Lectures on Heat and Thermodynamics

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Lectures on Heat and Thermodynamics   Physics 152 Michael Fowler, University of Virginia 8/30/08 Contents HEAT...........................................................................................................................................................3 Feeling and seeing temperature changes.....................................................................................3 Classic Dramatic Uses of Temperature-Dependent Effects..........................................................4 The First Thermometer........................................................................................................................5 Newton’s Anonymous Table of Temperatures ...............................................................................7 Fahrenheit’s Excellent Thermometer ...............................................................................................7 Amontons’ Air Thermometer: Pressure Increases Linearly with Temperature .............................7 Thermal Equilibrium and the Zeroth Law of Thermodynamics......................................................8 Measuring Heat Flow: a Unit of Heat ...............................................................................................8 Specific Heats and Calorimetry .......................................................................................................9 A Connection With Atomic Theory ................................................................................................10 Latent Heat .......................................................................................................................................11 THERMAL EXPANSION AND THE GAS LAW.............................................................................................12 Coefficients of Expansion................................................................................................................12 Gas Pressure Increase with Temperature......................................................................................13 Finding a Natural Temperature Scale............................................................................................13 The Gas Law .....................................................................................................................................14 Avogadro’s Hypothesis....................................................................................................................15 EARLY ATTEMPTS TO UNDERSTAND THE NATURE OF HEAT .....................................................................16 When Heat Flows, What, Exactly is Flowing? ................................................................................16 Lavoisier’s Caloric Fluid Theory .......................................................................................................16 The Industrial Revolution and the Water Wheel ...........................................................................17 Measuring Power by Lifting.............................................................................................................18 Carnot’s Caloric Water Wheel .......................................................................................................18 How Efficient are these Machines? ...............................................................................................19 Count Rumford.................................................................................................................................20 Rumford’s Theory of Heat................................................................................................................22 THE DISCOVERY OF ENERGY CONSERVATION: MAYER AND JOULE ....................................................24 Robert Mayer and the Color of Blood...........................................................................................24 James Joule......................................................................................................................................26 But Who Was First: Mayer or Joule? ...............................................................................................27 The Emergence of Energy Conservation ......................................................................................27 KINETIC THEORY OF GASES: A BRIEF REVIEW.........................................................................................28 2 Bernoulli's Picture ..............................................................................................................................28 The Link between Molecular Energy and Pressure ......................................................................29 Maxwell finds the Velocity Distribution ..........................................................................................30 Velocity Space .................................................................................................................................31 Maxwell’s Symmetry Argument......................................................................................................32 What about Potential Energy?.......................................................................................................37 Degrees of Freedom and Equipartition of Energy .......................................................................39 Brownian Motion ..............................................................................................................................39 IDEAL GAS THERMODYNAMICS: SPECIFIC HEATS, ISOTHERMS, ADIABATS..........................................39 Introduction: the Ideal Gas Model, Heat, Work and Thermodynamics ....................................39 The Gas Specific Heats C and C .................................................................................................40 V P Tracking a Gas in the (P, V) Plane: Isotherms and Adiabats ......................................................42 Equation for an Adiabat .................................................................................................................44 HEAT ENGINES: THE CARNOT CYCLE .....................................................................................................46 The Ultimate in Fuel Efficiency ........................................................................................................46 Step 1: Isothermal