Introduction to Thermodynamics

CHAPTER 1 INTRODUCTION 1.1 What is thermodynamics? Thermodynamics is the science which has evolved from the original investiga- tions in the 19th century into the nature of “heat.” At the time, the leading theory of heat was that it was a type of fluid, which could flow from a hot body to a colder one when they were brought into contact. We now know that what was then called “heat” is not a fluid, but is actually a form of energy – it is the energy associated with the continual, random motion of the atoms which compose macroscopic matter, which we can’t see directly. This type of energy, which we will call thermal energy, can be converted (atleastinpart)tootherformswhichwecan perceive directly (for example, kinetic, gravitational, or electrical energy), and which can be used to do useful things such as propel an automobile or a 747. The principles of thermodynamics govern the conversion of thermal energy to other, more useful forms. For example, an automobile engine can be though of as a device which first converts chemical energy stored in fuel and oxygen molecules into thermal en- ergy by combustion, and then extracts part of that thermal energy to perform the work necessary to propel the car forward, overcoming friction. Thermody- namics is critical to all steps in this process (including determining the level of pollutants emitted), and a careful thermodynamic analysis is required for the design of fuel-efficient, low-polluting automobile engines. In general, thermody- namics plays a vital role in the design of any engine or power-generating plant, and therefore a good grounding in thermodynamics is required for much work in engineering. If thermodynamics only governed the behavior of engines, it would probably be the most economically important of all sciences, but it is much more than that. Since the chemical and physical state of matter depends strongly on how much thermal energy it contains, thermodynamic principles play a central role in any description of the properties of matter. For example, thermodynamics allows us to understand why matter appears in different phases (solid, liquid, or gaseous), and under what conditions one phase will transform to another. 1 CHAPTER 1. INTRODUCTION 2 The composition of a chemically-reacting mixture which is given enough time to come to “equilibrium” is also fully determined by thermodynamic principles (even though thermodynamics alone can’t tell us how fast it will get there). For these reasons, thermodynamics lies at the heart of materials science, chemistry, and biology. Thermodynamics in its original form (now known as classical thermodynam- ics) is a theory which is based on a set of postulates about how macroscopic matter behaves. This theory was developed in the 19th century, before the atomic nature of matter was accepted, and it makes no reference to atoms. The postulates (the most important of which are energy conservation and the impos- sibility of complete conversion of heat to useful work) can’t be derived within the context of classical, macroscopic physics, but if one accepts them, a very powerful theory results, with predictions fully in agreement with experiment. When at the end of the 19th century it finally became clear that matter was composed of atoms, the physicist Ludwig Boltzmann showed that the postu- lates of classical thermodynamics emerged naturally from consideration of the microscopic atomic motion. The key was to give up trying to track the atoms in- dividually and instead take a statistical, probabilistic approach, averaging over the behavior of a large number of atoms. Thus, the very successful postulates of classical thermodynamics were given a firm physical foundation. The science of statistical mechanics begun by Boltzmann encompasses everything in classical thermodynamics, but can do more also. When Introduction to Thermodynamics Introduction to Thermodynamics Bởi: OpenStaxCollege A steam engine uses heat transfer to work Tourists regularly ride this narrow-gauge steam engine train near the San Juan Skyway in Durango, Colorado, part of the National Scenic Byways Program (credit: Dennis Adams) Heat transfer is energy in transit, and