Statistical Mechanifesto or Funky Statistical Mechanics Concepts The Anti-Textbook* A Work In Progress. See elmichelsen.physics.ucsd.edu for the latest versions of the Funky Series. Please send me comments. Eric L. Michelsen h * Physical, conceptual, geometric, and pictorial physics that didn’t fit in your textbook. Please do NOT distribute this document. Instead, link to elmichelsen.physics.ucsd.edu/FunkyStatMechConcepts.pdf Please cite as: Michelsen, Eric L., Funky Statistical Mechanics Concepts, elmichelsen.physics.ucsd.edu, 10/10/2019. 2006 values from NIST. For more physical constants, see http://physics.nist.gov/cuu/Constants/ . Speed of light in vacuum c ≡ 299 792 458 m s–1 (exact) Boltzmann constant k = 1.380 6504(24) x 10–23 J K–1 = 8.61734 x 10–5 eV/K Stefan-Boltzmann constant σ = 5.670 400(40) x 10–8 W m–2 K–4 –6 Relative standard uncertainty ±7.0 x 10 23 –1 Avogadro constant NA, L = 6.022 141 79(30) x 10 mol Relative standard uncertainty ±5.0 x 10–8 Molar gas constant R = 8.314 472(15) J mol–1 K–1 = 0.0820575 L-atm/(mol-K) calorie 4.184 J (exact) –31 Electron mass me = 9.109 382 15(45) x 10 kg –27 Proton mass mp = 1.672 621 637(83) x 10 kg Proton/electron mass ratio mp/me = 1836.152 672 47(80) Atomic mass unit (amu) 1.660 538 86 10–27 kg Elementary charge e = 1.602 176 487(40) x 10–19 C Electron g-factor ge = –2.002 319 304 3622(15) Proton g-factor gp = 5.585 694 713(46) Neutron g-factor gN = –3.826 085 45(90) –28 Muon mass mμ = 1.883 531 30(11) x 10 kg Inverse fine structure constant α–1 = 137.035 999 679(94) Planck constant h = 6.626 068 96(33) x 10–34 J s Planck constant over 2π ħ = 1.054 571 628(53) x 10–34 J s –10 Bohr radius a0 = 0.529 177 208 59(36) x 10 m –26 –1 Bohr magneton μB = 927.400 915(23) x 10 J T Other values and conversions: 1 eV/particle = 96.485 kJ/mole ≈ 100 kJ/mole kiloton (of TNT) ≡ 4.184 x 1012 J = 1 Teracalorie bar ≡ 100,000 N/m2 atm ≡ 101,325 N/m2 = 1.013 25 bar torr ≡ 1/760 atm ≈ 133.322 N/m2 Typical chemical bond energy: 1-10 eV elmichelsen.physics.ucsd.edu Statistical Mechanifesto emichels at physics.ucsd.edu Contents 1 Introduction .............................................................................................................................. 5 Why Funky? ........................................................................................................................................ 5 How to Use This Document ................................................................................................................. 5 What’s wrong with existing SM expositions? .............................................................................. 5 My Story .............................................................................................................................................. 6 Thank You .................................................................................................................................... 6 Scope ................................................................................................................................................... 6 What This Text Covers ................................................................................................................. 6 What This Text Doesn’t Cover ..................................................................................................... 6 What You Already Know ............................................................................................................. 6 Notation ............................................................................................................................................... 6 2 Basic Concepts.......................................................................................................................... 9 Meet Mr. Mole ..................................................................................................................................... 9 What Is Everything? ...........................................................................................................................11 The Fundamental Postulate of Statistical Mechanics..........................................................................11 The Four Concepts of Statistical Mechanics .......................................................................................12 What Is ...? ..........................................................................................................................................12 What is a System?........................................................................................................................12 What is an Ensemble? ..................................................................................................................13 What is a State? ...........................................................................................................................13 What is an Energy Level? ............................................................................................................13 What is Heat? ..............................................................................................................................13 The Thermodynamic Limit .................................................................................................................14 That’s Intense-ive ...............................................................................................................................14 Don’t Be Dense...................................................................................................................................14 Energy Density of States Per Particle ..........................................................................................14 Energy Density of States For a System .......................................................................................15 Energy Density of States Per Unit Volume For a System ...........................................................15 Phase-Space Density of States .....................................................................................................16 μ-Space Particle Density ..............................................................................................................17 Impartial Derivatives and the Constants of Nature .............................................................................17 Statistics Refresher .............................................................................................................................18 The Nature of Nature ..........................................................................................................................19 Introduction to Entropy .......................................................................................................................19 Equilibrium, Temperature, and the One True Entropy .......................................................................20 General Equilibrium ....................................................................................................................25 Spontaneous Processes .......................................................................................................................25 Seeing Entropy ............................................................................................................................26 Nuts and Boltzmann: The Canonical Distribution ..............................................................................27 Multiplicity Defies Energy .................................................................................................................29 Example: The Statistical Chain ...........................................................................................................31 3 Classical Statistical Mechanics ............................................................................................. 35 Terms of Endearment .........................................................................................................................35 Free Energy .........................................................................................................................................36 Example of Free Energy: Water Absorbed in a Paper Towel ......................................................36 Physical View of Free Energy .....................................................................................................36 Helmholtz Free Energy ................................................................................................................37 What’s Free About Energy? ........................................................................................................38 Gibbs Free Energy and Enthalpy .................................................................................................39 Generalized Free Energy .............................................................................................................40 How Much Energy Is Multiplicity Worth? ..................................................................................41 Enthalpy ..............................................................................................................................................42 Schrodinger’s Cat Says “mu”: Chemical Potential .............................................................................43 10/10/2019 17:26 Copyright 2002 - 2019 Eric L. Michelsen. All rights reserved. Page 3 of 83 elmichelsen.physics.ucsd.edu Statistical Mechanifesto