Expansion ..........................................................................................................47 Step 2: Adiabatic Expansion...........................................................................................................48 Steps 3 and 4: Completing the Cycle ...........................................................................................49 Efficiency of the Carnot Engine .....................................................................................................51 THE LAWS OF THERMODYNAMICS AND LIMITS ON ENGINE EFFICIENCY.............................................53 The Laws of Thermodynamics.........................................................................................................53 How the Second Law Limits Engine Efficiency .............................................................................55 A NEW THERMODYNAMIC VARIABLE: ENTROPY ...................................................................................57 Introduction ......................................................................................................................................57 Heat Changes along Different Paths from a to c are Different ................................................57 But Something Heat Related is the Same: Introducing Entropy.................................................59 Finding the Entropy Difference for an Ideal Gas..........................................................................61 Entropy in Irreversible Change: Heat Flow Without Work............................................................62 Entropy Change without Heat Flow: Opening a Divided Box....................................................62 The Third Law of Thermodynamics.................................................................................................64 ENTROPY AND THE KINETIC THEORY: THE MOLECULAR PICTURE ..........................................................64 Searching for a Molecular Description of Entropy.......................................................................64 Enter the Demon ..............................................................................................................................65 Boltzmann Makes the Breakthrough..............................................................................................66 Epitaph: S = k ln W............................................................................................................................68 But What Are the Units for Measuring W ? ....................................................................................68 A More Dynamic Picture.................................................................................................................68 The Removed Partition: What Are the Chances of the Gas Going Back?...............................69 Demon Fluctuations.........................................................................................................................70 Entropy and “Disorder”....................................................................................................................71 Summary: Entropy, Irreversibility and the Meaning of Never......................................................71 Everyday Examples of Irreversible Processes ................................................................................72 MOLECULAR COLLISIONS.......................................................................................................................73 Difficulties Getting the Kinetic Theory Moving..............................................................................73 How Fast Are Smelly Molecules?....................................................................................................73 The Mean Free Path.........................................................................................................................74 3 Gas Viscosity Doesn’t Depend on Density...................................................................................74 Gas Diffusion: the Pinball Scenario; Finding the Mean Free Path in Terms of the Molecular Diameter ...........................................................................................................................................74 But the Pinball Picture is Too Simple: the Target Molecules Are Moving..................................78 If Gases Intermingle 0.5cm in One Second, How Far in One Hour? ..........................................78 Actually Measuring Mean Free Paths............................................................................................79 Why did Newton get the Speed of Sound Wrong?.....................................................................80 BROWNIAN MOTION...............................................................................................................................80 Introduction: Jiggling Pollen Granules...........................................................................................80 Einstein’s Theory: the Osmosis Analogy .........................................................................................81 An Atmosphere of Yellow Spheres.................................................................................................83 Langevin’s Theory ............................................................................................................................84 References........................................................................................................................................86 HEAT TRANSPORT: CONDUCTION, CONVECTION, RADIATION ............................................................87 Conduction.......................................................................................................................................87 Microscopic Picture of Conduction...............................................................................................87 American Units..................................................................................................................................88 Convection.......................................................................................................................................88 Radiation...........................................................................................................................................89 Heat Feeling and seeing temperature changes Within some reasonable