Thermodynamics of Materials

314: Thermodynamics of Materials L. Lauhon, K. R. Shull March 24, 2021 Contents 1 314: Thermodynamics of Materials4 2 Introduction5 3 Big Ideas of Thermo5 3.1 Classification of Systems....................... 5 3.2 State Variables............................. 7 3.3 Process Variables ........................... 7 3.3.1 Work Done by an External Force.............. 8 3.3.2 Work Done by an Applied Pressure ............ 9 3.4 Extensive and Intensive Props. ................... 9 4 The Laws of Thermodynamics 11 4.1 First Law................................ 11 4.2 Second Law .............................. 12 4.2.1 Reversible and Irreversible Processes ........... 13 4.2.2 Entropy and Heat....................... 13 4.3 Third Law ............................... 14 4.4 Differential Quantities ........................ 15 5 Thermodynamic Variables 17 5.1 Enthalpy................................ 17 5.2 Helmholtz Free Energy ....................... 18 5.3 Gibbs Free Energy .......................... 18 5.4 Other Material Properties ...................... 19 5.4.1 Volume thermal expansion coefficient........... 19 5.4.2 Isothermal Compressibility................. 19 5.4.3 Heat Capacity......................... 19 5.5 Coefficient Relations ......................... 20 5.5.1 Internal Energy........................ 23 1 CONTENTS CONTENTS 5.5.2 Enthalpy............................ 23 5.5.3 Helmholtz Free Energy ................... 23 5.5.4 Gibbs Free Energy ...................... 23 5.6 Equation of State ........................... 23 5.6.1 Expression for dS ....................... 23 5.6.2 Expression for dV ....................... 24 5.6.3 Relationship between CP and CV . 24 6 The Ideal Gas 25 6.1 Thermal Expansion Coefficient ................... 25 6.2 Compressibility............................ 26 6.3 Internal Energy ............................ 26 6.4 Specific Heat.............................. 27 6.5 Examples................................ 27 6.5.1 Free expansion of an ideal gas ............... 27 6.5.2 Isothermal Compression of an Ideal Gas.......... 29 6.5.3 Reversibility requires work/heat ............. 29 6.5.4 Some useful relations for heat flow............. 31 7 Solids and Liquids 34 8 Conditions for Equilibrium 40 8.1 Finding equilibrium conditions................... 43 8.2 Energy Minima ............................ 44 8.3 Maxwell Relations .......................... 46 9 From Microstates to Entropy 47 9.1 Microstates and Macrostates..................... 48 9.1.1 Postulate of Equal Likelihood ............... 50 9.1.2 The Boltzmann Hypothesis (to be derived) . 51 9.2 Conditions for equilibrium ..................... 51 10 Noninteracting (Ideal) Gas 54 10.1 Ideal gas Law ............................. 55 11 Diatomic Gases 56 12 Einstein Model 57 12.1 Thermodynamic Properties ..................... 57 13 Unary Heterogeneous Systems 59 13.1 Phase Diagrams............................ 60 13.2 Chemical Pot. Surfaces........................ 60 13.3 µ(T;P ) Surfaces............................ 63 13.4 Clasius-Clapeyron Eq. ........................ 64 13.4.1 Step 1: Find ∆H(T;P ) .................... 65 2 CONTENTS CONTENTS 13.4.2 Step 2: Find ∆V α!β(T;P ) ≡ V β(P; T ) − V α(P; T ) . 66 13.4.3 Justification for Approximations.............. 67 13.4.4 Trouton’s Rule......................... 67 14 Open Multicomponent Systems 69 14.1 Partial Molal Properties ....................... 69 14.2 Gibbs-Duhem Eq............................ 70 14.3 The Mixing Process.......................... 71 14.4 Graphical Determination....................... 73 14.5 Relationships ............................. 74 14.6 Chemical Potential .......................... 75 14.7 Activity................................. 75 14.8 Ideal Solutions............................. 76 14.9 Dilute Solutions............................ 77 14.10Excess Properties ........................... 79 14.11Regular Solutions........................... 79 14.11.1 Mixtures of Real Gases: Fugacity.............. 83 15 Multicomponent Heterogeneous Systems 85 15.1 Equilibrium .............................. 86 15.2 Gibbs Phase Rule........................... 86 16 Phase Diagrams 87 16.1 Unary Phase Diagrams........................ 88 16.2 Alternative Representations..................... 88 16.3 Binary Phase Diagrams........................ 89 16.4 Alternative Representations: T vs. X................ 89 16.5 Interpreting Phase Diagrams: Tie Lines .............. 90 16.6 The Lever Rule ............................ 91 16.7 Applications of Phase Diagrams: Microstructure Evolution . 91 16.8 Review................................. 92 16.9 Types of α − L phase diagrams................... 93 17 Phase Diagrams 93 17.1 Calculation of ∆Gmix ......................... 94 17.2 Common Tangent Construction................... 