21. Thermodynamics
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Fundamentals of Physics I21 / 24
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CogenerationCartridge (firearms)Control rodBird vocalizationMeasurementGlassTemperatureThermodynamic equilibriumBending (metalworking)Club (weapon)Orbital periodOncotic pressureMeasuring cupAngeregter ZustandHose couplingHeatEnergy levelTypesettingLiquidAtomismWater vaporWeekShip classMassSpare partTheodoliteMechanicWeightPistonGasYearFocus (optics)FahrgeschwindigkeitHauptsatz der Thermodynamik 2HeißwasserspeicherKey (engineering)YardstickMonitorüberwachungAtmospheric pressureLeadMetreStandard cellCapital shipThermometerFoot (unit)FACTS (newspaper)WednesdayPerturbation theoryShake (shingle)Lecture/Conference
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RankingWater vaporTemperatureThermometerStellar atmosphereCogenerationExpansionsturbineAlcohol proofAtmospheric pressureEnergy levelSpare partFACTS (newspaper)MeasurementGas thermometerWeightMelting pointArbeitszylinderLiquidGasContainment buildingFord MercuryIceBubble chamberControl rodSummer (George Winston album)ThermometerSEEDDrehmasseSynthesizerMonitorüberwachungMountainDayBook designVideoTiefgangWindRailroad carHauptsatz der Thermodynamik 2Fuel tankLecture/Conference
17:52
TemperatureWater vaporKitchen stoveMonitorüberwachungGasPlatingUniverseAlcohol proofKickstandAtmospheric pressureVideoNanotechnologyIcePump (skateboarding)JointCollar (clothing)MeasurementCommand-line interfaceStandard cellMetreRoll formingThermometerDrehmasseThermometerBoatYearAmmeterLeistungsanpassungKette <Zugmittel>Mitsubishi A6M ZeroCooling towerDayPlanetPartial pressureYardstickCogenerationHydrogen atomArbeitszylinderFord MercuryGas thermometerWeightBird vocalizationNeon lampLecture/Conference
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Alcohol proofMaterialCogenerationMassFoot (unit)Standard cellKopfstützeDampfbügeleisenWoodHeatTemperatureRulerContainment buildingWater vaporKitchen stoveProzessleittechnikField-effect transistorKilogramMeasurementLiquidControl rodApparent magnitudeLinear motorYardstickEffects unitIceCylinder blockFACTS (newspaper)Absolute zeroBird vocalizationBook designCocktail party effectGameExpansionsturbineCorporal (liturgy)H-alphaCommitteeKette <Zugmittel>KelvinBoatTiefdruckgebietCabin (truck)MountainPorcelainScrew capVideoChemical substanceWeightAngeregter ZustandLecture/Conference
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Alcohol proofSeparation processRefrigeratorMitsubishi A6M ZeroRulerIceSource (album)SunriseOrbital periodWater vaporMassMaterialInitiator <Steuerungstechnik>LeadMeasurementField-effect transistorThermometerAbsolute zeroCalorimeterBig BangMelting pointTemperatureKopfstützeUniverseHeatYearAtomismPower (physics)Roman calendarFACTS (newspaper)FahrgeschwindigkeitLastRoll formingContainment buildingProzessleittechnikRocketKelvinKilogramSpeed of lightAtmospheric pressureHull (watercraft)CogenerationMechanicNightTissue paperCardboard (paper product)Cartridge (firearms)Thermodynamic equilibriumBlast furnaceLimiterLeistungsanpassungMonitorüberwachungTurningCaliberKette <Zugmittel>Vacuum tubeFoot (unit)WatchPump (skateboarding)Sport utility vehicleWoodturningGround (electricity)AC power plugs and socketsBasis (linear algebra)Car seatH-alphaAstronomisches FensterNarrow gauge railwayPotentiometerSteckverbinderBahnelementElectronic mediaLecture/Conference
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IceCylinder headKilogramOrbital periodApparent magnitudeData conversionMeltingMassNegationHose couplingCommand-line interfaceMinuteBending (metalworking)Hot workingChemical substanceAlcohol proofPotentiometerLiquidRoll formingConvertibleSolidFord MercuryWater vaporTemperatureHeatRegentropfenFACTS (newspaper)Cartridge (firearms)Key (engineering)Pair productionDuty cycleYearSunriseDayWashiKopfstützeSingle (music)HochbahnTurningLecture/Conference
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GaleMeasurementIceControl rodMeasuring cupField-effect transistorLimiterHeatTemperatureWater vaporSilveringMaterialForceAerodynamicsKilogramHochfrequenzübertragungHeat exchangerKopfstützeMassCross section (physics)SunlightEnergiesparmodusSpare partGround stationMagnetspuleVakuumphysikActive laser mediumDensityPlatingCork (material)CogenerationStellar atmosphereWoodGround (electricity)SunriseSteelFACTS (newspaper)Atmosphere of EarthConvectionWhiteBird vocalizationBinocularsKardierenWeightProzessleittechnikAlcohol proofKitchen stovePerturbation theoryConvertiblePackaging and labelingTextileCartridge (firearms)Volumetric flow rateDayElectronic mediaCosmic distance ladderGentlemanSeparation processRulerEffects unitRegentropfenLecture/Conference
