Thermodynamics Henri J.F. Jansen Department of Physics Oregon State University December 14, 2003 II Contents 1 Basic Thermodynamics. 1 1.1 Introduction. 1 1.2 Some de¯nitions. 4 1.3 Zeroth Law of Thermodynamics. 9 1.4 First law: Energy. 9 1.5 Second law: Entropy. 13 1.6 Third law of thermodynamics. 25 1.7 Ideal gas and temperature. 26 1.8 Extra equations. 30 1.9 Problems for chapter 1 . 31 2 Thermodynamic potentials and 39 2.1 Internal energy. 39 2.2 Free energies. 45 2.3 Euler and Gibbs-Duhem relations. 49 2.4 Maxwell relations. 52 2.5 Response functions. 53 2.6 Relations between partial derivatives. 56 2.7 Conditions for equilibrium. 60 2.8 Stability requirements on other fee energies. 65 2.9 A magnetic puzzle. 67 2.10 Role of fluctuations. 71 2.11 Extra equations. 77 2.12 Problems for chapter 2 . 79 3 Phase transitions. 89 3.1 Phase diagrams. 89 3.2 Clausius-Clapeyron relation. 97 3.3 Multi-phase boundaries. 101 3.4 Binary phase diagram. 103 3.5 Van der Waals equation of state. 106 3.6 Spinodal decomposition. 116 3.7 Generalizations of the van der Waals equation. 120 3.8 Extra equations. 121 III IV CONTENTS 3.9 Problems for chapter 3 . 122 4 Landau-Ginzburg theory. 129 4.1 Introduction. 129 4.2 Order parameters. 131 4.3 Landau theory of phase transitions. 137 4.4 Case one: a second order phase transition. 139 4.5 Order of a phase transition. 144 4.6 Second case: ¯rst order transition. 145 4.7 Fluctuations and Ginzburg-Landau theory. 151 4.8 Extra equations. 158 4.9 Problems for chapter 4 . 159 5 Critical exponents. 165 5.1 Introduction. 165 5.2 Mean ¯eld theory. 170 5.3 Model free energy near a critical point. 178 5.4 Consequences of scaling. 182 5.5 Scaling of the pair correlation function. 187 5.6 Hyper-scaling. 188 5.7 Validity of Ginzburg-Landau theory. 189 5.8 Scaling of transport properties. 192 5.9 Extra equations. 195 5.10 Problems for chapter 5 . 196 6 Transport in Thermodynamics. 197 6.1 Introduction. 197 6.2 Some thermo-electric phenomena. 201 6.3 Non-equilibrium Thermodynamics. 204 6.4 Transport equations. 206 6.5 Macroscopic versus microscopic. 212 6.6 Thermo-electric e®ects. 219 A Questions submitted by students. 229 A.1 Questions for chapter 1. 229 A.2 Questions for chapter 2. 232 A.3 Questions for chapter 3. 235 A.4 Questions for chapter 4. 238 A.5 Questions for chapter 5. 240 B Summaries submitted by students. 243 B.1 Summaries for chapter 1. 243 B.2 Summaries for chapter 2. 245 B.3 Summaries for chapter 3. 247 B.4 Summaries for chapter 4. 248 B.5 Summaries for chapter 5. 250 CONTENTS V C Solutions to selected problems. 253 C.1 Solutions for chapter 1. 253 C.2 Solutions for chapter 2. 269 C.3 Solutions for chapter 3. 291 C.4 Solutions for chapter 4. 301 VI CONTENTS List of Figures 1.1 Carnot cycle in PV diagram . 15 1.2 Schematics of a Carnot engine. 16 1.3 Two engines feeding eachother. 17 1.4 Two Carnot engines in series. 20 2.1 Container with piston as internal divider. 41 2.2 Container where the internal degree of freedom becomes ex- ternal and hence can do work. 42 3.1 Model phase diagram for a simple model system. 92 3.2 Phase diagram for solid Ce. .................... 93 3.3 Gibbs energy across the phase boundary at constant temper- ature, wrong picture. ........................ 94 3.4 Gibbs energy across the phase boundary at constant temper- ature, correct picture. ....................... 94 3.5 Volume versus pressure at the phase transition. 95 3.6 Model phase diagram for a simple model system in V-T space. 96 3.7 Model phase diagram for a simple model system in p-V space. 96 3.8 Gibbs energy across the phase boundary at constant temper- ature for both phases. ....................... 98 3.9 Typical binary phase diagram with regions L=liquid, A(B)=B dissolved in A, and B(A)=A dissolved in B. 104 3.10 Solidi¯cation in a reversible process. 105 3.11 Typical binary phase diagram with intermediate compound AB, with regions L=liquid, A(B)=B dissolved in A, B(A)=A dissolved in B, and AB= AB with either A or B dissolved. 106 3.12 Typical binary phase diagram with intermediate compound AB, where the intermediate region is too small to discern from a line. ..............................107 3.13 Impossible binary phase diagram with intermediate compound AB, where the intermediate region is too small to discern from a line. ..............................107 3.14 Graphical solution of equation 3.21. 111 3.15 p-V curves in the van der Waals model. 113 VII VIII LIST OF FIGURES 3.16 p-V curves in the van der Waals model with negative values of the pressure. 113 3.17 p-V curve in the van der Waals model with areas correspond- ing to energies. 