion. And that depends on the
34:53material. So, if you take copper,
34:56copper will have a certain α;
34:59iron will have a different α;
35:00wood will have a different α.
35:01Each material will have a different α.
35:03This is the property of the material.
35:05If you say, "Well, I had something and when I
35:08heated it up by one degree, it increased by nine inches;
35:11another one increased by two inches."
35:13Is it clear that the first one expands more readily?
35:16It's not, because the first one could have been a mile long,
35:19second one could have been a foot long.
35:21So, you have to take out certain factors that are
35:23universal, and the rest of it you put into a property of the
35:26material. Similarly, when you come to
35:29specific heat, you ask how much heat does it
35:32take to heat some object, it depends on the mass.
35:35It doesn't matter what you're heating.
35:37Depends on the increase in temperature, because that's the
35:39whole purpose of adding heat; it's always going to be linear
35:42in the ∆T. This one is the property of the
35:46material, and by definition, c equal one calorie per
35:50gram, or one kilocalorie per kilogram for water.
35:59Once you've got--So remember, one calorie per gram for water
36:05is the definition. Once you define water to have a
36:09specific heat of one calorie per gram, you can define specific
36:13heat for other materials by the following process.
36:17So, what do you do? You take a container with some
36:22water in it. Let's assume the container has
36:25zero mass, so I don't have to worry about it.
36:28It's an approximation. If you are worried about that,
36:30you know, take a huge container so that the volume of water
36:33dominates the surface area of the container.
36:36Anyway, container's neglected; you've got some water.
36:38This water is of some initial temperature
36:41T_1, and I have some new material,
36:46lead, and I want to find its specific heat.
36:49So, I take the lead in the form of pellets and I heat the lead
36:53pellets to some temperature T_2,
36:56and I drop these guys into this water.
37:01That'd be an example where initially, the lead is in
37:03equilibrium, maybe on a furnace, at temperature
37:06T_2; water's in equilibrium,
37:08maybe in the room, at temperature
37:10T_1. Then, I put the pellets into
37:12the water, and there will be a period when the temperature is
37:15not defined. Then, soon they'll settle down
37:17to some common temperature called T_f.
37:24We will now postulate--this is a postulate, or a law.
37:29The total change in Q is zero.
37:35In other words, if Q is lost by one body
37:37and gained by another body; the loss and the gain must
37:40equal. It's a new law.
37:43You can make up all the new laws you want.
37:45You don't know if they're right, but this is the law you
37:47first make up. In that case,
37:49what can you say in this particular problem?
37:52In any of these heat problems, I urge you to draw the
37:55following picture. Here is one temperature,
37:57here is another temperature, here is the final one,
38:00which we don't know, but we can measure with a
38:03thermometer and measure it. Then, you say the mass of the
38:07water, and specific heat of water, which is 1 times
38:12∆T, which is the final temperature
38:16minus initial temperature. Ditto for the lead pellet;
38:21mass of the lead, lead has got a symbol Pb,
38:25times specific heat which I don't know,
38:28times a change in temperature which is T_f-
38:34T_2 = 0. The sum of all the mc
38:38∆Ts is zero.
38:45This is the gain of heat, of the water.
38:49This, if you work it out, will be a negative number,
38:51because you can see T_f is below
38:54initial T. This will turn out to be
38:56negative, and the positive and negative will add up to zero.
38:58So, what is it you don't know? Well, you know the mass of the
39:02water. Specifically,
39:03the water is 1 by definition; T_f and
39:06T_1 are measured by thermometers.
39:07Mass of lead is for you to measure;
39:09these are known; you can find c.
39:12So, this is a birthday present for you guys.
39:15If you ever see this in an exam, jump on this first because
39:18you've been doing this in high school,
39:20and I know kids love this kind of calorimeter problems.
39:22Yes? Student: Looking at the
39:26volume in that equation it expands linearly but wasn't the
39:29problem with the liquid, measuring liquid,
39:32changing volume, but it didn't expand
39:35[inaudible] Professor Ramamurti
39:37Shankar: Yes. That's correct.
39:41So, the real point is, if everything expanded
39:43linearly, we wouldn't have the disagreement between different
39:47thermometers. So, it turns out to an
39:50excellent approximation, the change of length is
39:53proportional to the length, but it's not exactly
39:56proportional to the length. There will be terms involving
39:59higher powers of length. Not only that,
40:02specific heated materials is also not a constant.