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Antenna diversityMembrane potentialReaction (physics)MechanicSteelRailroad carWoodCogenerationField-effect transistorCannonTransmission towerRegentropfenBallpoint penHeatJoule heatingTemperatureRulerLocherContainment buildingWater vaporWeightTagesschachtPaddle steamerDrill bitCross section (physics)Radiant energyRopePercussion malletFACTS (newspaper)Roll formingSunriseSummer (George Winston album)CaliperFinger protocolMaterialDirect currentVolumetric flow rateData conversionContinuous trackMassFullingEffects unitFahrgeschwindigkeitControl rodGlobal warmingEnergy levelGasPlane (tool)AtomismBird vocalizationLastMetreFoot (unit)JuneCartridge (firearms)Tin canWoodturningWeekGemstoneEnergiesparmodusElectronic mediaRelative datingTire balanceStock (firearms)Disc brakeCollisionKardierenSoundElectrical conductorDayLecture/Conference
Transcript: English(auto-generated)
00:01
All right, 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 you enough time so you can plan your moves. Then I will probably give you
00:23
one last problem set with two or three problems on whatever I do near the end. We'll have to play it by year. Okay, so this is another new topic on thermodynamics, a fresh beginning for those who want a fresh beginning.
00:43
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, the heat transfer, things like that. So, we're going to start with the intuitive definition of
01:03
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.
01:22
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.
01:41
So, I'll tell you what equilibrium is with a concrete example. If you take a cup of hot water, then 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
02:01
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've not defined what temperature is precisely, but we can talk about whether whatever it is has changed or not changed.
02:24
So, it'll settle down to some temperature. It'll 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
02:42
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
03:03
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 in the top, the cold stuff is in the bottom. That'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.
03:22
But if you wait long enough until the two parts have gotten to know each other, they will turn into some un-drinkable mess. But the nice thing is it'll 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
03:41
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, a certain velocity.
04:02
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,
04:21
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, there's some gas inside, you put some weights, and everything is in equilibrium. We say this is in equilibrium because the macroscopic things, things you can see with
04:40
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
05:01
see anything with naked eyes that looks like anything is happening. In between, you will see the piston 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
05:25
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 it is.
05:41
One of the laws of thermodynamics is called the zeroth law. The zeroth law, because they wrote down the first law, then they went back and had an idea which is 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
06:02
temperature. Now, I see disbelief in the audience. 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,
06:20
and it reads the same number, then I may conclude these two entities, which never met each other directly, are also at the same temperature. That's not, that seems pretty obvious to you, but that is, 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,
06:42
but met a third system. 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.
07:02
All we have right now is an ocean 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 particular that seem to depend on temperature. One thing that seems to depend on temperature is the following.
07:23
You take this meter stick in the National Bureau of 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 if the room was hotter than the glass case, this rod then expands to a new length.
07:41
So, one rod is outside the case, one rod is inside the glass 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.
08:04
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, and if it's not gross, we can say the temperature is gone up by some amount. So, there we need units for temperature. That's completely arbitrary.
08:23
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.
