# THERMAL PHYSICS RALPH BAIERLEIN PDF

download Thermal Physics on ritipulmama.cf ✓ FREE SHIPPING on qualified orders. Cambridge Core - General and Classical Physics - Thermal Physics - by Ralph Ralph Baierlein, Wesleyan University, Connecticut . PDF; Export citation. Feb 28, Ralph Baierlein- Thermal Physics - Free ebook download as PDF File .pdf), Text File .txt) or read book online for free. Thermal Physics.  Showing Rating details. Sort order. Feb 11, Jonayhan rated it did not like it. This book is awful. There are very few problems at the end of each chapter.

The actual material is explained poorly and the equations lack meaning behind them. The book also says away from the mathematics necessary to understand this type of physics.

Beyond that, the step from 1. Often that assumption is a good approximation. For example, it is fine for diatomic oxygen and nitrogen under typical room conditions. If the temperature varies greatly, however, Cy will change; sections If you glance back through the chapter, from its opening paragraph up to equation 1.

Historically, the word "heat" has been used as a noun as well, but such use— although common—is often technically incorrect.

The reason is this: In sequence a , hot steam doubles in volume as it expands adiabatically into a vacuum. Because the water molecules hit only fixed walls, they always rebound elastically. The steam loses energy and also drops in temperature. To compensate for the energy loss, in the last stage a burner heats the water vapor, transferring energy by heating until the steam's energy returns to its initial value. If a definite amount or kind of energy in the water vapor could be identified as "heat," then there would have to be more of it at the end of sequence b than at the end of sequence a , for only in sequence b has any heating occurred. But in fact, there is no difference, either macroscopic or microscopic, between the two final states.

An equivalent way to pose the problem is the following. The challenge is to identify a definite amount or kind of energy in a gas that 1 increases when the gas is heated by conduction or radiation and 2 remains constant during all adiabatic processes. No one has met that challenge successfully.

Such a concept is untenable, as the analysis with figure 1. Such time is incorporated into the sequences. The literal meaning of the phrase would be "capacity for holding heat," but that is not meaningful. The ratio in expression 1. From time to time, we will calculate a "heat capacity. Even the words "heating" and "cooling" are used in more than one sense. Thus far, I have used them exclusively to describe the transfer of energy by conduction or radiation.

The words are used also to describe any process in which the temperature rises or drops. Thus, if we return to figure 1. If one were to push the piston back in and return the gas to its initial state, one could describe that process as "heating by adiabatic compression" because the temperature would rise to its initial value. In both cases the adjective "adiabatic" means "no cooling or heating by conduction or radiation," but the temperature does change, and that alone can be the meaning of the words "cooling" and "heating.

For a moment, imagine that you are baby-sitting your niece and nephew, Heather and Walter, as they play at the beach.

The 4-year-olds are carrying water from the lake and pouring it into an old rowboat. When you look in the rowboat, you see clear water filling it to a depth of several centimeters. While Heather is carrying water, you can distinguish her lake water from Walter's— because it is in a green bucket—but once Heather has poured the water into the rowboat, there is no way to distinguish the water that she carried from that which Walter poured in or from the rainwater that was in the boat to start with.

The same possibilities and impossibilities hold for energy transferred by heating, energy transferred by work done by or on the system , and internal energy that was present to start with.

Energy that is being transferred by conduction or radiation may be called "heat. Once such energy has gotten into the physical system, however, it is just an indistinguishable contribution to the internal energy. Only energy in transit may correctly be called "heat.

Beyond that, thermodynamics notes that some of those attributes can be calculated from others for example, via the ideal gas law. Such attributes are called state functions because, collectively, they define the macroscopic state and are defined by that state. As we reasoned near the beginning of this section, one may not speak of the "amount of heat" in a physical system. These are subtle, but vital, points if one chooses to use the word "heat" as a noun.

In this book, I will avoid the confusion that such usage invariably engenders and will continue to emphasize the process explicitly; in short, I will stick with the verb-like forms and will speak of "energy input by heating. It is neither. Historical usage, however, cannot be avoided, especially when you read other books or consult collections of tabulated physical properties.

Stay alert for misnomers. While we are on the subject of meanings, you may wonder, what is "thermal physics"? Broadly speaking, one can define thermal physics as encompassing every part of physics in which the ideas of heating, temperature, or entropy play an essential role. If there is any central organizing principle for thermal physics, then it is the Second Law of Thermodynamics, which we develop in the next chapter.

