VALERIU FILIP INTRODUCTORY THERMAL PHYSICS Editura Universitatii din Bucuresti - 2006 - Preface Thermal Physics is one of the basic fields of study in the curricula of any sound training in Physical Sciences. Many of the concepts operating in this branch of Physics have also everyday homonyms. Therefore, it is always tempting for beginning students to overuse intuitive shortcuts in dealing with practical problems and to think with vulgar arguments. However, while intuition is definitely a basic tool for a physicist, it should clearly rely on a solid conceptual background. The present set of lectures aims to develop the basic ideas and concepts of Thermal Physics for a beginning reader, starting from the intuitive counterparts of terms and adding the necessary amount of rigor needed for both an effective use in solving the practical work and in preparing for more advanced approaches of the field. The presented material assumes some basic knowledge of elementary physics and mathematics, at the usual level of high-school curricula. Nevertheless, it is essentially self-contained and includes some mathematical interludes on basic calculus, linear differential forms and elementary probability theory. At the end of every chapter a specialized bibliography and a set of exercises are offered. The exercises are intended for immediate practice on the topics treated in the corresponding chapter and are of various degrees of difficulty. Answers are provided to all of them. The most difficult problems are fully solved and hints are offered for the ones of intermediate difficulty. The lectures, which are currently followed by the first year of English-speaking students at the Faculty of Physics of the University of Bucharest, can be profitably used by those studying any related technical field and even by interested terminal high-school students. The Author Contents Preface 3 1. Introduction 7 1.1 Macroscopic and microscopic theories 7 1.2 Measurable and hidden parameters 7 1.3 The main purpose of Thermal Physics 9 1.4 Some fundamental concepts of Thermodynamics 9 Bibliography 13 2. Zeroth Law of Thermodynamics. The concept of empirical temperature. 14 2.1 Zeroth Law 14 2.2 The empirical temperature 15 Bibliography 17 Exercises 18 3. The concept of internal energy and the First Law of Thermodynamics 20 3.1 Extensive and intensive parameters 20 3.2 Variations of state parameters 20 3.3 The concept of internal energy 21 3.4 The mechanical work 23 3.5 The chemical work 25 3.6 Constraints, envelopes and walls 26 3.7 The First Law of Thermodynamics 28 Bibliography 30 Exercises 30 4. Mathematical interlude 31 4.1 Linear differential forms 31 4.2 Holonomic LDFs 33 4.3 Integration of LDFs 35 4.4 Implicit dependence 37 Bibliography 39 Exercises 39 5. The restraint statement of The Second Law of Thermodynamics. The concept of entropy. 40 5.1 Preliminary concepts 40 5.2 The restraint statement of The Second Law of Thermodynamics 40 5.3 The theorem of Carnot 41 5.4 Some properties of the function f(TX1,TX2) 43 5.5 The absolute thermodynamic temperature 46 5.6 The relation of Clausius 48 5.7 The efficiency of a Carnot motor 51 5.8 The concept of entropy 52 5.9 Example of a non-thermodynamic system 54 Bibliography 57 Exercises 57 6. The general statement of The Second Law of Thermodynamics. The fundamental equation of Thermodynamics. The Third Law of Thermodynamics and the scaling of the entropy 61 6.1 The general statement of The Second Law of Thermodynamics 62 6.2 The fundamental equation of Thermodynamics 63 6.3 The entropy variation in a heating or cooling process through a pure thermal contact 64 6.4 The Third Law of Thermodynamics and the entropy scaling 66 Bibliography 68 Exercises 69 7. Statistical description of thermal phenomena 71 7.1 Elements of the probability theory and mathematical statistics 71 7.1.1 The idea of probability 71 7.1.2 The classical probability 75 7.1.3 The statistical probability 75 7.1.4 The probability of statistically independent events 76 7.1.5 Statistics and distributions 77 7.1.6 The normal distribution 81 7.1.7 Functions of random parameters 82 7.1.8 Associated random parameters 83 7.1.9 Associated independent random variables 86 7.1.10 Covariance of associated random variables 88 7.1.11 Example: Maxwell’s derivation of the distribution of gas molecules upon their translation velocities 89 7.2 The statistical viewpoint in Physics. 93 7.2.1 The space of microscopic states of a system of particles. 93 7.2.2 The space of microscopic states of a classical (non-quantum) system. 