08:41
Because if this stick is as long as that one and as long as that one, then those two are equal in length. For temperature, you're similarly going to use this rod and say, this rod has a certain length when it's kept on top of this bucket of some fluid, and at the same length when I keep it on that bucket, then the two buckets are at the same temperature. So, we can use markings on
09:04
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,
09:21
you have to leave some room at the top so that 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 on hot
09:40
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,
10:01
a very thin tube evacuated at the top, and the fluid then is here. So, what's clever about this thermometer is that if this expands by 1%, your eyes should be good enough to see 1% increase in height. If this fluid expands by 1% in volume, that 1% in volume
10:23
and it climbs up this narrow tube can climb to quite a bit, 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 expanded fluid climb up this extremely narrow tube. In fact, the tube is so narrow
10:42
you cannot probably even see it well, which is why they have a little prism that magnifies the mercury or alcohol in a 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 0, that can be 5,
11:00
that can be 19. You just got to make sure 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
11:21
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
11:41
of water, so that the freezing point of water, the melting point of ice or the 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,
12:01
we will postulate to be 0°C. 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've figured out when parts of the world where you had ice, this is a very good
12:22
definition. You get ice, you've got 0°. 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
12:41
be called 100°, 100°C. Then, you take this column between 0 and 100, and you divide it into 100 equal parts. And that is postulated to be the temperature anywhere between
13:03
0 and 100. If you've gone 79% of the way to the top, from here to here, the temperature is 79°. That's how the degrees are introduced, and that's the centigrade scale. And you guys know there are different scales. You can have the Fahrenheit scale, you can have any other
13:21
scale in which what you want to call the freezing point is different. Somebody thinks it's 0, somebody thinks it's 32, and you can again call this something else. And you can divide this by 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
13:42
region between them into some number of equal steps. If it's 100 equal steps, you say it's a centigrade scale, provided the lowest one is called 0. 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
14:02
to be very reliable, because if you boil water on an 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 try to
14:21
cook something, cook some rice, or 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,
14:42
because things don't boil, they don't seem to boil at a certain predictable and fixed temperature. This is a very deep argument. I've never appreciated it fully when I was learning the subject. I said, it's all cyclic definition, because you may not know that the temperature is changing,
15:01
because this is thermometer by postulate, it's going to be the temperature by definition. How can it be wrong? What's wrong is that it's not, we 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, doesn't cook in the mountains, so we know the boiling point is to blame. The rice is the rice.
15:23
That's how we know that that's not a good measure. So, nowadays, people are 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 in different altitudes, you could go to sea level. And that's a good enough definition.
15:41
Sea level is pretty much constant all over the world, and you can say the pressure at 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,
16:01
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 75 if my fluid has climbed three-fourths of the way to the top. At that point,
16:22
yours may not have climbed three-fourths of the way. In other words, you've got two things, two graphs, which at zero and 100 agree, one graph may be like this, one may be like that. So that when I think it is 75, you may think it's 72, or 100 people agree because we have cooked it up that way. In other words, it's not true that all
16:43
liquids expand at the same rate. So, you will have to then pick one liquid and say, okay, we swear by that liquid, and when that liquid's gone halfway, we will say it's 50 degrees. So, you'll have to pick a liquid, you'll have to have an international convention, you know. There's an alcohol lobby and there's no alcohol lobby,
17:03
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 gases, which have some very, very nice properties. And this is the gas thermometer that I'm going to tell you now.
17:22
So, here is how you build a gas thermometer. You take some gas in a container, a typical container for me, and 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.
17:45
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
18:02
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.