40:05We said specific heated water is 1.
40:07Turns out at a certain temperature range it'll be 1;
40:10at a different range in fact, it's not quite 1.
40:12I told you long back. Everything I tell you is wrong.
40:16The question is, "How many decimal places do you
40:18have to go to before you honor my fallacies?"
40:21Specific heat of materials is not a constant,
40:23with the big industry calculating the specific
40:26materials starting from atoms and quantum mechanics.
40:29So, none of the things treated as constants are ever constant,
40:34including those alphas and betas.
40:37I can always fudge it by saying α itself may depend on
40:40the temperature, and also the dependence on
40:42L may not be linear. But you should also look at
40:45dimensional considerations and say if it's not L,
40:48if you want to put an L^(2) as a correction to
40:51the formula to match the units, L^(2) has to be divided
40:54by another length to keep the units.
40:56What other length do we have? It may turn out to be the
40:59inter-atomic spacing. So, once the atomic properties
41:02come into play, then you can find ways to
41:05calculate corrections. So, all these laws are,
41:08in fact, very tentative and approximate.
41:10These are pretty ancient physics.
41:12I think the way I do the physics course here,
41:14sometimes I'm in the 1600s, sometimes in the 1400s,
41:18sometimes in the year 2000, but going back and forth.
41:20This is way back when people did not even know about atoms.
41:23So, they were trying to do the best they can,
41:25and what you found empirically is that once you found a
41:29specific heat for lead, right, you solve for it,
41:32then you can do another experiment using that value and
41:35you find if you use the right values,
41:37∆Q does add up to zero. Again, when it adds up to zero,
41:41it adds up to zero to a very good approximation,
41:43during the epoch. Another epoch when people do
41:46more and more accurate experiments, everything is shot
41:48down. In fact, specific heats of all
41:51materials seem to go to zero when you approach absolute
41:55temperature. But you have to understand the
41:58laws of quantum physics to know why that happens.
42:01So, this is in a period when people are probing temperature
42:05ranges which are around room temperature,
42:09or boiling or freezing point of water, which is a very narrow
42:12window in temperature. If you look at the history of
42:14the universe, you've got incredibly high
42:15temperatures near the Big Bang, and even now the rest of the
42:18universe is bathed at some temperature that happens to be
42:21very, very low, which is near three
42:23degrees; it's called a blackbody
42:25radiation from the Big Bang. So, the temperature of the
42:28universe goes through huge ranges, and only when you probe
42:31different ranges you see different physics.
42:33If you come to Sloan Lab, you can go to temperatures way
42:36below 1 degree Kelvin or hundredth of a Kelvin,
42:39and we heard a talk last year, physics at one billionth of a
42:43Kelvin. If you want to cool them and
42:45cool them and cool them, by zero degree Kelvin,
42:48see, there I go. Zero Kelvin is a barrier we're
42:50not able to cross, just like the velocity of light
42:53is something we're not able to cross.
42:55These are all big surprises. The fact that velocity has an
42:59upper limit, not obvious even to Newton.
43:01Why not? Why not put rockets on top of
43:04rockets? Likewise, why not build better
43:06and better refrigerators? The reason you cannot go below
43:09zero is when you go to zero, all the mechanical attributes
43:13of pressure simply vanish, and they cannot have negative
43:16values. You will see more about this
43:18when you understand heat in greater depth.
43:20Anyway, right now, ∆Q = 0 is the rule you
43:23use. I'm sure you guys know how to
43:25do these problems. Now, there's a little twist
43:27that comes in, I just want to mention that to
43:30you. The twist is the following.
43:33So, I take some ice--ice, by the way, is not always at
43:37zero. You know, you can go below zero.
43:38Your refrigerator is several degrees below several tens below
43:42zero. So, let's take ice,
43:44and let me measure--I take this container, I put some ice at,
43:50say, minus 30 degrees. I've gone to centigrade now so
43:55we can relate to ice. And I put it on some source of
44:00heat, and I watch how many calories are coming in.
44:03Let me arrange a device that will pump in a fixed number of
44:06calories every second. So, as a function of time,
44:09I'm expecting the temperature of this to go up.