it can be used to work It can also be converted to any other form of energy A car engine, for example, burns fuel for heat transfer into a gas Work is done by the gas as it exerts a force through a distance, converting its energy into a variety of other forms—into the car’s kinetic or gravitational potential energy; into electrical energy to run the spark plugs, radio, and lights; and back into stored energy in the car’s battery But most of the heat transfer produced from burning fuel in the engine does not work on the gas Rather, the energy is released into the environment, implying that the engine is quite inefficient It is often said that modern gasoline engines cannot be made to be significantly more efficient We hear the same about heat transfer to electrical energy in large power stations, whether they are coal, oil, natural gas, or nuclear powered Why is that the case? Is the inefficiency caused by design problems that could be solved with better engineering and superior materials? Is it part of some money-making conspiracy by those who sell energy? Actually, the truth is more interesting, and reveals much about the nature of heat transfer 1/2 Introduction to Thermodynamics Basic physical laws govern how heat transfer for doing work takes place and place insurmountable limits onto its efficiency This chapter will explore these laws as well as many applications and concepts associated with them These topics are part of thermodynamics—the study of heat transfer and its relationship to doing work 2/2 Eyal Buks Introduction to Thermodynamics and Statistical Physics (114016) - Lecture Notes April 13, 2011 Technion Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Preface to be written Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Con tents 1. The Principle of Larg est Uncertain ty 1 1.1 EntropyinInformationTheory 1 1.1.1 Example- TwoStatesSystem 1 1.1.2 SmallestandLargest Entropy 2 1.1.3 Thecompositionproperty 5 1.1.4 Alternative Definitionof Entropy 8 1.2 LargestUncertaintyEstimator 9 1.2.1 Useful Relations 11 1.2.2 TheFreeEntropy 13 1.3 The Principle of Largest Uncertainty in Statistical Mechanics 14 1.3.1 Microcanonical Distribution 14 1.3.2 CanonicalDistribution 15 1.3.3 Grandcanonical Distribution 16 1.3.4 TemperatureandChemicalPotential 17 1.4 Time Evolutionof EntropyofanIsolatedSystem 18 1.5 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.1 ExternallyAppliedPotential Energy 20 1.6 FreeEntropyandFreeEnergies 21 1.7 ProblemsSet1 21 1.8 SolutionsSet1 29 2. Ideal Gas 45 2.1 AParticleinaBox 45 2.2 GibbsParadox 48 2.3 FermionsandBosons 50 2.3.1 Fermi-DiracDistribution 51 2.3.2 Bose-EinsteinDistribution 52 2.3.3 ClassicalLimit 52 2.4 IdealGasintheClassicalLimit 53 2.4.1 Pressure 55 2.4.2 Useful Relations 56 2.4.3 HeatCapacity 57 2.4.4 InternalDegrees of Freedom 57 2.5 ProcessesinIdealGas 60 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Contents 2.5.1 IsothermalProcess 62 2.5.2 IsobaricProcess 62 2.5.3 IsochoricProcess 63 2.5.4 Isentropic Process 63 2.6 CarnotHeatEngine 64 2.7 Limits Imposed Upon the Efficiency 66 2.8 ProblemsSet2 71 2.9 SolutionsSet2 79 3. Bosonic and Fermionic Systems 97 3.1 ElectromagneticRadiation 97 3.1.1 ElectromagneticCavity 97 3.1.2 PartitionFunction 100 3.1.3 CubeCavity 100 3.1.4 AverageEnergy 102 3.1.5 Stefan-Boltzmann Radiation Law 103 3.2 PhononsinSolids 105 3.2.1 OneDimensionalExample 105 3.2.2 The3DCase 107 3.3 FermiGas 110 3.3.1 OrbitalPartitionFunction 110 3.3.2 PartitionFunctionof theGas 110 3.3.3 EnergyandNumberofParticles 112 3.3.4 Example: ElectronsinMetal 112 3.4 Semiconductor Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.5 ProblemsSet3 115 3.6 SolutionsSet3 117 4. Classical Limit of Statistical M echanics 127 4.1 ClassicalHamiltonian 127 4.1.1 Hamilton-Jacobi Equations 128 4.1.2 Example 128 4.1.3 Example 129 4.2 Density Function 130 4.2.1 EquipartitionTheorem 130 4.2.2 Example 131 4.3 NyquistNoise 132 4.4 ProblemsSet4 136 4.5 SolutionsSet4 138 5. Exam Wint er 2010 A 147 5.1 Problems 147 5.2 Solutions 148 Eyal Buks Thermodynamics and Statistical Physics 6 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Con tents 6. Exam Wint er 2010 B 155 6.1 Problems 155 6.2 Solutions 156 References 163 Index 165 Eyal Buks Thermodynamics and Statistical Physics 7 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 1. The Principle of Largest Uncertainty In this chapter w e discuss relations between information theory and statistical mechanics. We show that the canonical and grand canonical distributions can be obtained from Shannon’s principle of maximum uncertainty [1, 2, 3]. Moreover, the tim e evolution of the entropy of an isolated system and the H theorem are discussed. 