Appendix a Thermodynamic Functions and Legendre Transforms
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Thermodynamics Notes
Thermodynamics Notes Steven K. Krueger Department of Atmospheric Sciences, University of Utah August 2020 Contents 1 Introduction 1 1.1 What is thermodynamics? . .1 1.2 The atmosphere . .1 2 The Equation of State 1 2.1 State variables . .1 2.2 Charles' Law and absolute temperature . .2 2.3 Boyle's Law . .3 2.4 Equation of state of an ideal gas . .3 2.5 Mixtures of gases . .4 2.6 Ideal gas law: molecular viewpoint . .6 3 Conservation of Energy 8 3.1 Conservation of energy in mechanics . .8 3.2 Conservation of energy: A system of point masses . .8 3.3 Kinetic energy exchange in molecular collisions . .9 3.4 Working and Heating . .9 4 The Principles of Thermodynamics 11 4.1 Conservation of energy and the first law of thermodynamics . 11 4.1.1 Conservation of energy . 11 4.1.2 The first law of thermodynamics . 11 4.1.3 Work . 12 4.1.4 Energy transferred by heating . 13 4.2 Quantity of energy transferred by heating . 14 4.3 The first law of thermodynamics for an ideal gas . 15 4.4 Applications of the first law . 16 4.4.1 Isothermal process . 16 4.4.2 Isobaric process . 17 4.4.3 Isosteric process . 18 4.5 Adiabatic processes . 18 5 The Thermodynamics of Water Vapor and Moist Air 21 5.1 Thermal properties of water substance . 21 5.2 Equation of state of moist air . 21 5.3 Mixing ratio . 22 5.4 Moisture variables . 22 5.5 Changes of phase and latent heats . -
Computational Thermodynamics: a Mature Scientific Tool for Industry and Academia*
Pure Appl. Chem., Vol. 83, No. 5, pp. 1031–1044, 2011. doi:10.1351/PAC-CON-10-12-06 © 2011 IUPAC, Publication date (Web): 4 April 2011 Computational thermodynamics: A mature scientific tool for industry and academia* Klaus Hack GTT Technologies, Kaiserstrasse 100, D-52134 Herzogenrath, Germany Abstract: The paper gives an overview of the general theoretical background of computa- tional thermochemistry as well as recent developments in the field, showing special applica- tion cases for real world problems. The established way of applying computational thermo- dynamics is the use of so-called integrated thermodynamic databank systems (ITDS). A short overview of the capabilities of such an ITDS is given using FactSage as an example. However, there are many more applications that go beyond the closed approach of an ITDS. With advanced algorithms it is possible to include explicit reaction kinetics as an additional constraint into the method of complex equilibrium calculations. Furthermore, a method of interlinking a small number of local equilibria with a system of materials and energy streams has been developed which permits a thermodynamically based approach to process modeling which has proven superior to detailed high-resolution computational fluid dynamic models in several cases. Examples for such highly developed applications of computational thermo- dynamics will be given. The production of metallurgical grade silicon from silica and carbon will be used to demonstrate the application of several calculation methods up to a full process model. Keywords: complex equilibria; Gibbs energy; phase diagrams; process modeling; reaction equilibria; thermodynamics. INTRODUCTION The concept of using Gibbsian thermodynamics as an approach to tackle problems of industrial or aca- demic background is not new at all. -
Lecture 2 the First Law of Thermodynamics (Ch.1)
Lecture 2 The First Law of Thermodynamics (Ch.1) Outline: 1. Internal Energy, Work, Heating 2. Energy Conservation – the First Law 3. Quasi-static processes 4. Enthalpy 5. Heat Capacity Internal Energy The internal energy of a system of particles, U, is the sum of the kinetic energy in the reference frame in which the center of mass is at rest and the potential energy arising from the forces of the particles on each other. system Difference between the total energy and the internal energy? boundary system U = kinetic + potential “environment” B The internal energy is a state function – it depends only on P the values of macroparameters (the state of a system), not on the method of preparation of this state (the “path” in the V macroparameter space is irrelevant). T A In equilibrium [ f (P,V,T)=0 ] : U = U (V, T) U depends on the kinetic energy of particles in a system and an average inter-particle distance (~ V-1/3) – interactions. For an ideal gas (no interactions) : U = U (T) - “pure” kinetic Internal Energy of an Ideal Gas f The internal energy of an ideal gas U = Nk T with f degrees of freedom: 2 B f ⇒ 3 (monatomic), 5 (diatomic), 6 (polyatomic) (here we consider only trans.+rotat. degrees of freedom, and neglect the vibrational ones that can be excited at very high temperatures) How does the internal energy of air in this (not-air-tight) room change with T if the external P = const? f ⎡ PV ⎤ f U =Nin room k= T Bin⎢ N room = ⎥ = PV 2 ⎣ kB T⎦ 2 - does not change at all, an increase of the kinetic energy of individual molecules with T is compensated by a decrease of their number. -