18:22
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. If you connect the dots, you find the product PV is linear in this temperature
18:43
variable. And this is 0 degrees, and this is 100 degrees. Now, here is the beauty of this. This is the gas thermometer. If you take a different gas and you put a different amount
19:02
of a different gas in a different cylinder, you will get some other graph. It may look like this. For you, that is 0 and that's 100. But the most important thing is that's also a straight line. If that's also a straight line, it has the following
19:22
implication you guys can prove at your own leisure, which is that if I think that my gas has climbed 56% of the way of this height to the top, so the temperature is 56 degrees, I ask, what's your gas done? You'll find it's also climbed
19:40
56% of the way. It's a property of straight lines. You can show that if you took two straight lines, whatever be their slope, if they agree, if this is 0, this is 100, it's 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 would all have climbed to
20:01
the halfway point from the 0 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 that is one requirement. This gas has to be very dilute. The more dilute it is,
20:26
the better it comes up. 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 gases have the property that if you calibrate them at 0 and 100,
20:42
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, you call that 0. Whenever you're on boiling water, you call it 100. You find the joint that's
21:03
connected by a straight line. Then every point in between, if divided equally, leads to equal increase in the product P times V. So, 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.
21:20
So, everybody can use the gas thermometer. That's why we prefer the gas thermometer. This is the interesting issue about measurement or definition of temperature. 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
21:40
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
22:01
doesn't expand, but we don't know what that means, because that meter stick is the standard. By definition, it's right. But then you will soon find out that's not really that simple, because there are good and bad meter sticks. For example, the same meter stick at one time 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,
22:21
and people arrive on the gas thermometer this way. 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 nowadays we can go
22:42
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.
23:03
Because if you took another gas, well, I'm going to do a little cheating here, that also extrapolates to that same temperature. So, all gases, 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
23:22
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
23:42
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 it can go to zero.
24:01
That's it. You 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 to this temperature.
24:20
So, people decided, you know what, calling this the zero is kind of artificial. That's based on human obsession with water. But if you think the laws of science describe the whole universe, what about planets where there's no water, right? You cannot describe,
24:40
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, it's the stuff you drink. Well, you don't know what these apes are drinking. Maybe they're drinking methane or hydrogen gas, or maybe they're drinking liquid hydrogen.
25:00
We don't know. On the other hand, you say, take any vapor and wait till the product of the pressure and volume go to zero. Let's call that the 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
25:21
thermometers. So, zero temperature is going to be set from here. Once they did that, they called that the 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.
25:42
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 will be called 273.16. So, this point is called the triple point of water. What's the triple point of water?
26:02
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
26:24
three options. Ice floating on water is when water is not decided whether to be ice or to be water. That's the coexistence point of two things. And when the water starts boiling in your stove, that's when water and steam coexist. But I'm saying there are certain conditions of pressure and temperature and volume so that water,
26:42
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
27:02
zero to a more natural point where all grabs meet. Then, you define the 1 degree Kelvin to be so that 273.16 of that Kelvin brings you to the triple point of water. So, if you found that 273.16 of water is not a fixed number, you go to the mountains,
27:20
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, and that will be called 273.16 Kelvin. Now, there is a rule, apparently.
27:40
You can say degree centigrade. You're not supposed to say degree centigrade, you're supposed to say degree Kelvin. That's a big deal made in a lot of books. I keep forgetting and say, 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
28:02
language is to have no ambiguities. But when I say degree Kelvin, and I find that you guys don't get confused, I don't think that's a big deal. 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 the standards. I just don't feel any affiliation to this particular, completely artificial and empty convention.
28:24
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
28:42
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. Now, I'm going to talk about heat. So, heat is denoted by the symbol Q. And you've got to ask
29:01
yourself, what are we talking about when we talk about heat? Again, let's use your intuitive sense of what heat is. You 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
29:21
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
29:42
the caloric fluid. They imagined that it 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 that process heats up the
30:01
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
30:25
mass of water times the change in temperature. And that's going to be calories. In other words, if I had a container with 10 grams of water, and the temperature went up,
30:42
I'm sorry, this is mass of water in grams. If you have one gram of water and you did something to it and the temperature went up by 7 degrees, you have, by definition, pumped in 7 calories.
31:01
If this was a kilogram of water, this would be called a kilocalorie. So, sometimes I use grams and calories, sometimes I use kilograms and kilocalories, but the definitions are consistent. So, you have a kilogram of water, and 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
31:24
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.
31:41
Pick your own favorite material, gold. Then the amount of heat, I think we can all appreciate, must be proportional 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,
32:02
it is clear that whatever this calorie 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 an increase in temperature. But this is true for any substance, whether you're eating copper or wood or gold.