115 3.18 Unstable and meta-stable regions in the van der Waals p-V diagram. ...............................117 3.19 Energy versus volume showing that decomposition lowers the energy. ................................119 4.1 Heat capacity across the phase transition in the van der Waals model. .................................130 4.2 Heat capacity across the phase transition in an experiment. 130 4.3 Continuity of phase transition around critical point in p-T plane. .................................133 4.4 Continuity of phase around singular point. 134 4.5 Continuity of phase transition around critical point in H-T plane. .................................135 4.6 Magnetization versus temperature. 140 4.7 Forms of the Helmholtz free energy. 141 4.8 Entropy near the critical temperature. 142 4.9 Speci¯c heat near the critical temperature. 143 4.10 Magnetic susceptibility near the critical temperature. 144 4.11 Three possible forms of the Helmholtz free energy in case 2. 146 4.12 Values for m corresponding to a minimum in the free energy. 147 4.13 Magnetization as a function of temperature. 148 4.14 Hysteresis loop. 149 4.15 Critical behavior in ¯rst order phase transition. 151 6.1 Essential geometry of a thermocouple. 202 C.1 m versus T-H ............................311 C.2 Figure 1. ...............................314 INTRODUCTION IX Introduction. Thermodynamics??? Why? What? How? When? Where? Many questions to ask, so we will start with the ¯rst one. A frequent argument against studying thermodynamics is that we do not have to do this, since everything follows from statistical mechanics. In principle, this is, of course, true. This argument, how- ever, assumes that we know the exact description of a system on the microscopic scale, and that we can calculate the partition function. In practice, we can only calculate the partition function for a few simple cases, and in all other cases we need to make serious approximations. This is where thermodynamics plays an invaluable role. In thermodynamics we derive basic equations that all systems have to obey, and we derive these equations from a few basic principles. In this sense thermodynamics is a meta-theory, a theory of theories, very similar to a study of non-linear dynamics. Thermodynamics gives a framework for the results derived in statistical mechanics, and allows us to check if approximations made in statistical mechanical calculations violate some of these basic results. For example, if the calculated heat capacity in statistical mechanics is negative, we know we have a problem! Thermodynamics also gives us a language for the description of experimen- tal results. It de¯nes observable quantities, especially in the form of response functions. It gives the de¯nitions of critical exponents and transport properties. It allows analyzing experimental data in the framework of simple models, like equations of state. It provides a framework to organize experimental data. To say that we do not need this is quite arrogant, and assumes that if you can- not follow the (often very complicated) derivations in statistical mechanics you might as well give up. Thermodynamics is the meeting ground of experimenters and theorists. It gives the common language needed to connect experimental data and theoretical results. Classical mechanics has its limits of validity, and we need relativity and/or quantum mechanics to extend the domain of this theory. Thermodynamics and statistical mechanics do not have such a relation, though, contrary to what peo- ple claim who believe that we do not need thermodynamics. A prime example is the concept of entropy. Entropy is de¯ned as a measurable quantity in ther- modynamics, and the de¯nition relies both on the thermodynamic limit (a large system) and the existence of reservoirs (an even larger outside). We can also de¯ne entropy in statistical mechanics, but purists will only call this an entropy analogue. It is a good one, though, and it reproduces many of the well known results. The statistical mechanical de¯nition of entropy can also be applied to very small systems, and to the whole universe. But problems arise if we now also want to apply the second law of thermodynamics in these cases. Small system obey equations which are symmetric under time reversal, which contra- dicts the second law. Watch out for Maxwell's demons! On the large scale, the entropy of the universe is probably increasing (it is a very large system, and X INTRODUCTION by de¯nition isolated). But if the universe is in a well de¯ned quantum state, the entropy is and remains zero! These are very interesting questions, but for a di®erent course.