temperature range, we can get a rough idea how warm something is by touching it. But this can be unreliable—if you put one hand in cold water, one in hot, then plunge both of them into lukewarm water, one hand will tell you it’s hot, the other will feel cold. For something too hot to touch, we can often get an impression of how hot it is by approaching and sensing the radiant heat. If the temperature increases enough, it begins to glow and we can see it’s hot The problem with these subjective perceptions of heat is that they may not be the same for everybody. If our two hands can’t agree on whether water is warm or cold, how likely is it that a group of people can set a uniform standard? We need to construct a device of some kind that responds to temperature in a simple, measurable way—we need a thermometer. The first step on the road to a thermometer was taken by one Philo of Byzantium, an engineer, in the second century BC. He took a hollow lead sphere connected with a tight seal to one end of a pipe, the other end of the pipe being under water in another vessel. 4 To quote Philo: “…if you expose the sphere to the sun, part of the air enclosed in the tube will pass out when the sphere becomes hot. This will be evident because the air will descend from the tube into the water, agitating it and producing a succession of bubbles. Now if the sphere is put back in the shade, that is, where the sun’s rays do not reach it, the water will rise and pass through the tube …” “No matter how many times you repeat the operation, the same thing will happen. In fact, if you heat the sphere with fire, or even if you pour hot water over it, the result will be the same.” Notice that Philo did what a real investigative scientist should do—he checked that the experiment was reproducible, and he established that the air’s expansion was in response to heat being applied to the sphere, and was independent of the source of the heat. Classic Dramatic Uses of Temperature-Dependent Effects This expansion of air on heating became widely known in classical times, and was used in various dramatic devices. For example, Hero of Alexandria describes a small temple where a fire on the altar causes the doors to open. 5 The altar is a large airtight box, with a pipe leading from it to another enclosed container filled with water. When the fire is set on top of the altar, the air in the box heats up and expands into a second container which is filled with water. This water is forced out through an overflow pipe into a bucket hung on a rope attached to the door hinges in such a way that as the bucket fills with water, it drops, turns the hinges, and opens the doors. The pipe into this bucket reaches almost to the bottom, so that when the altar fire goes out, the water is sucked back and the doors close again. (Presumably, once the fire is burning, the god behind the doors is ready to do business and the doors open…) Still, none of these ingenious devices is a thermometer. There was no attempt (at least none recorded) by Philo or his followers to make a quantitative measurement of how hot or cold the sphere was. And the “meter” in thermometer means measurement. The First Thermometer Galileo claimed to have invented the first thermometer. Well, actually, he called it a thermoscope, but he did try to measure “degrees of heat and cold” according to a colleague, and that qualifies it as a thermometer. (Technically, a thermoscope is a device making it possible to see a temperature change, a thermometer can measure the temperature change.) Galileo used an inverted narrow- necked bulb with a tubular neck, like a hen’s egg with a long glass tube attached at the tip. 6 He first heated the bulb with his hands then immediately put it into water. He recorded that the water rose in the bulb the height of “one palm”. Later, either Galileo or his colleague Santorio Santorio put a paper scale next to the tube to read off changes in the water level. This definitely made it a thermometer, but who thought of it first isn’t clear (they argued about it). And, in fact, this thermometer had problems. Question: what problems? If you occasionally top up the water, why shouldn’t this thermometer be good for recording daily changes in temperature? Answer: because it’s also a barometer But—Galileo didn’t know about the atmospheric pressure. Torricelli, one of Galileo’s pupils, was the first to realize, shortly after Galileo died, that the real driving force in suction was external atmospheric pressure, a satisfying mechanical explanation in contrast to the philosophical “nature abhors a vacuum”. In the 1640’s, Pascal pointed out that the variability of atmospheric pressure rendered the air thermometer untrustworthy. Liquid-in-glass thermometers were used from the 1630’s, and they were of course insensitive to barometric pressure. Meteorological records were kept from this time, but there was no real uniformity of temperature measurement until Fahrenheit, almost a hundred years later. 7 Newton’s Anonymous Table of Temperatures The first systematic account of a range of different temperatures, “Degrees of Heat”, was written by Newton, but published anonymously, in 1701. Presumably he felt that this project lacked the timeless significance of some of his other achievements. Taking the freezing point of water as zero, Newton found the temperature of boiling water to be almost three times that of the human body, melting lead eight times as great (actually 327C, whereas 8x37=296, so this is pretty good) but for higher temperatures, such as that of a wood fire, he underestimated considerably. He used a linseed oil liquid in glass thermometer up to the melting point of tin (232°C). (Linseed oil doesn’t boil until 343°C, but that is also its autoignition temperature) Newton tried to estimate the higher temperatures indirectly. He heated up a piece of iron in a fire, then let it cool in a steady breeze. He found that, at least at the lower temperatures where he could cross check with his thermometer, the temperature dropped in a geometric progression, that is, if it took five minutes to drop from 80° above air temperature to 40° above air temperature, it took another five minutes to drop to 20° above air, another five to drop to 10° above, and so on. He then assumed this same pattern of temperature drop was true at the high temperatures beyond the reach of his thermometer, and so estimated the temperature of the fire and of iron glowing red hot. This wasn’t very accurate—he (under)estimated the temperature of the fire to be about 600°C. Fahrenheit’s Excellent Thermometer The first really good thermometer, using mercury expanding from a bulb into a capillary tube, was made by Fahrenheit in the early 1720’s. He got the idea of using mercury from a colleague’s comment that one should correct a barometer reading to allow for the variation of the density of mercury with temperature. The point that has to be borne in mind in constructing thermometers, and defining temperature scales, is that not all liquids expand at uniform rates on heating—water, for example, at first contracts on heating from its freezing point, then begins to expand at around forty degrees Fahrenheit, so a water thermometer wouldn’t be very helpful on a cold day. It is also not easy to manufacture a uniform cross section capillary tube, but Fahrenheit managed to do it, and demonstrated his success by showing his thermometers agreed with each other over a whole range of temperatures. Fortunately, it turns out that mercury is well behaved in that the temperature scale defined by taking its expansion to be uniform coincides very closely with the true temperature scale, as we shall see later. Amontons’ Air Thermometer: Pressure Increases Linearly with Temperature A little earlier (1702) Amontons introduced an air pressure thermometer. He established that if air at atmospheric pressure (he states 30 inches of mercury) at the freezing point of water is enclosed then heated to the boiling point of water, but meanwhile kept at constant volume by increasing the 8 pressure on it, the pressure goes up by about 10 inches of mercury. He also discovered that if he compressed the air in the first place, so that it was at a pressure of sixty inches of mercury at the temperature of melting ice, then if he raised its temperature to that of boiling water, at the same time adding mercury to the column to keep the volume of air constant, the pressure increased by 20 inches of mercury. In other words, he found that for a fixed amount of air kept in a container at constant volume, the pressure increased with temperature by about 33% from freezing to boiling, that percentage being independent of the initial pressure. Thermal Equilibrium and the Zeroth Law of Thermodynamics Once the thermometer came to be widely used, more precise observations of temperature and (as we shall see) heat flow became possible. Joseph Black, a professor at the University of Edinburgh in the 1700’s, noticed that a collection of objects at different temperatures, if brought together, will all eventually reach the same temperature. As he wrote, “By the use of these instruments thermometers we have learned, that if we take 1000, or more, different kinds of matter, such as metals, stones, salts, woods, cork, feathers, wool, water and a variety of other fluids, although they be all at first of different heats, let them be placed together in a room without a fire, and into which the sun does not shine, the heat will be communicated from the hotter of these bodies to the colder, during some hours, perhaps, or the course of a day, at the end of which time, if we apply a thermometer to all of them in succession, it will point to precisely the same degree.” We say nowadays that bodies in “thermal contact” eventually come into “thermal equilibrium”— which means they finally attain the same temperature, after which no further heat flow takes place. This is equivalent to: The Zeroth Law of Thermodynamics: If two objects are in thermal equilibrium with a third, then they are in thermal equilibrium with each other. The “third body” in a practical situation is just the thermometer. It’s perhaps worth pointing out that this trivial sounding statement certainly wasn’t obvious before the invention of the thermometer. With only the sense of touch to go on, few people would agree that a piece of wool and a bar of metal, both at 0°C, were at the same temperature. Measuring Heat Flow: a Unit of Heat The next obvious question is, can we get more quantitative about this “flow of heat” that takes place between bodies as they move towards thermal equilibrium? For example, suppose I reproduce one 9 of Fahrenheit’s experiments, by taking 100 ccs of water at 100°F, and 100ccs at 150°F, and mix them together in an insulated jug so little heat escapes. What is the final temperature of the mix? Of course, it’s close to 125°F—not surprising, but it does tell us something It tells us that the amount of heat required to raise the temperature of 100 cc of water from 100°F to 125°F is exactly the same as the amount needed to raise it from 125°F to 150°F. A series of such experiments (done by Fahrenheit, Black and others) established that it always took the same amount of heat to raise the temperature of 1 cc of water by one degree. This makes it possible to define a unit of heat. Perhaps unfairly to Fahrenheit, 1 calorie is the heat required to raise the temperature of 1 gram of water by 1 degree Celsius. (Celsius also lived in the early 1700’s. His scale has the freezing point of water as 0°C, the boiling point as 100°C. Fahrenheit’s scale is no longer used in science, but lives on in engineering in the US, and in the British Thermal Unit, which is the heat required to raise the temperature of one pound of water by 1°F.) Specific Heats and Calorimetry First, let’s define specific heat: The specific heat of a substance is the heat required in calories to raise the temperature of 1 gram by 1 degree Celsius. As Fahrenheit continues his measurements of heat flow, it quickly became evident that for different materials, the amount of heat needed to raise the temperature of one gram by one degree could be quite different. For example, it had been widely thought before the measurements were made, that one cc of Mercury, being a lot heavier than one cc of water, would take more heat to raise its temperature by one degree. This proved not to be the case— Fahrenheit himself made the measurement. In an insulating container, called a “calorimeter” he added 100ccs of water at 100°F to 100ccs of mercury at 150°F, and stirred so they quickly reached thermal equilibrium. Question: what do you think the final temperature was? Approximately? Answer: The final temperature was, surprisingly, about 120°F. 100 cc of water evidently “contained more heat” than 100 cc of mercury, despite the large difference in weight 10 This technique, called calorimetry, was widely used to find the specific heats of many