44:12Do you understand that? In every second,
44:16I get some number of calories, and those number of calories
44:20are going to produce for me mc ∆T,
44:23m and c are constants, so ∆Q is
44:27proportional to ∆T. But if you divide both by the
44:30time elapsed, then the rate at which the
44:32temperature rises will be the rate at which the heat flows
44:35into the system. If heat is flowing at a steady
44:38rate, temperature should rise, and indeed it does.
44:41Temperature of the ice goes from minus 30 to minus 20 to
44:46minus 10 and so on. But once it hits zero,
44:50it gets stuck. I know heat is coming in,
44:54but it's not getting hotter. But I notice that the ice is
44:58beginning to melt. There will be a period between
45:02here and here when I pump in calories, I don't get any
45:06increase in temperature but I get conversion of ice into
45:11water. And there will be a period when
45:13this guy looks like some water with some chunks of ice floating
45:16on it.
45:20And until all the ice is converted to water,
45:24the whole system is stuck at that temperature.
45:29That's a very interesting property.
45:30Now, if you really took a real pot and you put a chunk of ice
45:33on it, you know what will happen, right?
45:36The bottom of the ice will melt; it may even evaporate.
45:38That's not what I'm talking about, because that's not a
45:40system where there's a globally defined temperature.
45:43I want you to heat the ice so slowly, the minute you put a
45:46little bit of calories, give it enough time for all
45:48these guys to share that heat, so that the whole system has
45:52one single common temperature. Let's watch the temperature
45:56rise. I'm saying it gets stuck at
45:58zero, but your calories are getting you something;
46:00they're converting ice into water.
46:02Then you can ask, okay, what penalty do I have to
46:05pay, that's called a latent heat of melting,
46:08and again, I know only in calories per gram,
46:10it's 80 calories per gram for water.
46:16Some of your ∆Q now goes not to raise the
46:20temperature, but to melt that amount of stuff at the latent
46:25heat of melting. That's how much Q you
46:28need to melt that amount of stuff and the L varies
46:33from substance to substance, but water is 80 calories per
46:37gram. If you want to melt mercury
46:39from solid mercury to liquid mercury, it will have a
46:42different number. Then, once everybody has become
46:46water, then that uniform system of water starts growing.
46:53And this is called a phase change.
46:56A phase change is when it changes its atomic arrangement
46:59from a regular array; for example,
47:01that forms a solid into a liquid.
47:04In a solid, everybody has its place;
47:06you can shake around where you are, but liquid you can run
47:08around. The specific heat of ice is not
47:11the same as the specific heat of water, so you've got to be
47:16careful. Even though it's still made up
47:18of water molecules, the calories needed to heat one
47:21gram of ice is roughly half what it takes to heat one gram of
47:24water. So, in these problems,
47:25don't make the mistake. Okay then, you go along and I
47:28guess you know what the next stopping point is.
47:31When you come to 100 degrees, again, it gets stuck until
47:35everybody vaporizes, and then you get steam.
47:38Then, you can have super-heated steam, which is at even higher
47:41than 100 degrees. So, that's the latent heat of
47:44vaporization. I really don't know what--you
47:46want to write something, I think it's 500 and something
47:49calories per gram. That's information I don't
47:52carry in my head.
47:57So, if I tell you I took some ice at minus 30 and I dumped in
48:025,000 calories, where will it end up?
48:05You've got to first spend a few calories going from here to
48:08here, you got some more money left you can start melting this,
48:11maybe you'll run out of stuff there, and that's what you will
48:14have. Some amount of water and some
48:16amount of ice. If you have even more calories
48:19at your disposal, you can melt it all and start
48:21heating it. You may come this way and you
48:23may be running out of calories; if not, keep going here and
48:26there and there, and you may end up there if you
48:29got enough calories. Or one can ask a question,
48:32"How many calories does it take to convert ice at minus 30 to,
48:36say, water at 100?" You'll have to do the mc
48:39∆T for that, m times latent heat for
48:42this, mc ∆T for that,
48:44and m times latent heat of vaporization for that.
48:50So, the kind of problems you can get are fairly simple most
48:54of the time. Only kind of problem where you
48:57can really get in trouble is the following.
49:00I will mention that to you. Suppose I take some water and
49:04some ice, so this is zero. The ice is at,
49:08say, minus 40, the water is at plus 80.