32:22
No matter what you're heating, it is true the heat needed is proportional to the mass and to the temperature. So, what is it that distinguishes one material from another? You put a number here, and that number is called the specific heat. The specific heat is a property of that material. You've got to understand,
32:46
certain formulas will depend on certain parameters in a generic way, and some things that depend on the actual material. In fact, that'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,
33:02
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? Do you think of what it may be proportional to? Yes? It depends on the original
33:22
length of the rod. Now, why is that? Why do we think it's got to be proportional to the length of the rod? Yeah, it's based on what it had before. Yes?
33:40
That's correct. I think one way to say that is, take a meter stick, expand for some amount, put another meter stick next to it, that expands by the same amount by definition of identical things. Therefore, the two-meter stick will expand by twice as much. So, we put the length of the rod here, and we expand that. So, no matter what you're heating, a block of wood,
34:01
a 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.
34:21
These are true no matter what you are heating. So, these specific numbers, these coefficients, these α's that come in, are going to come in all the time, and sometimes you should get used to that. 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,
34:41
will 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
35:00
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, the second one could have been a foot long. So, you have to take out
35:20
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. It 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 a property of the
35:43
material. And by definition, C equal to one calorie per gram, or one kilocalorie per kilogram for water. Once you've got, so remember, one calorie per gram for water
36:03
is a 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, put some water in it.
36:22
Let's assume the container has zero mass, so I don't have to worry about it. It's an approximation. If you're worried about that, 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 at some initial temperature T1, and I have some new material,
36:43
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'll be an example where initially the lead is in
37:02
equilibrium, maybe on a furnace, a temperature T2, water is in equilibrium, maybe in the room, a temperature T1. Then, I put the pellets into the water. There'll be a period when the temperature is not defined. Then soon, they will settle down to some common temperature called T. We will now postulate.
37:23
This is a postulate, or a log. 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 the new law. Now, you can make up all the
37:42
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 for the thermometer and measure it.
38:03
Then you say the mass of the water, that specific heat of water, which is 1, times ΔT, which is the final temperature minus the initial temperature, ditto for the lead pellet, mass of the lead,
38:21
that has got a symbol Pb, times specific heat, which I don't know, times the change in temperature, which is T final minus T two, is equal to zero. The sum of all the mcΔTs is zero. This is the gain of heat of
38:45
the water. This, if you work it out, will be a negative number, because you can see T final 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, specific heat of water is one by definition.
39:03
T final and T one 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.
39:21
Yes? Yes? That's correct. So, the real point is,
39:41
if everything expanded linearly, we won't have the disagreement between different thermometers. So, it turns out, to an excellent approximation, the change in 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 heat of materials is
40:01
also not a constant. We say specific heat of water is one. It turns out, at a certain temperature range, it'll be one. At a different range, in fact, it's not quite one. 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 unearth my fallacies? Specific heat of materials is
40:20
not a constant. It's a big industry calculating the specific heat of materials starting from atoms and quantum mechanics. So, none of the things treated as constants are ever constants, including those α's and β's. 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
40:43
dimensional considerations and say, if it's not L, if you want to put an L squared as a correction to the formula, to match the units, the L squared has to be divided by another length to keep the units. What other lengths 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
41:02
calculate corrections. So, all these laws are, in fact, very tentative and approximated. 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, so it's going back and forth. This is way back when people did not even know about
41:20
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, you found a specific heat for gas, right? You solved 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 you say adds up to zero, adds up to zero to a very good approximation during that epoch.
41:42
Another epoch when people do more and more accurate experiments, everything gets shot down. In fact, specific heats of all materials seem to go to zero when you approach absolute temperature. You have to understand the laws of quantum physics to know exactly why that happens. So, this is a period when
42:02
people are probing temperature ranges, which are around room temperature, or boiling and freezing point 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 3 degrees.
42:21
It's called the black body 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 100th of a Kelvin. And we heard a talk last year, physics said 1 billionth of a Kelvin.
42:41
You want to cool them and cool them and cool down. But 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?
43:02
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 like pressure simply vanish, and they cannot have negative values. You will see more about this when you understand heat in greater depth. Anyway, right now, ΔQ equal to zero is the rule you use.