different substances, and at first no clear pattern emerged. It was puzzling that the specific heat of mercury was so low compared with water As more experiments on different substances were done, it gradually became evident that heavier substances, paradoxically, had lower specific heats. A Connection With Atomic Theory Meanwhile, this quantitative approach to scientific observation had spread to chemistry. Towards the end of the 1700’s, Lavoisier weighed chemicals involved in reactions before and after the reaction. This involved weighing the gases involved, so had to be carried out in closed containers, so that, for example, the weight of oxygen used and the carbon dioxide, etc., produced would accounted for in studying combustion. The big discovery was that mass was neither created nor destroyed. This had not been realized before because no one had weighed the gases involved. It made the atomic theory suddenly more plausible, with the idea that maybe chemical reactions were just rearrangements of atoms into different combinations. Lavoisier also clarified the concept of an element, an idea that was taken up in about 1800 by John Dalton, who argues that a given compound consisted of identical molecules, made up of elementary atoms in the same proportion, such as H O (although that was thought initially to be HO). This 2 explained why, when substances reacted chemically, such as the burning of hydrogen to form water, it took exactly eight grams of oxygen for each gram of hydrogen. (Well, you could also produce H O under the right conditions, with exactly sixteen grams of oxygen to one of hydrogen, but the 2 2 simple ratios of amounts of oxygen needed for the two reactions were simply explained by different molecular structures, and made the atomic hypothesis even more plausible.) Much effort was expended carefully weighing the constituents in many chemical reactions, and constructing diagrams of the molecules. The important result of all this work was that it became possible to list the relative weights of the atoms involved. For example, the data on H O and H O 2 2 2 led to the conclusion that an oxygen atom weighed sixteen times the weight of a hydrogen atom. It must be emphasized, though, that these results gave no clue as to the actual weights of atoms All that was known was that atoms were too small to see in the best microscopes. Nevertheless, knowing the relative weights of some atoms in 1820 led to an important discovery. Two professors in France, Dulong and Petit, found that for a whole series of elements the product of atomic weight and specific heat was the same 11 Element Specific Heat Relative weights of Product of relative the atoms atomic weight and specific heat Lead 0.0293 12.95 0.3794 Tin 0.0514 7.35 0.3779 Zinc 0.0927 4.03 0.3736 Sulphur 0.1880 2.011 0.3780 The significance of this, as they pointed out, was that the “specific heat”, or heat capacity, of each atom was the same—a piece of lead and a piece of zinc having the same number of atoms would have the same heat capacity. So heavier atoms absorbed no more heat than lighter atoms for a given rise in temperature. This partially explained why mercury had such a surprisingly low heat capacity. Of course, having no idea how big the atoms might be, they could go no further. And, indeed, many of their colleagues didn’t believe in atoms anyway, so it was hard to convince them of the significance of this discovery. Latent Heat One of Black’s experiments was to set a pan of water on a steady fire and observe the temperature as a function of time. He found it steadily increased, reflecting the supply of heat from the fire, until the water began to boil, whereupon the temperature stayed the same for a long time. The steam coming off was at the same (boiling) temperature as the water. So what was happening to the heat being supplied? Black correctly concluded that heat needed to be supplied to change water from its liquid state to its gaseous state, that is, to steam. In fact, a lot of heat had to be supplied: 540 calories per gram, as opposed to the mere 100 calories per gram needed to bring it from the freezing temperature to boiling. He also discovered that it took 80 calories per gram to melt ice into water, with no rise in temperature. This heat is released when the water freezes back to ice, so it is somehow “hidden” in the water. He called it latent heat, meaning hidden heat. __________________________________________________________________________ Books I used in preparing this lecture: A Source Book in Greek Science, M. R. Cohen and I. E. Drabkin, Harvard university Press, 1966. A History of the Thermometer and its Uses in Meteorology, W. E. Knowles Middleton, Johns Hopkins Press, 1966. A Source Book in Physics, W. F. Magie, McGraw-Hill, New York, 1935. 12 Thermal Expansion and the Gas Law Coefficients of Expansion Almost all materials expand on heating—the most famous exception being water, which contracts as it is warmed from 0 degrees Celsius to 4 degrees. This is actually a good thing, because as freezing weather sets in, the coldest water, which is about to freeze, is less dense than slightly warmer water, so rises to the top of a lake and the ice begins to form there. For almost all other liquids, solidification on cooling begins at the bottom of the container. So, since water behaves in this weird way, ice skating is possible Also, as a matter of fact, life in lakes is possible—the ice layer that forms insulates the rest of the lake water from very cold air, so fish can make it through the winter. Linear Expansion The coefficient of linear expansion α of a given material, for example a bar of copper, at a given temperature is defined as the fractional increase in length that takes place on heating through one degree: 0 LL →+ΔL= 1 w +αL hen T→T+1 C () Of course, α might vary with temperature (it does for water, as we just mentioned) but in fact for most materials it stays close to constant over wide temperature ranges. −6 For copper, α=× 17 10 . Volume Expansion For liquids and gases, the natural measure of expansion is the coefficient of volume expansion,β . 0 VV →+ΔV= 1 w +βV hen T→T+1 C () Of course, on heating a bar of copper, clearly the volume as well as the length increases—the bar expands by an equal fraction in all directions (this could be experimentally verified, or you could just imagine a cube of copper, in which case all directions look the same). 