49:13In fact, let me make that water plus 40.
49:16I bring them together and I ask you what will happen.
49:20Now, this is a subtle problem. If you had two--If you had
49:26water at 40 and you had water at 20, you can easily guess that
49:30it'll end up somewhere in between;
49:32you can calculate it. Now it's more subtle.
49:36You've got water at 40, you've got ice at minus 40,
49:38you bring them together and ask what happens.
49:41Well, the answer will depend on how much of the stuff you have.
49:44If by water at 40 you mean the Atlantic Ocean,
49:47and by ice you mean a couple of ice cubes, we know what's going
49:51to happen. These guys are going to get
49:53clobbered; they're going to melt;
49:54you will end up somewhere here. Then, you can easily calculate
49:58the final temperature by saying mc times this ∆T
50:02for water, in magnitude,
50:04is going to be the heat given to this.
50:07Heat given to this is the mc ∆T to come here;
50:10then, the heat to melt this amount of ice,
50:12then the heat to raise this amount of water to that final
50:15temperature. Then, you can solve for the
50:18final temperature. So, if you want to solve this
50:21problem, and I give you some mass for this ice,
50:24of water, and I give you some mass for the ice,
50:27you can first make the optimistic assumption that you
50:30will end up as water, but at an unknown temperature.
50:33We call the unknown temperature T;
50:35this is the T_1,
50:36this is the T_2.
50:37Write your equations, except you'll have one more
50:40term there. That's the heat it takes to
50:42melt the ice. You solve for T.
50:45If you get a positive answer you can use it,
50:47because the assumption that you ended up on water meant you
50:51heated up the ice, you melted the ice into water,
50:54then heated up the water from zero to the final water.
50:57But if you did the calculation and got a negative value of
51:00T, that answer cannot be blindly used,
51:03because the assumption that you are on the other side of ice is
51:06wrong. Then, you can try something
51:08else; you can assume you're down here.
51:11If you think you're down here, then you've simply heated the
51:16ice from here to here. This water you brought down to
51:20zero, sucked out mc ∆T from that, then you've taken out
51:24now the latent heat of melting. You take out heat when you
51:28freeze, and then you've taken even more to come down here.
51:32Then, all those losses of the original water is equal to the
51:35gain of this ice. You can assume it here,
51:37you can solve for this T.
51:38When you solve for this T, if you've got a
51:40negative number, then you're okay.
51:42That will be a good assumption if I say I sprinkled two drops
51:45of water on a big iceberg; we know it's going to end up as
51:48ice and that's a good starting point.
51:50But if I give you numbers which are kind of wishy-washy,
51:53where I don't know whether this will win or that will win,
51:56there's a third possibility. The third possibility is at the
52:00end of the day, you end up here with some
52:04amount of water and some amount of ice at zero degrees.
52:09So, that's a third option you may have to consider,
52:11if neither of them works.
52:15Then, the question is not what is the final temperature.
52:18But what's the question then? What do you want to know in
52:23that case? How much is ice and how much is
52:26water? That's the question.
52:28And there are several ways to figure that out.
52:32Let me just say in words, I don't want to do this algebra
52:35because for you guys it would be fairly easy.
52:37If it's a question of--Suppose both of the things I try fail.
52:41I took a positive T, assumed I'm up here,
52:44and I assume the ice melted, and I get a negative answer;
52:47that's shot down. I take a negative T and
52:49assume everybody froze and that doesn't work.
52:51Then, I'm down to this option, which is some amount of water
52:54and some amount of ice. And the question is,
52:57"How much is left?" You solve that by doing the
53:00following. You say all this ice went from
53:04here to there. It does that by absorbing that
53:08mc ∆T; mass of the ice times specific
53:12heat of ice times ∆T. Maybe it was minus 40,
53:15the ∆T is plus 40. You give that heat to this guy;
53:20that heat you suck, out of this guy.
53:21When you suck that out of this guy, first you bring this to
53:25zero, then you still have some more heat you can extract from
53:28him, you will use that to convert
53:31water into ice at the price of 80 calories per gram.