94 7.2.3 The quantum corrections to the density of states in classical (non-quantum) systems 97 7.2.4 The state space of weakly interacting systems 98 7.2.5 The occupation probabilitie 99 7.2.6 The occupation probability of a union of weakly interacting systems 103 7.3 The statistical energy 104 7.4 The statistical entropy 106 7.5 The fundamental laws of the Equilibrium Statistical Physics 109 Bibliography 111 Exercises 111 8. Equations of state 113 8.1 Caloric and thermal equations of state 113 8.2 Functional relations between the equations of state of a thermodynamic system 114 8.3 Examples of equations of state for thermodynamic systems 116 Bibliography 120 Exercises 120 9. Response functions of thermodynamic systems 122 9.1 Definitions 122 9.2 Expressing the thermal response functions through The First Law of Thermodynamics 124 9.3 Expressing the thermal response functions through The Second Law of Thermodynamics (as entropy derivatives) 125 9.4 Relations between the response functions 126 9.5 The construction of state equations starting from the measured response functions 130 Bibliography 131 Exercises 131 10. Some general consequences of The Laws of Thermodynamics 135 10.1 The Euler relation 135 10.2 The Gibbs – Duheim equation 138 10.3 The chemical potential of an ideal gas of photons 138 10.4 The chemical potential of a classical ideal gas 140 10.5 The entropy of a classical ideal gas 141 Bibliography 142 Exercises 142 11. Characteristic functions 143 11.1 The general extremum method of Equilibrium Thermodynamics 143 11.2 Examples of typical constraints for thermodynamic systems 144 11.3 Natural variables of the characteristic functions and the equations of state 148 11.4 Mnemonic diagram 154 11.5 Simple applications of the extremum method: conditions for relative equilibrium 155 Bibliography 156 Exercises 156 12. The statistical approach in Equilibrium Thermodynamics 159 12.1 Recall of the probability principle 159 12.2 Obtaining thermodynamic information from the distribution function of a given system 160 12.3 Particular case: system of non-interacting particles 164 12.4 Application: statistical derivation of the equations of state of a mass-points classical ideal gas 166 12.5 Maxwell distribution of a gas molecules upon their translation velocities 168 12.6 Classical ideal gas in a non-homogeneous external potential: the barometric formula 170 12.7 System with interacting particles. The Van der Waals equation of state. 172 12.8 More insight into the classical ideal gas: the molecular currents 175 Appendix 12.1: Statistical averages of some important one-particle quantities 179 Appendix 12.2: Stirling's approximation 180 Bibliography 181 Exercises 181 13. Fluctuations and stability of the equilibrium states of thermodynamic systems 194 13.1 The stability of the equilibrium state of a thermodynamic system 195 13.2 The statistical theory of fluctuations in a thermodynamic system 198 Bibliography 202 Exercises 203 14. Equilibrium between phases 204 14.1 The equilibrium between two phases of a substance. The Clausius-Clapeyron equation. 204 14.2 Number of phases of a mixture of several components: the Gibbs phase rule 207 Bibliography 208 Exercises 208 7 1. Introduction 1.1 Macroscopic and microscopic theories Physics is a science dealing with experimental facts that is with events which can be systematically reproduced by preparing each time analogous environmental conditions. The experimental facts are further organized in various theories attempting to fit our mental reasoning on the obtained experimental results. Physical theories can belong essentially to two important categories: microscopic or macroscopic. Generally speaking, a macroscopic theory makes use of a restraint set of parameters, namely the ones which are more or less available to our direct senses. On the contrary, a microscopic theory may involve quite abstract representations (never met in our current sensitive experience) and therefore needs a much higher number of physical parameters. Usually it is useful to attach both a macroscopic and a microscopic point of view to every field of physics. One particularly striking example in this sense is The Physics of Thermal Phenomena: it can be described both macroscopically, in the so-called field of Thermodynamics, and microscopically in Statistical Physics. Those two different theories are actually not completely distinct since they share the same starting points and the same purposes. There are also several shared physical parameters and concepts, which allow a close interaction between the two "thermal" theories. 1.2 Measurable and hidden parameters It is now a very well established fact that every macroscopic body (that is a body whose presence may be evidenced through usual senses) is actually a system of a huge number (of the order of 1023) of interactive microscopic units or particles (upon the case, these may be atoms, molecules, electrons, even virtual particles like phonons, which result from the quantization of the atomic lattice vibrations in a crystal lattice, or photons, that is confined electromagnetic field particles). These units are far from being still: they are in a permanent, rapid and apparently disordered motion.