43:21
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 can go below zero. Your refrigerator is several degrees below, several tenths below zero. So, let's take ice and let me measure.
43:45
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.
44:01
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 mcΔt.
44:20
So, 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
44:44
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
45:01
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
45:23
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.
45:40
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. Then, 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.
46:00
Then, you can ask, okay, what penalty do I have to pay? That's called the 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 some amount of stuff at the latent heat of melting.
46:26
That's how much Q you need to melt that amount of stuff, and the L varies from substance to substance. For water, it's 80 calories per gram. If you want to melt mercury from solid mercury to liquid mercury, it'll have a different number.
46:41
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.
47:01
In a solid, everybody has its place. You can shake it on 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
47:20
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 till everybody vaporizes, and then you get steam. Then you can have super-heated steam, which is even higher than 100 degrees. So, that's the latent heat of vaporization.
47:43
I really don't even 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, you can, if I tell you I took some ice at minus 30 and I dumped in 5,000 calories,
48:01
where will it end up? You've got to first spend a few calories going from here to here. If you've 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've got 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.
48:22
If not, keep going here and there and there, and you may end up there if you've 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 heat, you'll have to do the mcΔt for that, m times latent heat for this,
48:40
mcΔt 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. The 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.
49:02
So, this is 0. The ice is at, say, minus 40. Water is at plus 80. In fact, let me make that also at plus 40. I bring them together and I ask you what will happen. Now, this is a subtle problem. If you had two,
49:23
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 this stuff you have.
49:42
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
50:01
going to be the heat given to this. Heat given to this is the mcΔt 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,
50:22
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. You call that unknown temperature T. This is the T1, this is the T2. Write your equations, except you will have one more term there. That's the heat it takes to melt the ice. You solve for T.
50:41
So, 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 you 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
51:01
that you are on the other side of ice is wrong. Then, you can try something else. You can assume you are down here. If you think you're down here, then you simply heated the ice from here to here. This water you brought down to zero, sucked out mcΔT from that,
51:20
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 got a negative number, then you're okay. That would be a good assumption
51:41
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, that's the third possibility. The third possibility is at the end of the day, you end up here with some
52:01
amount of water and some amount of ice at zero degrees. So, that's the 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?
52:21
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 the question of suppose both of the things I try fail. I take a positive T, assume I'm up here,
52:41
and I assume the ice melted, and I get a negative answer, that 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
53:01
from here to there. It does that by absorbing that mcΔT, mass of the ice and 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
53:20
guy, first you'll bring this to zero, then you still have some more heat you can extract from it. 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 the water. That will be the extra ice, the rest will be whatever water you started with.
53:43
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.
54:03
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.
54:21
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. That one more term is this.
54:43
Okay, so next question we ask is, what's the manner in which heat manages to flow? We say you've got this calories. I mean, how does it flow? What's the rate? What makes it flow? So, it turns out there are three popular ways for heat transfer. One is called radiation.
55:07
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
55:21
electromagnetic radiation that comes from hot, glowing objects and directly comes to you. And 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,
55:42
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, because the radiation will keep coming to you, okay? That's radiation heat. There are lots of laws for
56:03
radiation. I don't want to give them to you because they're 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, it can come in vacuum. It doesn't need a medium is the key. Then, the second way of heat
56:21
transfer is called convection. So, convection is explained by this 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.
56:41
So, the water gets hot. Therefore, by loss of buoyancy, it will start rising 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 a lower density. They're like a piece of
57:01
cork. They will rise to the top. When they rise to the top, the cold water with a higher density will fall down. So, you'll set up a current here, and the 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
57:22
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 the medium is not even present in radiation.
57:41
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 is a little more quantitatively, is conduction. So, heat conduction is
58:05
something you've all experienced. I mean, if you have a skillet, why does it have a wooden handle? The 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,
58:21
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 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,
58:40
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.
59:03
If you drop an ice cube in the Atlantic, you lower the temperature of the Atlantic, but by a negligible amount. So, take the limit of Atlantic, it 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,
59:22
you say it'll come to room temperature. Actually, the room temperature meets the coffee, not halfway but slightly up, right? But the room is large enough so that we can attribute to the room a 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
59:42
100°C. This is a water-steam mixture, maybe at 100°C. 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.