This section collects essential ideas and results from the entire chapter. It is neither a summary of everything nor a substitute for careful study of the chapter. Its purpose is to emphasize the absolutely essential items, so that—as it were—you can distinguish the main characters from the supporting actors. Think of heating as a process of energy transfer, a process accomplished by conduction or radiation.

No change in external parameters is required. Whenever two objects can exchange energy by heating or cooling , one says that they are in thermal contact. Any macroscopic environmental parameter that appears in the microscopic mech- anical expression for the energy of an atom or electron is an external parameter.

Volume and external magnetic field are examples of external parameters. Pressure, however, is not an exteijial parameter. That is, the goal of the "temperature" notion is to order objects in a sequence according to their "hotness" and to assign to each object a number—its temperature—that will facilitate comparisons of "hotness. The phrase thermal equilibrium means that a system has settled down to the point where its macroscopic properties are constant in time.

Surely the microscopic motion of individual atoms remains, and tiny fluctuations persist, but no macroscopic change with time is discernible. When the conditions of temperature and number density are such that a gas satisfies the ideal gas law, we will call the gas a classical ideal gas.

The shorter phrase, ideal gas, means merely that intermolecular forces are negli- gible. The questions of whether classical physics suffices or whether quantum theory is needed remain open. The First Law of Thermodynamics is basically conservation of energy: A general definition of heat capacity is the following: Thus the last expression is not general.

A process in which no heating occurs is called adiabatic. Attributes that, collectively, define the macroscopic state and which are defined by that state are called state functions. Examples are internal energy E, temperature T, volume V, and pressure P. One may not speak of the "amount of heat" in a physical system. Max Planck introduced the symbols h and k in his seminal papers on blackbody radiation: Leipzig 4, and Thomas S.

Kuhn presents surprising historical aspects of the early quantum theory in his book, Black- body Theory and the Quantum Discontinuity, Oxford University Press, New York, Zemansky in The Physics Teacher 8, Appendix A provides physical and mathematical data that you may find useful when you do the problems.

A fixed number of oxygen molecules are in a cylinder of variable size. Someone compresses the gas to one-third of its original volume. Simultaneously, energy is added to the gas by both compression and heating so that the temperature increases five fold: The gas remains a dilute classical gas. By what numerical factor does each of the following change: Be sure to show your line of reasoning for each of the three questions.

Radiation pressure. Adapt the kinetic theory analysis of section 1. There are TV photons, each of energy hv, where h is Planck's constant and v is a fixed frequency. The volume V has perfectly reflecting walls.

Express the pressure in terms of N, V, and the product hv. Section 6. Relativistic molecules. Suppose the molecules of section 1. Suppose also that the molecules survive collision with the wall! Eliminate the speed v entirely. If you know the pressure exerted by a photon gas, compare the limit here with the photon gas's pressure.

Adiabatic compression. A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder. For air, the ratio of heat capacities is y — 1. Adiabatic versus isothermal expansion. In figure 1. Ruchardts experiment: You take a stainless steel sphere of radius ro and lower it— slowly—down the tube until the increased air pressure supports the sphere. Assume that no air leaks past the sphere an assumption that is valid over a reasonable interval of time and that no energy passes through any walls.

Determine the distance below the tube's top at which the sphere is supported. Provide both algebraic and numerical answers. You have determined an equilibrium position for the sphere while in the tube. Numerical data: Take the ratio of heat capacities to lie in the range 1. Ruchardt's experiment: This question carries on from "Ruchardt's experiment: In your quantitative calculations, ignore friction with the walls of the tightly fitting tube, but describe how the predicted evolution would change if you were to include friction.

Note that the equilibrium location for the "lowered" ball is about half-way down the tube. After you have worked things out algebraically, insert numerical values. Present an expression for y in terms of the oscillation frequency together with known or readily measurable quantities. A monatomic classical ideal gas of N atoms is initially at temperature 7b in a volume Vo. The gas is allowed to expand slowly to a final volume 7Fo in one of three different ways: For each of these contexts, calculate the work done by the gas, the amount of energy transferred to the gas by heating, and the final temperature.

Express all answers in terms of N9 7b, VQ9 and k. Chapter 2 examines the evolution in time of macroscopic physical systems. This study leads to the Second Law of Thermodynamics, the deepest principle in thermal physics. To describe the evolution quantitatively, the chapter introduces and defines the ideas of multiplicity and entropy. Their connection with temperature and energy input by heating provides the chapter's major practical equation.

Simple things can pose subtle questions. A bouncing ball quickly and surely comes to rest. Why doesn't a ball at rest start to bounce? There is nothing in Newton's laws of motion that could prevent this; yet we have never seen it occur. If you are skeptical, recall that a person can jump off the floor.