1.1 Entropy in Information Theory The possible states of a given system are denoted as e m ,wherem =1, 2, 3, , and the An Introduction to Thermodynamics and Statistical Mechanics Second Edition Keith Stowe California Polytechnic State University ,:~ CAMBRIDGE UNIVERSITY PRESS Contents Preface List of constants, conversions, and prefixes Part I Setting the scene 1 Introduction Part 11 Small systems 2 Statistics for small systems 3 Systems with many elements Part III Energy and the first law 4 Internal energy 5 Interactions between systems Part IV States and the second law 6 Internal energy and the number of accessible states 7 Entropy and the second law 8 Entropy and thermal interactions Part V Constraints 9 Natural constraints 10 Models 11 Choice of variables 12 Special processes 13 Engines 14 Diffusive interactions Part VI Classical statistics 15 Probabilities and microscopic behaviors 16 Kinetic theory and transport processes in gases 17 Magnetic properties of materials 18 The partition function page vii xii 1 3 23 25 40 63 65 79 99 101 117 135 153- 155 186 210 226 252 287 327 329 352 369 382 v vi Contents Part VII Quantum statistics 399 19 Introduction to quantum statistics 401 20 Quantum gases 422 21 Blackbody radiation 438 22 The thermal properties of solids 457 23 The electrical properties of materials 477 24 Low temperatures and degenerate systems 504 Appendices 531 Further reading 537 Problem solutions 538 Index 551 Index absolute zero approaching, 506-510 behaviors near, 175-176 acceptors, 483,488 activation energy, 297 adiabatic demagnetization, 508-509 adiabatic processes, 229-232 in ideal gas, 231-232 in photon gas, 449 temperature changes, 229 air, liquefaction of, 312-314 alloys, 314-315 angular momentum, 9,11-14,531 atomic magnets, 371-372 atomic vibrations, 457 average molecular speed, 355 band-edge equivalent states, 485-486 band structure and width, 477-479 bands density of states, 486 in divalent metals, 480 overlapping, 479 unfilled, 479-481 valence and conduction, 192,477-479 Bardeen-Cooper-Schrieffer (BCS) theory distribution, 518 bell curve, see Gaussian beta (Ilk]), 332 bias, forward and reverse, 497 Big Bang, radiation from, 441,443 binary mixtures, 308-316 binomial expansion, 29,42 black holes, 520 blackbody emissivity, 442 blackbody radiation, 438-449 distribution, 439 energy density, 439-440 energy flux, 441-444 Bohrmagneton, 371 boiler, 273 Boltzmann statistics, 333 Boltzmann's constant, 72, 126, 138 Bose-Einstein condensation, 512-513 for fermion pairs, 517-518 Bose-Einstein statistics, 403, 423 bosons, 13 degenerate, 176,394,413,429,510, 512-519 occupation number, 403,423-425 relativistic and nonrelativistic, 403, 423-425 bound states, 15-17 Brownian motion, 342 canonical ensemble, 330 Carnot, Sadi, 265 Carnot cycle, 265 efficiency, 266, 267 engine, 265-267 catalysts, 297 chain rule, 215 charge carriers, 483 mobility, 484 thermal excitation, 482 chemical equilibrium, 295-297 chemical potential, 81,83-84, 137, 288-291 and concentration, 288, 290 and heat released, 83-84 and nunlber of particles and osmosis, 293-294 and particle distributions, 84-86 and phase space, 84 and potential energy, 288, 290 and the second law, 288 at all temperatures, 431-433 calculation of, 391-392,411-414 classical limit, 431 dependence on T,p, 168,288 from partition function, 388,394, 406-407,413 in low- and high-density limits, 394, 412-414 of degenerate systems, 401,413-414, 429,430 bosons, 429, 513 fermions, 429, 430 of nearly degenerate fermions, 432-433 chemical reactions, heat transfer, 144 classical1irnit, 431 classical probabilities, 333-334,336-337 classical statistics, 329-342 examples, 336-337 limits of, 386,409 needed ingredients, 