53:35Maybe you can freeze 5 grams or 5 kilograms of water;
53:39that will be the extra ice, the rest will be the water you
53:43started with. The total mass will be the
53:45same, but if you got 60 grams of water, you bring the 60 grams to
53:50zero and you still have some more heat to be extracted;
53:54maybe you'll convert 10 grams to ice and 50 will remain as
53:58water. So, the final answer will be 50
54:00grams of water, 10 grams of ice plus whatever
54:03grams of ice you started with. That's about the most complex
54:08heat-exchange problem. If you guys want me to tell you
54:14some more I will, or I can move on.
54:16I don't know what your view on this is.
54:19Do you understand what you have to do in each problem?
54:23Okay. So, it's the conservation of
54:25heat that's applied. So, the most tricky part is
54:29phase change, when you've got a phase change,
54:31you've got to remember that the formula mc
54:34∆T--∆Q has one more term,
54:37the one more term is this.
54:46Okay, so next question we ask is, "What's the manner in which
54:52heat manages to flow?" We say you got these calories,
54:56I mean, how does it flow, what's the rate at--what makes
54:58it flow. So, it turns out there are
55:01three popular ways of heat transfer;
55:03one is called radiation.
55:10Radiation is when the heat energy leaves some hot body and
55:15comes to you without the benefit of any medium,
55:19like heat from the Sun. So, that's really
55:22electromagnetic radiation that comes from hot,
55:26glowing objects, and directly comes to you.
55:29Electromagnetic radiation doesn't need air,
55:32doesn't need anything. In fact, if it needed air,
55:34we would not get any heat from the Sun because there is no
55:37medium between the Earth and the Sun.
55:39Most of it is just vacuum. So, if you took one of these
55:43space heaters, you know, with glowing red
55:45coils, and you feel warm. If I start pumping the air out
55:50of that room, of course, you will be dying
55:53very rapidly, but your last thoughts will be,
55:57"I am still warm" [laughter] because the radiation will keep
56:00coming to you. Okay?
56:02That's radiation heat. There are lots of laws for
56:06radiation; I don't want to give them to
56:07you because there are formulas you memorize,
56:09and you don't understand too much of the physics right now.
56:12Other than to say it's electromagnetic radiation,
56:14whatever that means--we haven't gotten to that yet.
56:17That's what comes from there to here and can come in vacuum.
56:20It doesn't need a medium, is the key.
56:22Then, the second way of heat transfer is called convection.
56:29So, convection is explained by the following example.
56:33You've got water; you put it on a hot plate.
56:35Then, in the lower part of it, the water gets hot.
56:40When it gets hot it expands, and when it expands the density
56:44goes down; therefore, by loss of buoyancy
56:47it will start raising up. Remember, a chunk of water
56:52belongs in water. A chunk of something else with
56:55lower density will float to the top.
56:56But the point is, water doesn't have a fixed
56:59density. If you heat it up,
57:00the density goes down, so the water guys downstairs
57:03have lower density-- they're like a piece of cork,
57:05they will rise to the top. When they rise to the top,
57:08the cold water with the higher density will fall down.
57:12So, you set up a current. Hot rises to the top and cold
57:16comes down. And this also happens in the
57:19atmosphere. On a hot day,
57:20the air next to the ground gets really heated up and it rises,
57:23and the cold air comes down and you set up these thermal
57:26currents. So, here you're trying to
57:28equalize the temperature between a region which is cold and a
57:32region which is hot by the actual motion of some material.
57:35In radiation, you don't have the medium
57:39transferring heat because a medium is not even present in
57:43radiation. In convection,
57:46the medium actually moves. The hot guys physically move to
57:49the other place and the cold guys come here,
57:51and by that process, the heat is transferred.
57:53The heat transfer I want to focus on a little more
57:57quantitatively, is conduction.
58:07So, heat conduction is something you've all
58:09experienced. I mean, if you have a skillet,
58:12why does it have a wooden handle?
58:13Simple reason; if you had a steel handle,
58:16you put it on a hot stove and you put your hand here,
58:21the fact that your body is at whatever, 98 degrees,
58:24and this one is God knows, 200 degrees,
58:27you're going to have heat flow from here to here.
58:30So, we want to understand what's the rate at which heat
58:33flows from the hot end to the cold end.
58:35So, you can imagine a rod of some cross-section A,
58:38one end of the rod is in some reservoir at some temperature
58:42T_1, other end is at temperature
58:45T_2. By the way, I'm now introducing
58:48a new term called reservoir. Reservoir is another body like
58:51you and me, except it's not like you and me.