Similarly, a ball could—in principle—spontaneously rise from the ground, especially a ball that had just been dropped and had come to rest. Or let us look at a simple experiment, something that seems more "scientific.

When the clamp is opened, the bromine diffuses almost instantly into the evacuated flask. The gas fills the two flasks about equally.

The molecules seem never to rush back and all congregate in the first flask. You may say, "That's not surprising. But let's consider this "not surprising" phenomenon more deeply. Could the molecules all go back? There is nothing in Newton's laws, the irregular molecular motion, and the frequent collisions to prevent the molecules from all returning—and, indeed, from then staying in the original container.

The collisions. Figure 2. Each flask has a volume of approximately one liter. The bromine is visible as a rusty brown gas. In practice, the "bromine-filled" flask contains air also, but we focus exclusively on the bromine because it is visible.

And yet, although the collisions could be just right, such a situation is not probable. If there were just three bromine molecules, we would expect a simultaneous return to the original container to occur from time to time, perhaps in a matter of minutes.

But if we consider 6, then 60, next , and finally bromine molecules, the event of simultaneous return shrinks dramatically in probability. Table 2. We expect a more-or-less uniform spatial distribution of bromine molecules, and that is what we actually find.

We can say that a more-or-less uniform distribution of the molecules is much more probable than any other distribution. The simple phrase, "a more-or-less uniform distribution," is an instance of a large-scale, macroscopic characterization, as distin- guished from a microscopic characterization which would focus on individual mol- ecules.

Boldly, we generalize from the bromine experiment to macroscopic physics in general: Macroscopic regularity. When a physical system is allowed to evolve in isolation, some single macroscopic outcome is overwhelmingly more probable than any other. This property, of course, is what makes our macroscopic physical world reproducible and predictable to the large extent that it actually has those characteristics.

The inference is another way of saying that a liter of water or a roomful of air has reproducible physical properties. Multiplicity Let us sharpen the inference.

We distinguish between the "state of affairs" on the microscopic and on the macroscopic level. Here are two definitions: Microscopic state of affairs, abbreviated microstate: If you toss a penny 60 times, you expect about 30 heads and 30 tails. For all 60 tosses to yield heads is possible but quite improbable.

The situation with tosses would be extreme. A related and even more relevant issue is this: In short, can we be pretty sure that we will find a more-or-less even distribution of heads and tails?

The table below shows some probabilities for a more-or-less even distribution. As soon as the number of tosses is , or so, the probability of a more-or-less even distribution of heads and tails is nearly unity, meaning that such a distribution is a pretty sure outcome.

By a million tosses, the more-or-less even distribution is overwhelmingly more probable than any and all significant deviations from that distribution, and the numbers quickly become unimaginable. If we were to consider a number of tosses comparable to the number of bromine atoms—10 20 tosses—then the probability of a more-or-less even distribution would be so close to a certainty as to make any deviation not worth accounting for.

Fortunately, there are some tricks for doing that arithmetic efficiently, but to go into them would take us too far afield. Macroscopic state of affairs, abbreviated macrostate: Here is an analogy. Given four balls labeled A, B, C, D and two bowls, what are the different ways in which we can apportion the balls to the two bowls? Some macrostates have many microstates that correspond to them; others, just a few. This is a vital quantitative point. It warrants a term:. The even distribution of balls has the largest multiplicity, but, because the number of items is merely four here, the multiplicity for the even distribution is not yet over- whelmingly greater than the multiplicity for the quite uneven distributions.

The next step in our reasoning is easiest to follow if I draw the picture first and make the connections with physics later. Imagine a vast desert with a few oases. A "mindless" person starts at an oasis and wanders in irregular, thoughtless motion.

With overwhelming probability, the person wanders into the desert because there is so much of it around and remains there for the same reason. If we make some correspondences, the desert picture provides an analogy for the behavior of a molecular system. With the aid of the analogy, we can understand the diffusion of bromine. Common sense and the example with the four balls tell us that the macrostate with a more-or- less uniform distribution of molecules has the largest multiplicity, indeed, over- whelmingly so.

When the clamp was opened, the bromine—in its diffusion—evolved through many microstates and the corresponding macrostates. By molecular collisions, the bromine was almost certain to get to some microstate corresponding to the macrostate of largest multiplicity.