402,413 when to use, 333 Clausius-Clapeyron equation, 299-300 coefficient of performance, 258 coefficient of utility, 267 cold packs, 309 7/10/07 Intro- 1 An Introduction to Thermodynamics Classical thermodynamics deals with the flow of energy under conditions of equilibrium or near-equilibrium and with the associated properties of the equilibrium states of matter. It is a macroscopic theory, ignoring completely the details of atomic and molecular structure, though not the existence of atoms and molecules to the extent required for writing chemical reactions. Time is not recognized as a variable and cannot appear in thermodynamic equations. For students who have become familiar with atoms and molecules, it may be surprising to find how far one can go toward treating chemical and physical equilibria without employing any simplified models or delving into theories of molecular structure. The detachment of thermodynamics from molecular theory is an important asset. The fundamental principles of thermodynamics were developed during the 19 century on the th foundation of two principal axioms, supplemented by a small number of definitions, long before atomic structure was understood. Because of this lack of dependence of theory on models, even today we need not worry about our vast ignorance at the molecular level, especially in the areas of liquids and ionic solutions, in applying thermodynamics to real systems. It has been said, with some justification, that if you can prove something by thermodynamics you need not do the experiment. Such a strong statement must be handled with care, but it should become clear in the following pages that common practice is quite consistent with this assumption. Two developments associated primarily with the 20 century introduced substantial new th insights into thermodynamics. Statistical thermodynamics, or statistical physics, originated with the efforts of Maxwell and of Boltzmann in the late 19 century and grew with additions by th Gibbs, Planck, Einstein, and many others into a companion science to thermodynamics. Because statistical thermodynamics relies on specific models of atomic and molecular structure and interactions, it provides important tests of those models, at the expense of substantially greater mathematical complexity than classical thermodynamics. More important for present purposes, statistical mechanics provides much greater insight into the quantities that appear in thermodynamic equations, and thus a clearer view of why things happen. Thus we will not hesitate to introduce some basic principles of statistical mechanics (without the extensive mathematics) when necessary to explain what is going on. The other new development, largely responsible for the change in physics from what is generally considered purely Newtonian to relativistic and quantum physics, arose from the introduction of operational definitions at the end of the 19 century. This viewpoint requires that th any definition (of energy, position, or time, for example) must include a statement of how we can measure the quantity. Application of this criterion demands clarification of some quantities that were introduced casually, without a solid foundation, in the early days of thermodynamics. We will try to be more careful in explaining what is meant by our symbols, and what can or cannot be measured, than has been customary in thermodynamic textbooks. One of the characteristics of thermodynamics is that most of the terms are familiar. Everyone has heard of energy, of heat, and of work. The difficulty is that we must sharpen our definitions to distinguish between loosely associated ideas. We will therefore be particularly careful to define these familiar quantities carefully, often emphasizing what our technical meanings do not include as much as specifying the intended meanings. These are also called vacuum flasks, because the space between the silvered double walls 1 has been evacuated, a design developed by Sir James Dewar. Another common name for these and for containers of different design but for the same purpose is “Thermos” bottle, which is This scale was formerly called “centigrade” in English-speaking countries, a name that 5 can be confused with 1/100 degree in other languages. 