58:54It's enormous. It is so big that its
58:57temperature cannot be changed. You can sit on it,
59:00you will fry and you'll evaporate, but its temperature
59:03will not change. No body is really a reservoir.
59:06If you drop an ice cube in the Atlantic, you'll lower the
59:09temperature of the Atlantic but by a negligible amount.
59:12So, take the limit of Atlantic goes to infinity,
59:15then you have a reservoir. Reservoirs have one label,
59:18namely, what's our temperature. So, something big enough can
59:21be--this room is like a reservoir.
59:23You put a cup of coffee here, you say it will come to room
59:26temperature. Actually, the room temperature
59:29meets the coffee, not halfway but slightly up.
59:31But the room is large enough so that we can attribute to the
59:35room temperature quite independent of bodies that go in
59:38and out of it. So, this is connected on the
59:41left to an enormous tank of maybe a water-ice mixture at
59:45zero degrees; this is a water-steam mixture
59:48at maybe 100 degrees. You put a rod there.
59:50We know heat is going to flow from the hot body,
59:53from the hot end to the cold end.
59:54And we want to write a formula for how much heat flows per
59:59second. Again, I'm going to write these
1:00:02formulas over and over again. So, you've got to ask yourself,
1:00:06what will it depend on? What are the properties it will
1:00:09depend on, in general, independent of what the rod is
1:00:12made of? Can you think of one?
1:00:15Yes? Student: [inaudible]
1:00:17Professor Ramamurti Shankar: You said the
1:00:19cross-section. Now, why do we say--what reason
1:00:22can you give for cross-- Student: If you just want
1:00:25to consider a rod with twice the cross-section area,
1:00:28you're going to come up with [inaudible]
1:00:31two rods and twice [inaudible] Professor Ramamurti
1:00:33Shankar: Yes, okay let me look at this
1:00:36argument. You take one rod,
1:00:37and for convenience let's just take it to be a rectangular rod.
1:00:40Take another rod, rectangular rod;
1:00:43they will both transfer the same amount of heat for a given
1:00:45amount of time. Just glue them together and say
1:00:48here is my new rod. We know it's going to transmit
1:00:51twice the amount of heat. So, it's going to be
1:00:54proportional to the area. And why is the heat flowing?
1:00:58It's flowing because of a temperature difference.
1:01:00So, that's always there; that's the underlying force for
1:01:04heat transfer. That's the dynamics in
1:01:06thermodynamics; that's what makes the heat flow.
1:01:08But then, we find as an empirical fact,
1:01:12that if these two reservoirs are separated by that distance,
1:01:19then the heat flow is a lot less than when they are closer.
1:01:22It seems to depend on how much temperature difference is packed
1:01:26in spatially. So, you want to divide by a
1:01:30∆x is not infinitesimal;
1:01:33it's the length of the rod separating the hot and cold
1:01:36ends. In other words,
1:01:37if you dilute the temperature difference over one mile,
1:01:40the heat flow will be correspondingly reduced,
1:01:43whereas if there's huge temperature difference between a
1:01:45very small spatial separation, there will be very robust flow
1:01:48of heat; that's what we're saying.
1:01:50These happen to be true, you realize,
1:01:52independent of what material I'm talking about.
1:01:55When I said one rod plus one rod is two rods,
1:01:57it doesn't matter what it's made of.
1:02:00Again, having put all these factors which you can argue on
1:02:03general grounds, you have to now ask,
1:02:05"What happens when this is a copper rod versus silver rod
1:02:09versus wooden rod?" So, you've got to put one more
1:02:11number which is kappa [κ]
1:02:13here; not k you guys,
1:02:15it's κ, and it's called the thermal
1:02:18conductivity of that material.
1:02:29Sometimes you put a minus sign; minus sign just means it flows
1:02:33from hot to cold. I don't care whether you put
1:02:35the plus sign or don't put the minus sign;
1:02:38anybody knows that the heat is going to flow from hot to cold.
1:02:42So, just remember that direction of flow,
1:02:44and that's all I care about, this sign here.
1:02:47This κ is the property of the material.
1:02:49Once again, let me tell you--You can say,
1:02:52"Well, I have two reservoirs, hot and cold.
1:02:56I connected them with two different rods.
