K. P. N. Murthy† Lecture Notes on Thermal Physics for undergraduate students. †Adjunct Professor (1 August 2017 - 31 July 2018) Chennai Mathematical Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam, Chennai 603 103 Tamilnadu, INDIA. December 11, 2017 i Example isn’t another way to teach, it is the only way to teach. Albert Einstein Preface These notes are, almost, a verbatim reproduction of my lectures on thermal physics, to the undergraduate students of the Chennai Mathematical Institute, Chennai during August-November, 2017. Prior to this I have taught thermodynamics and statistical mechanics in the University of Hyderabad, Hyderabad from 2006-2015, ◦ the Rajeev Gandhi University for Knowledge Technology (RGUKT) (Video ◦ lectures) 2010, the School of Basic Sciences, Indian Institute of Technology ◦ Bhubaneswa during January - April 2017 and Chennai Mathematical Institute, ◦ in the years 2004 and 2005. While preparing this document, I have extensively drawn from the notes, assignment sheets, tutorial papers, and question papers generated and distributed to the students of these courses. Each chapter can be taught in some three to five hours. The material in the entire document can be covered comfortably in one semester - forty to forty five hours or so, including one problem - solving session every other week. A major problem of teaching thermal physics is not about what to teach; it is about what not to teach. Thermal Physics encompasses, the entire thermody- namics, and a good part of statistical mechanics. While making this statement, I am fully aware that strictly thermodynamics does not need any model or any assumptions about atoms and molecules - that make up matter. Thermodynamics does not require any help from statistical mechanics; nor does it require help from any other disciplines. Thermodynamics is a stand-alone subject, self-contained with a coherent struc- ture and inner consistency and with concepts that are well defined and well-knit. One can teach traditional thermodynamics following, for example the book of Weinreich1 or Callen2 without involving statistical mechanics, stochastic-cum- kinetic heat or atomic matter. But then in such an approach, a beginner shall most likely, face huge difficulties. Defining internal energy in terms of adiabatic work is a bit odd, to say the least. So is describing heat as difference between adiabatic and actual work; entropy would remain enigmatic when defined in the context of converting an inexact differential to an exact one; a beginner is likely get more puzzled than wise if thermodynamic is taught this way3. 1 Weinreich, Fundamental Thermodynamics, Addison-Wesley (1968) 2 H B Callen, Thermodynamics, John wiley (1960) 3 It calls for a certain maturity, a deep knowledge, and a love for historical and traditional approach, to appreciate, and enjoy thermodynamics in its pure form. iv Thermodynamics peppered with a bit of statistical mechanics, kinetic heat and atomic matter, is a different ball-game altogether. The subject wold become know- able, and easily at that; the subject would become transparent and even a bit more interesting. I would take the attitude that the notion of internal energy becomes transparent when it is taught invoking atoms, their properties and their interactions; entropy is best described and its tendency to increase, is best understood by invoking its statistical moorings. All said and done, is isn’t that a macroscopic system chooses that value of its macroscopic property, which is overwhelmingly most probable - the one that has maximum entropy ? Hence bringing in a little bit of statistical mechanics right at the beginning of teaching thermodynamics would be very helpful. But then we should be careful to keep it to a bare minimum, lest statistical mechanics should push thermodynamics to its appendix-pages. I think I have maintained this balance in these lecture notes. I have chosen to publish what I taught as lecture notes rather than a book. A book would demand a reasonably complete discussion of various issues and call for some serious efforts toward tying-up of all loose ends. Such problems are not there for lecture notes. You simply write up what you teach, and the way you teach; nothing more; and nothing less. Leave it to the readers to complete the narrative you have initiated in the lecture notes; and they can do it by consulting other books and articles. To facilitate such an enterprise I have listed several books and articles, at the end of the first chapter. The list, I must admit, is, by no means, exhaustive. I have picked up for listing, the books I have studied and the ones that caught my fancy. A reasonably good number of worked examples, and a set of practice problems, are included in separate sections. These are drawn mostly