3 The volume of a cube of copper of side L is V = L . Suppose we heat it through one degree. Putting together the definitions of α,β above, 33 33 VV → 1, +βαL→+ 1L,L→+ 1αL or V→+ 1αV. ( ) ( ) () () 13 3 So 11 +=β +α . But remember α is very, very small—so even though ()( ) 3 2 3 11 +=α +3αα+3 +α , the last two terms are completely negligible (check it out) so to a () fantastically good approximation: β=3.α The coefficient of volume expansion is just three times the coefficient of linear expansion. Gas Pressure Increase with Temperature In 1702, Amontons discovered a linear increase of P with T for air, and found P to increase about 33% from the freezing point of water to the boiling point of water. That is to say, he discovered that if a container of air were to be sealed at 0°C, at ordinary atmospheric pressure of 15 pounds per square inch, and then heated to 100°C but kept at the same volume, the air would now exert a pressure of about 20 pounds per square inch on the sides of the container. (Of course, strictly speaking, the container will also have increased in size, that would lower the effect—but it’s a tiny correction, about ½% for copper, even less for steel and glass.) Remarkably, Amontons discovered, if the gas were initially at a pressure of thirty pounds per square inch at 0°C, on heating to 100°C the pressure would go to about 40 pounds per square inch—so the percentage increase in pressure was the same for any initial pressure: on heating through 100°C, the pressure would always increase by about 33%. Furthermore, the result turned out to be the same for different gases Finding a Natural Temperature Scale In class, we plotted air pressure as a function of temperature for a fixed volume of air, by making several measurements as the air was slowly heated (to give it a chance to all be at the same temperature at each stage). We found a straight line. On the graph, we extended the line backwards, to see how the pressure would presumably drop on cooling the air. We found the remarkable prediction that the pressure should drop to zero at a temperature of about −273°C. In fact, if we’d done the cooling experiment, we would have found that air doesn’t actually follow the line all the way down, but condenses to a liquid at around −200°C. However, helium gas stays a gas almost to −270°C, and follows the line closely. 14 We shall discuss the physics of gases, and the interpretation of this, much more fully in a couple of lectures. For now, the important point is that this suggests a much more natural temperature scale than the Celsius one: we should take −273°C as the zero of temperature For one thing, if we do that, the pressure/temperature relationship for a gas becomes beautifully simple: PT∝ . This temperature scale, in which the degrees have the same size as in Celsius, is called the Kelvin or absolute scale. Temperatures are written 300K. To get from Celsius to Kelvin, just add 273 (strictly speaking, 273.15). An Ideal Gas Physicists at this point introduce the concept of an “Ideal Gas”. This is like the idea of a frictionless surface: it doesn’t exist in nature, but it is a very handy approximation to some real systems, and makes problems much easier to handle mathematically. The ideal gas is one for which PT∝ for all temperatures, so helium is close to ideal over a very wide range, and air is close to ideal at ordinary atmospheric temperatures and above. The Gas Law We say earlier in the course that for a gas at constant temperature PV = constant (Boyle’s Law). PT∝ Now at constant volume, . We can put these together in one equation to find a relationship between pressure, volume and temperature: PV = CT where C is a constant. Notice, by the way, that we can immediately conclude that at fixed pressure, V∝T , this is called Charles’ Law. (Exercise: prove from this that the coefficient of volume expansion of a gas varies significantly with temperature.) But what is C? Obviously, it depends on how much gas we have—double the amount of gas, keeping the pressure and temperature the same, and the volume will be doubled, so C will be doubled. But notice that C will not depend on what gas we are talking about: if we have two separate one-liter containers, one filled with hydrogen, the other with oxygen, both at atmospheric pressure, and both at the same temperature, then C will be the same for both of them. 15 One might conclude from this that C should be defined for one liter of gas at a specified temperature and pressure, such as 0°C and 1 atmosphere, and that could be a consistent scheme. It might seem more natural, though, to specify a particular mass of gas, since then we wouldn’t have to specify a particular temperature and pressure in the definition of C. But that idea brings up a further problem: one gram of oxygen takes up a lot less room than one gram of hydrogen. Since we’ve just seen that choosing the same volume for the two gases gives the same constant C for the two gases, evidently taking the same mass of the two gases will give different C’s. Avogadro’s Hypothesis The resolution to this difficulty is based on a remarkable discovery the chemists made two hundred years or so ago: they found that one liter of nitrogen could react with exactly one liter of oxygen to produce exactly two liters of NO, nitrous oxide, all volume measurements being at the same temperature and pressure. Further, one liter of oxygen combined with two liters of hydrogen to produce two liters of steam. These simple ratios of interacting gases could be understood if one imagined the atoms combining to form molecules, and made the further assumption, known as Avogadro’s Hypothesis (1811): Equal volumes of gases at the same temperature and pressure contain the same number of molecules. One could then understand the simple volume results by assuming the gases were made of diatomic molecules, H , N , O and the chemical reactions were just molecular recombinations 2 2 2 given by the equations N + O = 2NO, 2H + O = 2H O, etc. 2 2 2 2 2 Of course, in 1811 Avogadro didn’t have the slightest idea what this number of molecules was for, say, one liter, and nobody else did either, for another fifty years. So no-one knew what an atom or molecule weighed, but assuming that chemical reactions were atoms combining into molecules, or rearranging from one molecular pairing or grouping to another, they could figure out the relative weights of atoms, such as an oxygen atom had mass 16 times that of a hydrogen atom—even though they had no idea how big these masses were This observation led