95 17.3 A Monotonic Two-Phase Field.................... 97 17.4 Non-monotonic Two-Phase Field.................. 97 17.5 Regular Solutions........................... 98 17.6 Misciblity Gap............................. 98 17.7 Spinodal Decomposition....................... 99 18 Summary of 314 102 18.1 Chapters 2 and 3 ........................... 102 18.2 Chapter 4................................ 102 18.3 Chapter 5................................ 103 3 1 314: THERMODYNAMICS OF MATERIALS 18.4 Chapter 6................................ 103 18.5 Chapter 7................................ 103 18.6 Chapter 8................................ 104 18.7 Chapter 9................................ 104 18.8 Chapter 10............................... 105 19 314 Problems 106 19.1 Phases and Components....................... 106 19.2 Intensive and Extensive Properties . 106 19.3 Differential Quantities and State Functions . 107 19.4 Entropy................................. 107 19.5 Thermodynamic Data ........................ 108 19.6 Temperature Equilibration...................... 111 19.7 Statistical Thermodynamics..................... 111 19.8 Single Component Thermodynamics . 112 19.9 Mulitcomponent Thermodynamics . 114 19.10Computational Exercises....................... 114 Nomenclature 120 Catalog Description Classical and statistical thermodynamics; entropy and energy functions in liquid and solid solutions, and their applications to phase equilibria. Lectures, problem solving. Materials science and engineering degree candidates may not take this course for credit with or after CHEM 342 1. (new stuff) Course Outcomes 1 314: Thermodynamics of Materials At the conclusion of the course students will be able to: 1. articulate the fundamental laws of thermodynamics and use them in ba- sic problem solving. 2. discriminate between classical and statistical approaches. 3. describe the thermal behavior of solid materials, including phase transi- tions. 4. use thermodynamics to describe order-disorder transformations in materials . 4 3 BIG IDEAS OF THERMO 5. apply solution thermodynamics for describing liquid and solid solution behavior. 2 Introduction The content and notation of these notes is based on the excellent book, “Ther- modynamics in Materials Science”, by Robert DeHoff [2]. The notes provided here are a complement to this text, and are provided so that the core curricu- lum of the undergradaute Materials Science and Engineering program can be found in one place (http://msecore.northwestern.edu/). These notes are designed to be self-contained, but are more compact than a full text book. Readers are referred to DeHoff’s book for a more complete treatment. 3 Big Ideas of Thermodynamics The following statements sum up some of foundational ideas of thermodynamics: • Energy is conserved • Entropy is produced • Temperature has an absolute reference at 0 - at which all substances have the same entropy 3.1 Classification of Systems Systems can be classified by various different properties such as constituent phases and chemical components, as described below. • Chemical components – Unary system has one chemical component – Multicomponent system has more than one chemical component • Phases – Homogeneous system has one phase – Heterogeneous system has more than one phase 5 3.1 Classification of Systems 3 BIG IDEAS OF THERMO Figure 3.1: Unary system (left) and multicomponent system (right). ice Figure 3.2: Homogeneous system (left) and heterogeneous system (right). • Energy and mass transfer: closed and open systems: – An isolated system cannot change energy or matter with its surroundings. – Closed system can exchange energy but not matter with its surroundings. – Open system can exchange both energy and matter with its surroundings. Surroundings Surroundings Surroundings Energy Energy System System System Matter OPEN CLOSED ISOLATED Figure 3.3: Open, closed and isolated thermodynamic systems. • Reactivity – Non-reacting system contains components that do not chemically react with each other – Reacting system contains components that chemically react with each other Another classification is between simple and complex systems. For instance, complex systems may involve surfaces or fields that lead to energy exchange. 6 3.2 State Variables 3 BIG IDEAS OF THERMO AB AB AB C C C AB AB AB C C BC C A A BC BC AB+C AB+C A AB+C A+BC Figure 3.4: Non-reacting system (left) and reacting system (right). 3.2 Thermodynamic State Variables Thermodynamic state variables are properties of a system that depend only on current condition, not history. Temperature, T , is an obvious example, since we know intuitively that the temperature in a room has the same meaning to us, now matter what the previous temperature history actually was. Other state

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