1:02:58This rod carried twice the amount of heat per second as the
1:03:03other rod. Is it necessarily a better
1:03:05conductor?" No.
1:03:06Maybe it had 10,000 times the cross-section.
1:03:08So, what you want to do is to make the playing field level,
1:03:12and compare rods of the same cross-section,
1:03:14same temperature difference, same length,
1:03:17then ask who conducts more heat.
1:03:19That depends on the material and that's the thing you pulled
1:03:22out specific to the material. That is the property of wood or
1:03:26copper of steel; that's the heat conductivity.
1:03:31Okay. Now, the final topic is just
1:03:35going to be more hand-waving now.
1:03:38I don't want to get into too many details.
1:03:39It really has to do with what is heat.
1:03:45In the old days, people just said that it was a
1:03:47fluid, and they postulated the conservation law for the fluid.
1:03:50You can postulate what you want, you've got to make sure it
1:03:53works, and it seems to work, in the sense that all the
1:03:55∆Qs in any reaction add up to zero.
1:03:58But then, people are getting hints that maybe this thing that
1:04:02we call heat is not entirely independent of other things we
1:04:07have learned. So, where do you get the clue?
1:04:10One clue is, long back when we studied
1:04:12mechanics, we talked about two cars that come and collide;
1:04:16they slam into one big lump. Now, you've got no kinetic
1:04:20energy, no potential energy. Potential energy is always
1:04:23zero, they're moving on the same height, kinetic energy was ½
1:04:26mv^(2) for this, ½ mv^(2) for that;
1:04:29at the end there's nothing. No kinetic, no potential,
1:04:32we just gave up and said, "Look, conservation of energy
1:04:35does not apply to this problem." We just say it's inelastic.
1:04:41On the other hand, we find whenever that happens,
1:04:45we find the bodies become hot. Here's another thing you can
1:04:50do, you can take a cannonball, drop it from a big tower.
1:04:53This is how some people in the French army, I think,
1:04:55first detected this feature; you dropped cannonballs from a
1:04:58big height. When they hit the sand,
1:04:59they start heating up. Or you drill a hole in a
1:05:02cannon, that's what Count-somebody did,
1:05:04and he also noticed that you need to constantly pour water to
1:05:08keep the drill bit from heating up.
1:05:11You'll find very often, mechanical energy is lost and
1:05:15things heat up. So, you get a suspicion
1:05:18whatever the underlying mechanism, maybe there's a rule
1:05:21that says if you lose so much mechanical energy that you
1:05:24cannot account for, then it translates into a fixed
1:05:27number of calories. If that is the case,
1:05:30then we at least get a dictionary on--between calories
1:05:35and joules. So, joules is energy you can
1:05:39see, calories is energy you cannot see.
1:05:42That was going to be the premise.
1:05:43But first, you've got to prove that every time you lose some
1:05:46number of joules, you get a fixed amount of
1:05:48calories. And that experiment is due to
1:05:51Joule. Here is the Joule experiment.
1:05:56It's very, very simple and tells you the whole story.
1:05:58You have a little container in which there is a paddle.
1:06:04This is a shaft with a pulley, and there is a weight here.
1:06:10So look, try to imagine this guys.
1:06:13You got rope wrapped around the top pulley, and when you let
1:06:16this weight go down, it's going to go down like
1:06:19this; it's going to spin the shaft.
1:06:21And put some water here, and I have some fins that are
1:06:24sticking out, so they churn up the water.
1:06:27So, it's like this thing, the egg-beater,
1:06:31right? In fact, I tried to do the
1:06:32experiment with an egg-beater this summer to a bunch of high
1:06:35school kids, and I got thoroughly humiliated
1:06:38because nothing happened as planned.
1:06:41But the idea is the same. You agitate the water in some
1:06:44fashion. But this guy did it in a
1:06:46particularly simple way. My egg beating was not good
1:06:49enough; you will see maybe in a while
1:06:50why that's not good. What he did was to put these
1:06:53paddles, let the weight go down from there to here.
1:06:56Now, we can keep track of how much mechanical energy is lost,
1:07:02right? Because if this mass was at
1:07:04rest, and a drop to height mg drop to height
1:07:07h, it's supposed to have mgh kinetic energy.