Simply because there are so many such. A point on the map A specific microstate The desert The macrostate of largest multiplicity An oasis A macrostate of small multiplicity The "mindless" person A system of many molecules The person's path The sequence of microstates in the evolution of the system The initial oasis The initial macrostate.

Thereafter, changes of microstate will certainly occur, but further change of macrostate is extremely unlikely. The continual changes of microstate will almost certainly take the bromine from one to another of the many microstates that corre- spond to the more-or-less uniform distribution. The desert analogy suggests a refinement of our tentative inference about macro- scopic regularity, a refinement presented here along with its formal name:.

The Second Law of Thermodynamics. If a system with many molecules is permitted to change, then—with overwhelming probability—the system will evolve to the macrostate of largest multiplicity and will subsequently remain in that macrostate.

The stipu- lation includes the injunction, do not transfer energy to or from the system. As for the stipulation, let us note that we left the bromine alone, permitting it to evolve by itself once the clamp was opened, and so prudence suggests that we append the stipulation, which contains implicitly the injunction about no energy transfer.

Our statement of the Second Law may need a little more attention to make it more quantitative , but already we have a powerful law, and we can even use it to save money for the government. It costs a great deal of money to launch rockets for astronomical observations. We have to pay for a lot of energy in the form of combustible liquids and solids in order to put a satellite into orbit, and we have to pay even more to enable a satellite to escape the Earth's gravitational pull.

An economy-minded astronomer has an idea: The mechanism inside the building is not permitted to use up anything, such as batteries. Rather, after each satellite-carrying projectile has been launched, the mechanism must return fully to its original state. Only outside the building is net change permitted. Moreover, only change to the water and to the state of motion of the projectiles is permitted.

The extracted energy will be used to launch the satellite-carrying projectiles, perhaps by compressing a gigantic spring. Now I will admit that there are some engineering difficulties here, but people have solved engineering problems before.

Will the scheme work, even in principle? Should we give the astronomer a grant? We can tell the engineers not to exert themselves: By the First Law of Thermodynamics, the scheme is all right. Energy would be conserved. The kinetic energy of the projectiles would be provided by the energy extracted from the warm ocean water. The Second Law, however, says no, the proposal will not work. Here is the logic: Now we go through the logic in detail, as follows. The "system" consists of the projectiles, the building, and the water, in both liquid and solid form.

An engineer can push one button to set in motion the launching often projectiles; thus there is no need for continuous human intervention, and we can meet the conditions under which the Second Law may be applied. To bring the Second Law to bear on this issue, we need to compare multiplicities. Figures 2. How many ways can one arrange water molecules to get something that looks like sea water in liquid form?

How many ways for an ice crystal? The multiplicity for the liquid greatly exceeds that for the crystal. The projectile multiplicities, however, are unchanged, for the following reason. To each old microstate of a projectile, there corresponds one and only one new microstate, which differs from the old microstate in only one respect: The number of microstates does not change, and so the multiplicity for each projectile remains the same.

If the proposal worked, the system would go from a macrostate with high multi- plicity to a macrostate with low multiplicity. The Second Law, however, says that with overwhelming probability a system will evolve to the macrostate of largest multi- plicity and then remain there. Therefore the proposal will not work. We started with a bouncing ball and some bromine. By now we have a law, based on. Now we go on to establish the connection between the Second Law and energy transfer by heating.

We consider a confined classical ideal gas and study how the multiplicity changes when we allow the gas to expand by a small amount. To avoid a change in how the various molecular momenta or velocities contribute to the multiplicity a calculation that would be difficult to handle now , let us keep the gas at constant temperature T while it expands.

How can one do this? By heating the gas judiciously perhaps with a large warm brick. As we slowly put in energy by heating, we let the gas slowly expand, do work, and thereby neither drop nor rise in temperature. We will return to this equation, but first we need to learn how it can help us to make a connection with multiplicity. Succinctly, more space available to the gas implies more ways to arrange the molecules, and that, in turn, implies a larger number of microstates and hence a larger multiplicity.

Because we arrange to keep the temperature constant while the volume increases, the gas pressure will drop. Some device is used to control the piston's motion, and that device must take into account the change in gas pressure, but we need not concern ourselves with the device. If there were, say, 10 spatial locations for a single molecule, then there would be 10 X 10 different arrangements for two molecules, that is, For three molecules, there would be 10X 10X 10 different arrangements, yielding , and so on.

The gas—as an ideal gas with molecules of infinitesimal size—is so dilute that we need not worry about molecules getting in each other's way or using up "spatial locations. If we form the ratio of multiplicities, however, the unknown proportionality factor cancels out, and we have. The proportionality factor included whatever contribution the momenta make to the multiplicity.