01:00:00
So, you've got to ask yourself, what will it depend on? What are the properties it'll depend on in general, independent of what the rod is made of? Can you think of one? Yes? You said the cross section. Now, why do we say, what reason can you give for a cross?
01:00:31
Okay, let me repeat this argument. You take one rod, 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
01:00:41
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
01:01:00
for heat transfer. That's the dynamics and thermodynamics. That's what makes the heat flow. But then, we find as an example, if the temperature difference is separated by that distance, then the heat flow is a lot less than when you're closer. It seems to depend on how
01:01:20
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 a huge temperature difference between a
01:01:41
very small spatial separation, there'll be a 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,
01:02:02
what happens when there's a copper rod versus a silver rod versus a wooden rod? So, you've got to put one more number, which is κ here, not k, you guys. It's κ and it's called the thermal conductivity of that material.
01:02:25
Sometimes you put a minus sign. A 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
01:02:40
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 had 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
01:03:01
conductor? No, maybe it had 10,000 times the cross section. So, what you want to do is to make the plane feel level and compare rods of the same temperature, 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.
01:03:21
That is a property of wood or copper or steel. That's the heat conductivity. Okay, now, final topic. It's 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.
01:03:41
In the old days, people just said it's 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 were getting hints that maybe this thing that we call heat is not
01:04:02
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 talk about two cars. They 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
01:04:20
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 doesn't apply to this problem. We just say it's inelastic. On the other hand, we find whenever that happens,
01:04:40
we find the bodies become hot. Here's another thing you can do. You can take a cannon ball, drop it from a big tower. This is how some people in the French army, I think, first detected this feature. You drop cannon balls 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
01:05:00
somebody did. And he also noticed that you need to constantly pour water to keep the drill bit from heating up. You find very often, mechanical energy is lost and things heat up. So, you get a suspicion. Whatever the underlying mechanism, maybe there is a rule that says if you lose so much mechanical energy that you cannot account for,
01:05:21
then it translates into a fixed number of calories. If that is the case, then we'll 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
01:05:42
some number of joules, you get a fixed amount of calories. And that experiment is due to joule. Here is a joule experiment. It's very, very simple and tells you the whole story. You have a little container in which there is a paddle.
01:06:01
This is a shaft with a pulley, and there is a weight here. So, look, I tried to imagine this, guys. You've got rope wrapped around the top pulley, and when you let this weight go down, you're going to go down like this. It's going to spin the shaft. I put some water here, and I have some fins that are
01:06:21
sticking out so that it'll churn up the water. So, it's like this thing, the egg beater, right? In fact, I tried to do the experiment with the 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.
01:06:41
But this guy did it in a particular simple way. My egg beater was not good enough. You will see 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
01:07:00
rest, and it dropped a height mg, dropped a 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, but the water gets hot.
01:07:22
When the water gets hot, you can immediately ask, how many calories was supplied to the water? Because that water heats up the same way whether or not you put it on a hot plate, or whether or not you churn it. So, 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
01:07:41
calories. So, you can find out how many calories you put in by looking at the mass of the water, specifically the water is one, 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?
01:08:01
And you find there is. And that happens to be four points to 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
01:08:23
purposes. So, in the example of the colliding cars, this had some energy. There is some energy all measured in joules. They slam together, they come to rest. That means you can take those many joules, divide by 4.2 and get some number of calories. So, you can take the calories. Imagine the whole car is made out of copper. Then, those calories will
01:08:41
produce an increase in temperature, right, equal to ΔQ is mcΔT. 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 sparks are flying, that's light energy that's gone. If you subtract all that out, you find that in the end, the calories explain the missing
01:09:03
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. The law of conservation of energy is not in fact violated,
01:09:22
even in 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
01:09:40
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 this car starts climbing up the hill and slows down, we think it's got potential. 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,
01:10:03
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 kinetic energy of atoms is what we call heat.
01:10:23
But it got me 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,
01:10:41
yet it got emotional 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 will get exactly the same
01:11:01
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
01:11:21
motion is thermal motion. But it's still kinetic energy and that's what we will see next time.
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