from other books. Where ever I remember, I have cited the sources. The emphasis is on conceptual issues and on learning thermal physics by exam- ples. I have made extensive use of toy-problems for illustrating important concepts. I hope these lecture notes on Thermal Physics make one more useful addition to your bookshelf. If you find any mistakes, or find any portions that lack clarity, and if you have suggestions to improve the readability, please tell me4. I shall make use of your comments and suggestions while preparing subsequent edition(s) : I am hoping that there would be a demand for, atleast, one more edition ! Chennai Mathematical Institute, K. P. N. Chennai, Tamilnadu, India December, 2017 4 at [email protected] Acknowledgements First and foremost, I thank all the students who put up with my idiosyncratic, disorganized, and completely indisciplined way of teaching and still managed, I hope, to learn. The discussions on and off the class hours helped me fine-tune my lectures and improve them from year to year. Next I must thank my teachers, a very large number of them, from whom I learned thermodynamics and statistical mechanics; these include Sushantha Dattagupta, V Balakrishnan, Subodh R Shenoy, V S S Sastry, G Ananthakrishna, Klaus W Kehr, M C Valsakumar, and S L Naraimhan, amongst several others. I thank the Director, CMI for giving an adjunct position which provided me with an academic ambiance, and infrastructure to teach and write up these notes for publication. Chennai Mathematical Institute, K. P. N. Chennai, Tamilnadu, India December, 2017 Contents Preface .................................................. iii Acknowledgements ........................................ v 1. LECTURE 1 ............................................ 1 OPENING REMARKS ............................................. 1 1.1 NatureofHeat..................................... ...... 1 1.2 NatureofWork ..................................... ..... 1 1.2.1 SignConvention forWorkDone ........................ 2 1.2.2 OtherKindsofWork ................................. 2 1.3 Equality of Temperature Thermal Equilibrium . 3 ⇒ Thermal Equilibrium : Equivalence Relation . 3 1.4 Extensive and Intensive Thermodynamic Properties . .... 4 1.4.1 Fundamental Equation.............................. .. 4 1.4.2 EquationsofState ................................. .. 4 BernouilliandtheIdealGas .............................. .. 4 1.5 ThenCameSadiCarnot ............................... .... 5 1.5.1 BirthofThermodynamics ............................. 6 1.6 EndRemarks....................................... ..... 6 1.7 Books ............................................ ...... 7 H C Van Ness, Understanding Thermodynamics ................ 7 H B Callen, Thermodynamics ............................... 7 H B Callen, Thermodynamics and an Introduction to Thermostatistics 8 G Weinreich, Fundamental Thermodynamics ................... 8 N D Hari Dass, Principles of Thermodynamis .................. 8 Evelyn Guha, Basic Thermodynamics ........................ 9 C B P Finn, Thermal Physics ............................... 9 Max Planck, Treatise on Thermodynamics ..................... 9 Enrico Fermi, Thermodynamics ............................. 9 J S Dudgale, Entropy and its Physical Meaning ................. 9 David Goodstein, States of Matter ........................... 9 Felix Rief, Fundamentals of Statistical and Thermal Physics ...... 9 Joon Chang Lee, Entropy and Free Energies ................... 10 James P Sethna, Entropy, Order Parameters and Complexity ..... 10 viii Contents C Kittel, and H Kr¨omer, Thermal Physics ..................... 10 Daniel V Schr¨oder, An Introduction to Thermal Physics .......... 10 M W Zamansky and R H Dittman, Heat and Thermodynamics, an Intermediate Textbook ................................ 10 R Shantini, Thermodynamics for Beginners .................... 10 Dilip Kondepudi and Ilya Prigogine, Modern Thermodynamics : From Heat Engines to Dissipative Structures .............. 10 1.8 ExtraReading:Books ................................ .... 11 1.9 ExtraReading:Papers............................... ..... 11 2. LECTURES 2 - 5 ....................................... 13 U AND S IN STATISTICAL MECHANICS AND THERMODYNAMICS ............ 13 2.1 OpeningRemarks .................................... .... 13 2.2 Internal Energy : Statistical Mechanics ................. ...... 13 2.2.1 Lennard-Jones 6-12 Potential ....................... ... 14 2.3 Internal Energy : Thermodynamics ..................... ..... 15 2.4 Entropy : Statistical Mechanics........................ ...... 18 2.4.1 Volume of an N-Dimensional Sphere .................... 19 2.4.2 An N( ) Dimensional Sphere Lives on its Outer Shell . 21 → ∞ 2.4.3 CountingofMicro States............................