Because we kept the temperature constant, that contribution remained constant and canceled out. Equation 2. We can achieve that goal by relating A V to the energy input by heating, as follows. Section 1. Thus the energy balance equation, which was displayed as equation 2.

Substitution for P in equation 2. Whenever one is confronted with a large exponent, taking logarithms may make the expression easier to work with. So we take the natural logarithm of both sides of equation 2. The step to the second line is permitted by an excellent approximation: If you are not familiar with this approximation, you can find it derived in appendix A. If we multiply equation 2. We see that q is connected with the logarithms of multiplicities. The logarithm, we will find, is so useful that it merits its own symbol S,.

We can write the consequence of our slow expansion at constant temperature T as the equation. The German physicist Rudolf Clausius coined the word "entropy" in Looking for a word similar to the word "energy," Clausius chose the Greek word "entropy," which means in Greek "the turning" or "the transformation.

This conclusion is the essential content of our purely verbal statement of the Second Law, presented in section 2. The same is true, of course, if we describe the situation with the logarithm of the multiplicity. If we permit an otherwise-isolated system to change, it will evolve to the macrostate for which the logarithm of the multiplicity is largest, that is, to the macrostate of largest entropy.

A pause to consolidate is in order. Our detailed calculations in this subsection culminate in equation 2. The notion of multiplicity remains primary, but the language of "entropy" will be increasingly useful.

Some generalization While we have the context of this gaseous system, let us consider what would result if we increased the volume by AV suddenly by removing a partition and letting the gas rush into a vacuum and if we did not put in any energy by heating.

In the expansion into a vacuum, the molecules would strike only stationary walls and hence no energy would be lost from the gas.

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The major differences would be these: We set out to find a connection between change in multiplicity and energy input by heating. So far we have two specific instances, both for a classical ideal gas:. The two instances can be combined into a single mathematical form:. Could this relationship be valid in general? It is neat enough and simple enough for that to be plausible. Indeed, the generality of this relationship can be established by reasoning from our verbal statement of the Second Law of Thermo- dynamics; we will do that in section 2. Right now, however, it is better to interrupt the sequence of derivations and to study some examples.

So, provisionally, we take equation 2. Moreover, the temperature of both system and environment may change during the system's evolu- tion.

Such a broad domain, we will find, is the equation's scope of applicability. Working out some examples will help you to grasp the ideas, and so this section is devoted to that project. Example 1. Melting ice The multiplicity of liquid water, in comparison with that of ice, was crucial in our analysis of the astronomer's proposal.

By what factor does the multiplicity change? Recall that "entropy" is just the logarithm of a multiplicity. If we can calculate the change in entropy, then we can readily determine the change in multiplicity. So, to start with, let us look at the entropy change,. Working from the definition of entropy, we rewrite the difference as follows:. Thus, once we have calculated AS, we can determine the ratio of multiplicities.

Moreover, we can relate AS to the energy input by heating that occurs during the melting; our basic equation, equation 2. Recall the convention adopted in section 1. A cube from a typical refrigerator has a volume of approximately 18 cm3. So we can go on:. Combine this result with equation 2. There are vastly many more ways in which one can arrange the water molecules to look like a liquid than to look like an ice crystal.

You may find it worthwhile to glance back at figures 2. Before we leave this subsection, let us glance back over the logic of the calculation. Given that we study a slow process, the essential relationships are these:.

The next equality just spells out what AS symbolizes. Example 2. Slowadiabatic expansion Let us return to the classical ideal gas of section 2. Again we allow the gas to expand slowly, but now we specify no heating by an external source of energy; rather, the gas is thermally isolated. The expansion is both slow and adiabatic. As the gas does work against the piston, the energy of the gas decreases and the temperature of the gas drops.

Does the entropy change? Section 2. Thus, for every small increment in volume, the entropy change is zero. Yet we know that an increase in volume implies an increase in the number of possible spatial locations for molecules and hence an increase in the spatial part of the multiplicity. How, then, can the entropy not change?

We must remember the momentum part of the multiplicity, namely, the idea that the molecules may have different directions and magnitudes for their momenta. During the adiabatic expansion, the total kinetic energy of the molecules decreases. The less energy, the fewer the ways to share it as kinetic energy of individual molecules. Therefore the momentum part of the multiplicity decreases. The preceding two paragraphs provide qualitative reasons for why—in a slow, adiabatic expansion—the spatial part of the multiplicity increases and the momentum part decreases.