to defining the natural mass of a gas for setting the value of the constant C in the gas law to be a “mole” of gas: hydrogen was known to be H molecules, so a mole of 2 hydrogen was 2 grams, oxygen was O , so a mole of oxygen was 32 grams, and so on. 2 16 With this definition, a mole of oxygen contains the same number of molecules as a mole of hydrogen: so at the same temperature and pressure, they will occupy the same volume. At 0°C, and atmospheric pressure, the volume is 22.4 liters. So, for one mole of a gas (for example, two grams of hydrogen), we set the constant C equal to R, known as the universal gas constant, and equal to 8.3 J/(mol.K) and PV = RT. For n moles of a gas, such as 2n grams of hydrogen, the law is: PV = nRT and this is the standard form of the Gas Law. (Footnote: after the discovery of isotopes, nuclei of the same element having different masses, and in particular of a form of hydrogen called heavy hydrogen present in small quantities in nature, the definition of the mole was refined to be equal to precisely 12 grams of the carbon 12 isotope C . In practice, this is a tiny correction which doesn’t affect anything we’ve said here.) Early Attempts to Understand the Nature of Heat When Heat Flows, What, Exactly is Flowing? By the late 1700’s, the experiments of Fahrenheit, Black and others had established a systematic, quantitative way of measuring temperatures, heat flows and heat capacities—but this didn’t really throw any new light on just what was flowing. This was a time when the study of electricity was all the rage, led in America by Benjamin Franklin, who had suggested in 1747 that electricity was one (invisible) fluid (it had previously been suggested that there were two fluids, corresponding to the two kinds of electrical charging observed). Lavoisier’s Caloric Fluid Theory Perhaps heat was another of these invisible fluids? In 1787, Lavoisier, the French founder of modern chemistry, thought so, and called it the caloric fluid, from the Greek word for heat. (Lavoisier was the first to attempt to list a table of elements, to replace the ancient elements of earth, air, water and fire. His list of thirty-three elements included hydrogen, oxygen, sulphur, charcoal, etc., but he also included caloric—and light.) 17 Lavoisier and wife, by David, from Wikimedia Commons. The existence of such a fluid was really quite plausible—heat flowed from a hot body to a cold body, and the recent quantitative calorimetric experiments of Black and others seemed to establish that heat was a conserved quantity, as one would expect of a fluid. One could also understand some of the well-known effects of heat in terms of a fluid, and establish some of the fluid’s properties. For example, since it tended to flow from hot bodies into cold bodies and spread throughout the body, presumably its particles repelled each other, just like those of the electrical fluid. However, in contrast to electricity, which had no noticeable effect on the appearance of a charged object, when heat was added to a solid things changed considerably. First the material expanded, then it changed to a liquid and finally to a gas, if sufficient heat could be delivered. Further heating expanded the gas, or increased its pressure if it was held in a fixed container. To interpret this sequence of events in terms of a caloric fluid being fed into the material, one could imagine the fluid flowing between the atoms of the solid and lessening their attraction for each other, until the solid melted into a liquid, whereupon the caloric continued to accumulate around the atoms until they were pushed apart into a gas. It was thought that in the gas each atom or molecule was surrounded by a ball of caloric, like a springy ball of wool, and these balls were packed in a container like oranges in a crate, except that the caloric balls could expand indefinitely as heat was poured in. Various other effects could be explained by the caloric theory: when a gas is suddenly compressed, it gets hotter because the same amount of caloric is now occupying a smaller volume. When two solids are rubbed together, some caloric is squeezed out at the surfaces, or perhaps tiny pieces of material are rubbed off, and lose their caloric, so heat appears. Radiant heat was presumed to be caloric particles flying through space. Recall that at that time (just before 1800) it was generally accepted that light was a stream of particles. Painting by Joseph Wright, Wikimedia Commons. The Industrial Revolution and the Water Wheel In 1769 a Lancashire wigmaker, Richard Arkwright, patented a successful cotton-spinning machine. Lancashire had been for a long time a center of the textile trade, but before Arkwright the fabrics were woven on hand looms by skilled weavers. The new machines could be operated by less skilled workers, and 18 in fact were largely operated by children, although, in contrast to some of his competitors, Arkwright refused to employ any child younger than six. The motive power driving the machines was at first hor ses, but in 1771 Arkwright built a large factory containing many machines all driven by a water wheel. This was the beginning of the modern system of mass production. Prices fell, and the skilled hand weavers became impoverished. Our interest in this, however, is not the social consequences, but just the water wheel. Previously, water wheels had been used for centuries to grind flour, and for other purposes, but their efficiency had not been a major concern. In the factory, though, the more efficient the wheel, the more children could be spinning the cotton, and the bigger the profits. Twenty years earlier, John Smeaton (the first Englishman to call himself a civil engineer) had investigated different types of water wheels, and found the overshot type (in which the water pours on to the top of the wheel) to perform best. Measuring Power by Lifting The power output of a water wheel can be measured by using it to raise a load—in those days, it would be how many pounds could be raised through one foot per second, say (we would now just use watts, and it’s amusing to note that the first unit of power, the horsepower, was proposed in 1783 by James Watt to be 33,000 foot pounds per minute). The ultimate in efficiency would be a reversible water wheel, which could be run backwards, to raise the water back again. This is best visualized by having a wheel with a series of buckets attached. Suppose the wheel is run for some time and its power output