Thermal Physics David Roundy Contents 1 Spring 2020 Physics 441 4 Remote learning . .4 Introduction and course philosophy . .5 Brief review of Energy and Entropy ...................................6 2 Week 1: Gibbs entropy approach 8 Microstates vs. macrostates . .8 Probabilities of microstates . .9 Energy as a constraint . .9 Lagrange multipliers (for those who are curious) . 11 Maximizing entropy . 12 Homework for week 1 (PDF) . 13 3 Week 2: Entropy and Temperature (K&K 2, Schroeder 6) 15 Quick version . 15 Multiplicity of a paramagnet . 16 Entropy of our spins . 18 Thermal contact (probably skipped in class) . 19 Homework for week 2 (PDF) . 21 4 Week 3: Boltzmann distribution and Helmholtz (K&K 3, Schroeder 6) 22 Spring 2020: We skipped this section last week, and will skip this section this week . 22 Internal energy . 23 Pressure . 24 Helmholtz free energy . 24 Using the free energy . 25 Ideal gas with just one atom . 26 Ideal gas with multiple atoms . 28 Homework for week 3 (PDF) . 30 5 Week 4: Thermal radiation and Planck distribution (K&K 4, Schroeder 7.4) 32 Harmonic oscillator . 32 Summing over microstates . 34 1 Black body radiation . 35 Low temperature heat capacity . 37 Homework for week 4 (PDF) . 38 6 Week 5: Chemical potential and Gibbs distribution (K&K 9, Schroeder 7.1) 40 Chemical potential . 40 Gibbs factor and sum . 42 Homework for week 5 (PDF) . 47 7 Week 6: Ideal gas (K&K 6, Schroeder 6.7) 49 Midterm on Monday . 49 Motivation . 49 Quantum mechanics and orbitals . 49 Fermi-Dirac distribution . 51 Bose-Einstein distribution . 52 Entropy................................................... 53 Classical ideal gas . 53 Homework for week 6 (PDF) . 55 8 Week 7: Fermi and Bose gases (K&K 7, Schroeder 7)) 57 Density of (orbital) states . 57 Finding the density of states . 58 Using the density of states . 59 Fermi gas at finite temperature . 60 Bose gas . 62 Homework for week 7 (PDF) . 63 9 Week 8: Work, heat, and cycles (K&K 8, Schroeder 4) 65 Heat and work . 65 Homework for week 8 (PDF) . 67 10 Week 9: Phase transformations (K&K 10, Schroeder 5.3) 69 Coexistence . 69 Clausius-Clapeyron . 70 van der Waals . 71 van der Waals and liquid-vapor phase transition . 74 Examples of phase transitions . 75 Landau theory . 76 Homework for week 9 (PDF) . 78 11 Review 81 Equations to remember . 81 Equations not to remember . 82 12 Solutions 84 Solution for week 1 . 84 Solution for week 2 . 88 2 Solution for week 3 . 90 Solution for week 4 . 98 Solution for week 5 . 102 Solution for week 6 . 106 Solution for week 7 . 112 Solution for week 8 . 116 Solution for week 9 . 120 3 Chapter 1 Spring 2020 Physics 441 Office hours MWTRF 11:00-11:30. Office hours Remote learning will be held via zoom (see the new canvas site) or slack. You may message me at any time, but Welcome to thermal physics! As most of you are likely I can’t guarantee that I’ll respond except during aware, OSU will be employing remote teaching for at scheduled office hours. least the first few weeks of spring term. We are still Syllabus The syllabus is here. in the process of determining exactly what the course Textbook An Introduction to Thermal Physics by will look like, and we will post updates as we learn Schroeder or Thermal Physics by Kittel and more. Until then, below are some general things that Kroemer. A textbook is not required, but ei- we will try to follow: ther of these textbooks can provide an additional • The course will be delivered remotely. You will resource for learning the course material. need access to a computer/tablet/phone with Course notes If you wish, you may download my Internet, and you will need to download OSU’s entire course notes as a lengthy PDF file. Note video software, Zoom. (You might want to start that I do edit my notes as the class progresses, practicing how to use Zoom now!) You can so you may want to re-download the notes after get help at https://learn.oregonstate.edu/keep- a while. learning). Homework Homework will be due via Gradescope on Wednesday of each week (but not the first • Homework will be turned in through Canvas via week of class). You should be able to start each Gradescope. homework the week before it is due. See the syllabus for details on homework grading. You • The course will be delivered synchronously. You may use the solutions that I provide (or any should plan to “attend” class at the scheduled other resource you wish to use) but at the end time: MWF 12-1. I try to record class sessions of each problem, please cite what resources you for those who cannot attend, but best to plan on used (students you worked with, whether you showing up. looked at the solutions, etc). Note that verbatim • The course will be interactive, to the extent that copying from any source is plagiarism, and is not is possible. You will be working in groups for a permitted. during the course using Zoom. • I have set up a slack channel, #ph441 for discus- sion related to this class. You are welcome to use this channel as you would have used Weniger 304F to discuss homework with each other, or to 4 ask questions about the class. 2. Physics lectures can be active engagement, pro- vided students do the math in real time as the lecture is given. This works well for some stu- Introduction and course philoso- dents (including most of us who are now faculty). We can hope that in the senior year you are able phy to follow a lecture in real time. This is your second course in thermal physics. Energy 3. Some things you can’t figure out for yourself, and and Entropy took a thermodyamics-first approach, you just need to be told. (Counter argument: with primary emphasis on how you could measure maybe you should read this content. but it’s something, and only later introducing how you could also true that some things you won’t understand predict it. I strongly support this approach, but it is the first time you encounter, so reading and being not the most common approach to thermal physics. told beats.