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The effects operate in opposing directions. Only our use of equation 2. Initial Final Figure 2. In the ususal graphics, one draws the momentum vector of each molecule as emanating from the molecule's current spatial location. For the present purposes, move the tail of each momentum vector to a single, fixed location. Then, collectively, the momenta form a bristly object, like a sea urchin or a porcupine.

The slow, adiabatic expansion cools the gas and shrinks the bristly object, as illustrated here. In a significant sense, the momentum vectors occupy a smaller spherical volume in a "momentum space," and so the multiplicity associated with different arrangements of the momentum arrows decreases.

Sequence 1. Slow isothermal expansion. Detailed derivation produced the equation. Sequence 2. Extremely fast adiabatic expansion into a vacuum.

No heating: Sequence 3. Slow adiabatic expansion. Temperature drops: Can this be consistent with what we know? Lower energy and temperature imply smaller average molecular speed and hence a smaller value for the momentum part of the multiplicity.

That decrease compensates exactly for the increase in the spatial part of the multiplicity. Our route to full generalization proceeds in two stages.

First we learn how the entropy of a classical ideal gas changes when its temperature changes. Then we go on to entropy changes in much more general physical systems, not merely an ideal gas. From our analysis in section 2. How does the multiplicity depend on the temperature? We can reason as follows. The average kinetic energy of a gas atom is.

The important point is that the size of the range grows with temperature as T1'2. Now we reason by analogy with the spatial part of the multiplicity. For a cubical box of edge length L, the volume V is L3, and the spatial part of the multiplicity is proportional to L3 N. The sea urchin or porcupine diagrams with momentum arrows in figure 2. Each arrangement of momenta in the momentum space may be paired with each spatial arrangement of the atoms in the literal volume V Thus the spatial and momentum parts of the multiplicity combine as a product of two factors to give the full multiplicity.

Its numeri- cal value remains unknown to us, but we shall not need the value here; later, in section 5.

Next, suppose we heat the gas slowly and only a little bit but do so at constant volume; thereby we change the temperature by an amount AT. How much does the entropy change? In preparation, we expand the logarithm in 2. Then equations 2. Thus we find. A glance back at equation 2.

But our ultimate goal is to establish firmly that equation 2. Further generalization Our statement of the Second Law was restricted by the stipulation, "allow the system to evolve in isolation. To derive from our statement of the Second Law a consequence like that in equation 2.

Imagine removing some barrier that initially prevents change—such as permitting a small amount of the gases in the chemically reactive system to mix and to react. Simulta- neously, permit energy exchange by heating between the chemical system and the helium, a monatomic gas. Note that there is no chemical reaction with the helium. Then wait until things settle down. If there was real change in that finite time interval, our verbal form of the Second Law implies that the entropy of the combined system—the chemical system plus the helium—increases because the combined system evolves to a macrostate of larger multiplicity.

The chemically reactive system is in the small container which holds molecules of odd shapes , and it is the system of interest. The dashed outer wall filled with thermal insulation prevents any transfer of energy to the external world orfromit. You may wonder about the step from equation 2. The logarithm of a product equals the sum of the logarithms of the factors. Thus equation 2. The property that the entropy of a composite system is the sum of the entropies of macroscopic component subsystems is called the additivity of entropy.

Moreover, the helium remains close to thermal equilibrium. How is that energy related to "energy into the chemical system by heating," which we will denote by ql Energy conservation implies.

The minus sign is a nuisance to deal with, but thermodynamics is ultimately easier to comprehend if one sticks with "energy into. We use equation 2. Here is the evidence: Note that finite changes in a volume or in the number of molecules that have reacted chemically are permitted even in the limit of vanishing rate of change. One just has to wait a long time—in principle, infinitely long in the limit.

The limit of slow change is, of course, an idealization, but it is extremely fruitful. We incorporate the limit by generalizing equation 2.

Note that we used no special properties of the "chemical system;" it could be a dilute gaseous system or a dense one or a liquid or have some solids present. The change need not be a chemical reaction, but could be melting, say.

## Thermal Physics: Newtonian Dynamics, and Newton to Einstein: The Trail of Light

Hence one may generalize yet again, turning equation 2. Section 3. A few words should be said about the temperature T that appears in the denominator on the right-hand side of equation 2.

When the process is fast, however, the system of interest may be so far from equilibrium that we cannot ascribe a temperature to it. Nonetheless, the heating or cooling source has a well-defined temperature a supposition implicit in the derivation , and the symbol T refers to the source's temperature.