is used to lift a weight a given distance. Now reverse it, let the weight fall, running the wheel backwards, making sure the buckets now fill at the bottom and empty at the top. How much water is lifted back up? A truly reversible wheel would put all the water back. We know this isn’t going to happen, but if the reversed wheel manages to lift half the water back, say, then it’s 50% efficient. In building the first factory, the water wheel was not just placed under a waterfall. The water was channeled to it for maximum efficiency. Smeaton had established that the flow of water into the buckets must be as smooth as possible. Turbulence was wasted effort—it didn’t help the wheel go round. The water should flow onto the wheel, not fall from some height. Finally, the perfect wheel (not quite realizable in practice) would be reversible—it could be run backwards to put the water back up using the same amount of work it delivered in the first place. Carnot’s Caloric Water Wheel Arkwright’s factory was so successful that within a few years similar factories had been built wherever a water wheel could be operated economically in Northern England. The next step was to use steam power, which had been developed in the previous century to lift water out of mines. As steam engine design improved, the English economy mushroomed far ahead of European 19 competitors—but in contrast to the present day, these technological advances owed virtually nothing to basic science. It was all inspired tinkering. The first attempt to analyze the steam engine in a scientific way was by a Frenchman, Sadi Carnot, in 1820—and he relied heavily on an analogy with the water wheel. In the steam engine, heat is delivered to water to boil off steam which is directed through a pipe to a cylinder where it pushes a piston. The piston does work, usually by turning a wheel, the steam cools down, and the relatively cold vapor is expelled, so that the piston will be ready for the next dose of steam. Where is the analogy to a water wheel? Recall that heat was seen as an invisible fluid, impelled by its nature to flow from hot objects to cold objects. Water always flows from high places to low places. Carnot saw these as parallel processes—and, just as a water wheel extracts useful work from falling water, he saw the steam engine extracting work from the “falling” caloric fluid, as it cascaded from a hot object to a cold object. How Efficient are these Machines? As we’ve discussed, an ordinary water wheel is most efficient if the water flows in and out very smoothly, so no energy is wasted in turbulence or splashing. If we could make such a wheel with friction-free bearings, etc., then it could be made to drive a twin wheel going backwards, which could lift all the water back up again. This idealized wheel would be 100% efficient. Carnot’s idealized heat engine had gas in a cylinder which pushed a piston as it expanded, doing work. Heat was fed into the gas, it expanded, then the heat supply was cut off, but the hot gas continued to expand and cool down at the same time. The piston then reversed direction, and the heat generated by the compression was allowed to flow out into a heat sink, until a certain point was reached at which the sink was disconnected, and the further compression heated up the gas to its original temperature, at which point the cycle began again. We’ll be discussing this so- called “Carnot cycle” in much more detail later, all we need to take away from it at this point is that heat is supplied to the gas at a high temperature, and it flows out to the sink at a lower temperature. This “falling” of the “caloric fluid” from hot to cold is the analogy to the water wheel. Carnot argued that if all friction were eliminated, and the heat flow into and out of the gas were smooth—going from one place to another at the same temperature, just like the water moving smoothly on to the water wheel, not dropping on to it, then one could imagine a reversible heat engine: the work output could be used to drive a similar engine in reverse which would take heat from a cold place and expel it in a warmer place (that’s a refrigerator). 20 Carnot found, not surprisingly, that the amount of work a perfect engine could deliver for a given amount of heat increased as the temperature difference between heat source and heat sink increased. Obviously, water wheels get more energy from the same amount of water if the wheel is bigger so the water has further down to go. For a given temperature difference, then, a given amount of heat can only deliver so much work. And, this is quite independent of the materials used in constructing the engine, including the gas itself. As we shall discuss in detail later, he was able to find for such an engine just how much work the engine could perform for a given heat input, and the answer was surprisingly low. Furthermore, no engine could ever be more efficient than a reversible engine, because if it were, it could be used to drive the reversible engine backwards, replacing the heat in the furnace, with energy to spare, and would be a perpetual motion machine. Carnot’s basic assumption that heat is a fluid was flawed, but his reasoning was of sufficient generality that his conclusions about efficiency were correct, and proved to be a crucial step toward understanding engines, as we shall see. Count Rumford Picture from http://www.rumford.com/Rumfordpicture.html, original Gainsborough painting in Fogg Museum, Harvard. The first real attack on the caloric theory of heat took place in a cannon factory in Bavaria, under the direction of one Count Rumford of the Holy Roman Empire. This Count was actually born Benjamin Thompson in Woburn, in the English colony of Massachusetts, in 1753, which he left in a hurry after choosing the wrong side in the Revolutionary War. He was a brilliant man, extraordinarily inventive as a scientist and engineer—but it is difficult to form a coherent picture of his character. He seemed genuinely upset by the plight of the poor in Munich (see below) and made great personal efforts for years to ensure they were properly fed and clothed. Throughout his life, he invented practical devices to make daily living better: stoves, fitted kitchens, drip coffeepots, lighting, and many more. Yet, despite this love for humanity and his clear desire to make life better for everyone, Rumford did not apparently like—or get on with—actual people. The only

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