1:07:10Let's say it's got some kinetic energy, which is not equal to
1:07:14mgh. So, mgh minus kinetic
1:07:17energy is missing. So, some number of joules are
1:07:22gone. So, the water gets hot.
1:07:25When the water gets hot, you can immediately ask how
1:07:28many calories were supplied to the water.
1:07:31Because that water heats up the same way whether or not you put
1:07:35it on a hotplate, or whether or not you churn it.
1:07:38It doesn't seem to depend on how it got hot.
1:07:40This has the same effect. This water is hot in every real
1:07:43sense. So, you must have put some
1:07:45calories. You can find out how many
1:07:47calories you put in by looking at the mass of the water;
1:07:49specific heat of the water is 1; looking at the increase in
1:07:53temperature. So, some joules are missing,
1:07:56some calories have been pumped into the water.
1:08:01Then you ask, "Is there a proportionality
1:08:03between joules and calories?" And you find that it is.
1:08:06And that happens to be 4.2 joules per calorie.
1:08:16In other words, if you can expend 4.2 joules of
1:08:20mechanical energy, you got yourself one calorie to
1:08:25be used for whatever heating purposes.
1:08:28So, in the example of the colliding cars,
1:08:30this had some energy, that had some energy,
1:08:32all measured in joules; they slammed together,
1:08:34they come to rest. That means you can take those
1:08:37many joules, divide it by 4.2 and get some number of calories.
1:08:41Imagine the whole car is made out of copper.
1:08:44Then those calories will produce an increase in
1:08:46temperature, right, equal to ∆Q is mc
1:08:50∆T. That will be the rise in
1:08:52temperature of the car. In practice,
1:08:54there will be other losses, because you heard the sound,
1:08:56well, that's some energy gone; you won't get it back.
1:08:59Maybe some sparks are flying, that's light energy;
1:09:01that's gone. You subtract all that out,
1:09:03you find that in the end, the calories explain the
1:09:07missing joules. So, that made people think that
1:09:11this is just another form of energy.
1:09:14Because if you add this to your energy balance,
1:09:17there is no reason to go on apologizing for the Law of
1:09:21Conservation of Energy. Law of Conservation of Energy
1:09:25is not in fact violated, even at the inelastic
1:09:27collision, if you include heat as a form of energy.
1:09:30And the conversion factor is 4.2 joules per calorie.
1:09:34But the question is, "What right do you have to call
1:09:38it energy?" Energy, we think--primarily,
1:09:41when you say somebody's energetic, you mean that
1:09:43someone's running around mindlessly, back and forth.
1:09:46Energy is associated with motion.
1:09:48These two cars were moving, and we have every right to say
1:09:51they have energy. How about potential energy?
1:09:54Well, if the car starts climbing up a hill and slows
1:09:57down, we think it's got potential.
1:09:59If you let it go, it'll come back and give you
1:10:01the kinetic energy. So, most people's idea of
1:10:03energy is just kinetic energy. That is lost.
1:10:06And yet, you get calories in return, so you ask yourself,
1:10:10"What can it be?" Well, the correct answer to
1:10:13that came only when we understood that everything is
1:10:16made up of atoms. Once you grant that everything
1:10:19is made up of atoms, then it turns out that the
1:10:22kinetic energy of atoms is what we call heat.
1:10:26But you've got to be very careful.
1:10:28Take a tank full of gas. I throw it at you.
1:10:33That whole tank is moving, that's not what I call heat.
1:10:37Okay? That motion you can see.
1:10:40I'm talking about a tank of gas that doesn't seem to be going
1:10:43anywhere; yet, it got motional energy
1:10:45because the little guys are going back and forth.
1:10:49So, what we will find is what I'm going to show you next time,
1:10:53is that if you kept track of the kinetic energy of every
1:10:57single molecule in this car, every single molecule in that
1:11:01car, before and after, and you added them up,
1:11:04you would get exactly the same number.
1:11:06The only difference will be originally the car has got
1:11:11global common velocity; macroscopic velocity you can
1:11:15see. On top of it,
1:11:16it's got random motion of the molecules that make up the car.
1:11:20So does the other car. When they slam together,
1:11:22the macroscopic motion is completely gone,
1:11:24and all the motion is thermal motion.
1:11:27But it's still kinetic energy, and that's what we will see the
1:11:30next time.

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