Indeed, in the derivation in this section, the symbol r appears first in equation 2. Our analysis led from the notion of multiplicity to a special role for the logarithm of a multiplicity more specifically, the logarithm times Boltzmann's constant.

That log- arithmic quantity we called "the entropy," and it has a perfectly clear meaning in terms of the fundamental notion of "the multiplicity of a macrostate. The words "order" and "disorder" are colloquial and qualitative; nonetheless, they describe a distinction that we are likely to recognize in concrete situations, such as the state of someone's room.

The connection with multiplicity becomes clear if we use the notion of "correlation" as a conceptual intermediary, as indicated in figure 2. Imagine a bedroom with the usual complement of shoes, socks, and T-shirts.

Suppose, further, that the room is one that we intuitively characterize as "orderly. If we see one clean T-shirt, then the others are in a stack just below it. There are strong spatial correlations between the shoes in a pair or the T- shirts on the dresser.

Those correlations limit severely the ways in which shoes and T- shirts can be distributed in the room, and so the objects exhibit a small multiplicity and a low entropy. Now take the other extreme, a bedroom that we immediately recognize as "dis- orderly. Under the. The double-headed arrow signifies that each of the notions implies the other.

Behind the bed? Lost in the pile of dirty T-shirts? And, for that matter, what a about the T-shirts? If we see a clean one on the dresser, the next clean one may be on the desk or in the easy chair. Correlations are absent, and the objects enjoy a large multiplicity of ways in which they may find themselves distributed around the room.

It is indeed a situation of high entropy. There is usually nothing wrong with referring to entropy as "a measure of disorder. To gain precision and something quantitative, one needs to connect "disorder" with "absence of correlations" and then with multiplicity. It is multiplicity that has sufficient precision to be calculated and to serve as the basis for a physical theory.

Wright issues strictures on the use of "disorder" to characterize entropy in his paper, "Entropy and disorder," Contemporary Physics, 11, Wright provides examples where an interpretation of entropy as disorder is difficult at best; most notable among the examples is crystallization from a thermally isolated solution when the crystallization is accompanied by a decrease in tempera- ture.

In a private communication, Dr. Wright cites a supersaturated solution of calcium butanoate in water and also sodium sulfate provided that the solid crystallizing out is anhydrous sodium sulfate. When a process occurs rapidly, how does one calculate a definite numerical value for the entropy change?

Our central equation, equation 2. Indeed, chapter 3 fills in some gaps, provides practice with entropy, and introduces several vital new ideas. We turn to those items now. The primary concept is the multiplicity of a macrostate: Entropy is basically the logarithm of the multiplicity: The major dynamical statement is the Second Law of Thermodynamics.

It can be formulated in terms of multiplicity or, equivalently, in terms of entropy.

## Thermal physics

The latter formulation is the following: The stipulation includes the injunction, do not transfer energy to or from the system. For all practical purposes, the one-line version of the Second Law is this: An isolated macroscopic system will evolve to the macrostate of largest entropy and will then remain there. The chapter's major equation connects a change in entropy with the energy input by heating and with the temperature:.

The temperature T on the right-hand side in item 4 refers to the temperature of the source of energy input by heating. When the equality sign holds, the system and the source have the same temperature or virtually so , and hence one need not draw a distinction about "whose temperature? The Second Law of Thermodynamics has prompted several equivalent formulations and a vast amount of controversy. A fine introduction to the debates in the late nineteenth century is provided by Stephen G.

Brush, Kinetic Theory, Vol. There one will find papers by Boltzmann, Poincare, and Zermelo, ably introduced and set into context by Brush. Rudolf Clausius followed a route to entropy quite different from our "atomistic" route based on the idea of multiplicity.

That was, of course, almost an historical necessity. In a clear fashion, William H. Cropper describes Clausius's analysis in "Rudolf Clausius and the road to entropy," Am. More about the etymology of the word "entropy" can be found in the note, "How entropy got its name," Ralph Baierlein, Am.

Problems Just as the scaffolding around a marble sculpture is removed when its function has been served, so it is best to put equation 2.

Computer simulation of macroscopic regularity. Imagine a volume V partitioned into 10 bins of equal volume. Use a computer's random number generator to toss molecules into the bins randomly. Also, compare results with those gotten by your classmates. Do you find macroscopic regularity emerging from these simulations of an ideal gas? Work out the analog of table 2. Draw a bar graph of multiplicity versus macrostate, the latter being specified by the number of balls in the right-hand bowl together with the total number of balls, N.

Using symmetry will expedite your work. Using the known total number of microstates which you can reason to be 2 6 provides either a check on your arithmetic or a way to skip one multiplicity computation.