7/10/07 1- 14 temperature on the Celsius scale. Temperature differences are the same for the two scales. With 5 the appropriate numerical substitutions, ∆E = 2 mol x 21 J/mol·K x (323 - 298) K = 1.05 kJ/mol This is the amount of (thermal) energy that must be added to the gas to raise its temperature 25 C. o IDEAL-GAS EXPANSIONS. Now let 2 mol of hydrogen, at 3 atm pressure, expand at a constant temperature of 50 C through a pinhole (to maintain a slow expansion) and against a piston. o Assume the external pressure acting on the piston is 1 atm, as in Figure 1. The work done on the gas is the product of the force exerted on the gas (by the piston) and the distance through which the piston moves, with a negative sign because the pressure (of the gas) acts opposite to the pressure acting on the gas. The force, on the gas, times the displacement, is f dx = - (PA) dx = - P (Adx) = - P dV. Because of the unusual arrangement (effusion controlled), the pressure acting on the gas at the point of displacement is the external pressure, P . Therefore the work done on ext the gas is (Effusion controlled) W = I f dx = - I P dV (9) ext Because P is maintained constant, it may be removed from the integral. ext W = - P I dV = - P ∆V (9a) ext ext 3 5 1 m 0.0530 Pa 10 x 013 . 1 K 323K x J/mol .3148 x mol 2 = ⋅ =V 3 5 2 m 0.01767 Pa 10 x 1.013 x 3 K 323K x J/mol .3148 x mol 2 = ⋅ =V Kmol atmml 06.82 Kmol cal 987.1 Kmol J 314.8 ⋅ ⋅ = ⋅ = ⋅ =R The term “ideal” means simply that the substance obeys a certain equation. An ideal gas 6 obeys the equation PV = nRT; in later chapters we will encounter the “ideal solution”, which obeys an equation known as Raoult’s law. The ideal-gas equation combines Boyle’s law and Charles’, or Gay-Lussac’s, law into a single, more convenient expression. The temperature must be on an absolute scale, which we will always take as the Kelvin scale; n is the number of moles of gas; and R is a universal constant, whose value depends on the units chosen for pressure and volume. It should be noted that the product of pressure and volume has the dimensions and units of energy (although PV is not a measure of energy). Such gases as He, H , O , and N closely 2 2 2 follow the ideal-gas equation at room temperature; such easily condensable gases as CO or H O 2 2 vapor follow the equation less closely. 7/10/07 1- 15 The initial and final volumes can be calculated from the ideal-gas equation , PV = nRT. 6 Thus the work is W = - P ∆V = - P (V - V ) ext ext f i = - 1.013 x 10 Pa (0.0530 - 0.01767) m = - 3.58 kJ 5 3 The necessary constants are given in Table 1. Table 1 GAS CONSTANT AND CONVERSION FACTORS 1 joule = 1 newton·meter = 0.239 cal = 1 pascal·meter 3 1 cal (thermochemical) = 4.1840 joule = 41.3 ml·atm * 1 ml·atm = 0.1013 joule = 0.0242 cal 1 atm = 1.01325 x 10 N/m 5 2 * Various experimentally defined values of the calorie appear in the literature (including the dietician's Calorie = 1 kcal). Only the thermochemical calorie is defined exactly. In order that the temperature may remain constant it is necessary to supply thermal energy to the gas to compensate for the energy expended in doing work. From the first-law equation, 7/10/07 1- 16 Q = ∆E - W For the special case of an ideal gas, the energy depends only on the temperature, not on the pressure or volume. At constant temperature, therefore, ∆E = 0 and the thermal energy that must be supplied is Q = - W = 3.58 kJ A more important example than the expansion against a constant external pressure is the “reversible” and isothermal (constant temperature) expansion. A process is said to be thermodynamically reversible if it can be reversed at any stage by an infinitesimal increase in the opposing force or an infinitesimal decrease in the driving force. It should be clear that such a reversible process is an .. .Introduction to Thermodynamics Basic physical laws govern how heat transfer for doing work takes place and place insurmountable limits onto its efficiency This chapter... many applications and concepts associated with them These topics are part of thermodynamics the study of heat transfer and its relationship to doing work 2/2