Do you see the more-or-less even distribution growing in numerical significance? Let each region correspond to "a spatial location," in the sense used in section 2. A region may contain more than one molecule. Show the different spatial arrangements with sketches. Now you may skip the sketches—except as an aid to your thinking.

How, then, is the number of microstates for N molecules related to the volume— on the basis of your analysis here? The density of copper is 8. The energy needed to melt one gram of copper is joules.

You will need to estimate the volume of a penny. By what factor does the multiplicity of the copper change when the penny is melted? While the volume is kept fixed, the gas is gently heated to K. In this process, the entropy of the gas would not change. How can we understand this—that is, the "no change in entropy"—in terms of the various contributions to the total multiplicity?

On a cold winter day, a snowflake is placed in a large sealed jar in the sunshine the jar otherwise being full of dry air. There is no need—here—to concern yourself with the temperature of the sun's surface.

It takes 3, joules per gram to vaporize snow in this fashion. The physical system here is the water, existing first as ice and then as vapor. That is, give one or more qualitative reasons why a change in multiplicity should occur. Separated from the gas by a wall with a hole—initially closed—is a region of total vacuum. Here are some additional facts: When everything has settled down again, by what factor has the multiplicity changed? Briefly, why? Also, what is the change in entropy? Take the walls to be thermally insulating. Describe the process verbally and be quantitatively precise where- ever you can be. A cylindrical container of initial volume Vo contains N atoms of a classical ideal gas at room temperature: One end of the container is movable, and so we can compress the gas slowly, reducing the volume by 2 percent while keeping the temperature the same because the container's walls are in contact with the air in the room.

Provide first an algebraic expression and then a complete numerical evaluation. You do not have to do the four parts in the sequence in which the questions are posed, but the sequence a through d is a convenient route. For each part, be sure to provide a numerical answer. What happens for an intermediate process, specifically, for the following process? After the gaseous system has settled down, what is the size of A S relative to the entropy change in the entirely slow expansion discussed in section 2.

What is the energy input by heating relative to that for the slow expansion?

Do you find AS equal to "the energy input by heating divided by T7"? Should you expect to? Or what is the relationship?

In section 2. That was for 18 grams of ice. It takes 4. Can you think of a reason why? How does this change compare numerically with the entropy change in part a? Take the specific heat of water to have the constant value 4. Provide a qualitative explanation for the change that you calcu- late. The chapter begins with a classic topic: The topic remains relevant today—for environmental reasons, among others—and it also pro- vides the foundation for William Thomson's definition of absolute temperature, an item discussed later in chapter 4.

Next, the chapter develops a method for computing the entropy change when a process occurs rapidly. An extended example—the Otto cycle—and a discussion of "reversibility" conclude the chapter.

Engineers and environmentalists are interested in cyclic processes, because one can do them again and again. Figure 3. A simpler cycle, however, is more instructive theoretically, and so let us consider the cycle shown in figure 3.

We may take the substance to be some real gas, such as nitrogen as distinguished from an ideal gas. There are four stages, characterized as follows. Stage 1 to 2: From state 1 to state 2, the gas is allowed to expand while being maintained at constant, high temperature 7hot- The pressure drops because the volume increases. We denote by ghot the total amount of energy that is supplied by heating at temperature Thot- Stage 2 to 3: The pressure drops faster now as the volume increases; this is so because the temperature drops as the gas loses energy by doing work.

The gas is no longer heated. Containment structure Steam generator Steam line. Reactor vessel Condenser cooling water Pump. Nuclear fission heats water in the reactor vessel; that hot water, at temperature 7hOt, transfers energy to the liquid water in the central loop and produces steam. The steam, at high pressure, turns the blades of the turbine, doing work; that work is used to run the electrical generator and give electrical potential energy to electrons.

To ensure a difference of pressure across the turbine, the steam— after passing through the turbine—must be cooled and condensed back into liquid water.It would achieve the astronomer's dream of section 2.

Fromhold Jr. Both tradition and convenience suggest expressing Nk as the difference of two heat capacities; by 1. Recall that y, the ratio of heat capacities, is greater than 1, and so the exponent y — 1 is positive. For now, however, we may regard Tas simply what one gets by adding But, at other times, A may denote a large or finite change in some quantity.

Those expressions must be numerically equal, and so comparison implies 1. We now compress the gas at constant temperature rcoid. By the way, xenon atoms are not shaped like chocolate kisses; the conical appearance is an artifact of the technique.

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