High Temperature Enthalpies of the Lead Halides
HIGH TEMPERATURE ENTHALPIES OF THE LEAD HALIDES: ENTHALPIES AND ENTROPIES OF FUSION APPROVED: Graduate Committee: Major Professor Committee Member Min fessor Committee Member X Committee Member UJ. Committee Member Committee Member Chairman of the Department of Chemistry Dean or the Graduate School HIGH TEMPERATURE ENTHALPIES OF THE LEAD HALIDES: ENTHALPIES AND ENTROPIES OF FUSION DISSERTATION Presented to the Graduate Council of the North Texas State University in Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY By Clarence W, Linsey, B. S., M. S, Denton, Texas June, 1970 ACKNOWLEDGEMENT This dissertation is based on research conducted at the Oak Ridge National Laboratory, Oak Ridge, Tennessee, which is operated by the Uniofl Carbide Corporation for the U. S. Atomic Energy Commission. The author is also indebted to the Oak Ridge Associated Universities for the Oak Ridge Graduate Fellowship which he held during the laboratory phase. 11 TABLE OF CONTENTS Page LIST OF TABLES iv LIST OF ILLUSTRATIONS v Chapter I. INTRODUCTION . 1 Lead Fluoride Lead Chloride Lead Bromide Lead Iodide II. EXPERIMENTAL 11 Reagents Fusion-Filtration Method Encapsulation of Samples Equipment and Procedure Bunsen Ice Calorimeter Computer Programs Shomate Method Treatment of Data Near the Melting Point Solution Calorimeter Heat of Solution Calculations III. RESULTS AND DISCUSSION 41 Lead Fluoride Variation in Heat Contents of the Nichrome V Capsules Lead Chloride Lead Bromide Lead Iodide Summary BIBLIOGRAPHY ...... 98 in LIST OF TABLES Table Pa8e I. High-Temperature Enthalpy Data for Cubic PbE2 Encapsulated in Gold 42 II. High-Temperature Enthalpy Data for Cubic PbF2 Encapsulated in Molybdenum 48 III. -
Covariant Hamiltonian Field Theory 3
December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte COVARIANT HAMILTONIAN FIELD THEORY JURGEN¨ STRUCKMEIER and ANDREAS REDELBACH GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH Planckstr. 1, 64291 Darmstadt, Germany and Johann Wolfgang Goethe-Universit¨at Frankfurt am Main Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany [email protected] Received 18 July 2007 Revised 14 December 2020 A consistent, local coordinate formulation of covariant Hamiltonian field theory is pre- sented. Whereas the covariant canonical field equations are equivalent to the Euler- Lagrange field equations, the covariant canonical transformation theory offers more gen- eral means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinites- imal canonical transformation furnishes the most general form of Noether’s theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations. Keywords: Field theory; Hamiltonian density; covariant. PACS numbers: 11.10.Ef, 11.15Kc arXiv:0811.0508v6 [math-ph] 15 Dec 2020 1. Introduction Relativistic field theories and gauge theories are commonly formulated on the basis of a Lagrangian density L1,2,3,4. The space-time evolution of the fields is obtained by integrating the Euler-Lagrange field equations that follow from the four-dimensional representation of Hamilton’s action principle. -
Thermodynamic State Variables Gunt
Fundamentals of thermodynamics 1 Thermodynamic state variables gunt Basic knowledge Thermodynamic state variables Thermodynamic systems and principles Change of state of gases In physics, an idealised model of a real gas was introduced to Equation of state for ideal gases: State variables are the measurable properties of a system. To make it easier to explain the behaviour of gases. This model is a p × V = m × Rs × T describe the state of a system at least two independent state system boundaries highly simplifi ed representation of the real states and is known · m: mass variables must be given. surroundings as an “ideal gas”. Many thermodynamic processes in gases in · Rs: spec. gas constant of the corresponding gas particular can be explained and described mathematically with State variables are e.g.: the help of this model. system • pressure (p) state process • temperature (T) • volume (V) Changes of state of an ideal gas • amount of substance (n) Change of state isochoric isobaric isothermal isentropic Condition V = constant p = constant T = constant S = constant The state functions can be derived from the state variables: Result dV = 0 dp = 0 dT = 0 dS = 0 • internal energy (U): the thermal energy of a static, closed Law p/T = constant V/T = constant p×V = constant p×Vκ = constant system. When external energy is added, processes result κ =isentropic in a change of the internal energy. exponent ∆U = Q+W · Q: thermal energy added to the system · W: mechanical work done on the system that results in an addition of heat An increase in the internal energy of the system using a pressure cooker as an example. -
Math Background for Thermodynamics ∑
MATH BACKGROUND FOR THERMODYNAMICS A. Partial Derivatives and Total Differentials Partial Derivatives Given a function f(x1,x2,...,xm) of m independent variables, the partial derivative ∂ f of f with respect to x , holding the other m-1 independent variables constant, , is defined by i ∂ xi xj≠i ∂ f fx( , x ,..., x+ ∆ x ,..., x )− fx ( , x ,..., x ,..., x ) = 12ii m 12 i m ∂ lim ∆ xi x →∆ 0 xi xj≠i i nRT Example: If p(n,V,T) = , V ∂ p RT ∂ p nRT ∂ p nR = = − = ∂ n V ∂V 2 ∂T V VT,, nTV nV , Total Differentials Given a function f(x1,x2,...,xm) of m independent variables, the total differential of f, df, is defined by m ∂ f df = ∑ dx ∂ i i=1 xi xji≠ ∂ f ∂ f ∂ f = dx + dx + ... + dx , ∂ 1 ∂ 2 ∂ m x1 x2 xm xx2131,...,mm xxx , ,..., xx ,..., m-1 where dxi is an infinitesimally small but arbitrary change in the variable xi. nRT Example: For p(n,V,T) = , V ∂ p ∂ p ∂ p dp = dn + dV + dT ∂ n ∂ V ∂ T VT,,, nT nV RT nRT nR = dn − dV + dT V V 2 V B. Some Useful Properties of Partial Derivatives 1. The order of differentiation in mixed second derivatives is immaterial; e.g., for a function f(x,y), ∂ ∂ f ∂ ∂ f ∂ 22f ∂ f = or = ∂ y ∂ xx ∂ ∂ y ∂∂yx ∂∂xy y x x y 2 in the commonly used short-hand notation. (This relation can be shown to follow from the definition of partial derivatives.) 2. Given a function f(x,y): ∂ y 1 a. = etc. ∂ f ∂ f x ∂ y x ∂ f ∂ y ∂ x b. -
6CCP3212 Statistical Mechanics Homework 1
6CCP3212 Statistical Mechanics Homework 1 Lecturer: Dr. Eugene A. Lim 2018-19 Year 3 Semester 1 https://nms.kcl.ac.uk/eugene.lim/teach/statmech/sm.html 1) (i) For the following differentials with α and β non-zero real constants, which are exact and which are inexact? Integrate the equation if it is exact. (a) x dG = αdx + β dy (1) y (b) α dG = dx + βdy (2) x (c) x2 dG = (x + y)dx + dy (3) 2 (ii) Show that the work done on the system at pressure P d¯W = −P dV (4) where dV is the change in volume is an inexact differential by showing that there exists no possible function of state for W (P; V ). (iii) Consider the differential dF = (x2 − y)dx + xdy : (5) (a) Show that this is not an exact differential. And hence integrate this equation in two different straight paths from (1; 1) ! (2; 2) and from (1; 1) ! (1; 2) ! (2; 2), where (x; y) indicates the locations. Compare the results { are they identical? (b) Define a new differential with dF y 1 dG ≡ = 1 − dx + dy : (6) x2 x2 x Show that dG is exact, and find G(x; y). 2) This problem asks you to derive some derivative identities of a system with three variables x, y and z, with a single constraint x(y; z). This kind of system is central to thermodynamics as we often use three state variables P , V and T , with an equation of state P (V; T ) (i.e. the constraint) to describe a system. -
3 More Applications of Derivatives
3 More applications of derivatives 3.1 Exact & inexact di®erentials in thermodynamics So far we have been discussing total or \exact" di®erentials µ ¶ µ ¶ @u @u du = dx + dy; (1) @x y @y x but we could imagine a more general situation du = M(x; y)dx + N(x; y)dy: (2) ¡ ¢ ³ ´ If the di®erential is exact, M = @u and N = @u . By the identity of mixed @x y @y x partial derivatives, we have µ ¶ µ ¶ µ ¶ @M @2u @N = = (3) @y x @x@y @x y Ex: Ideal gas pV = RT (for 1 mole), take V = V (T; p), so µ ¶ µ ¶ @V @V R RT dV = dT + dp = dT ¡ 2 dp (4) @T p @p T p p Now the work done in changing the volume of a gas is RT dW = pdV = RdT ¡ dp: (5) p Let's calculate the total change in volume and work done in changing the system between two points A and C in p; T space, along paths AC or ABC. 1. Path AC: dT T ¡ T ¢T ¢T = 2 1 ´ so dT = dp (6) dp p2 ¡ p1 ¢p ¢p T ¡ T1 ¢T ¢T & = ) T ¡ T1 = (p ¡ p1) (7) p ¡ p1 ¢p ¢p so (8) R ¢T R ¢T R ¢T dV = dp ¡ [T + (p ¡ p )]dp = ¡ (T ¡ p )dp (9) p ¢p p2 1 ¢p 1 p2 1 ¢p 1 R ¢T dW = ¡ (T ¡ p )dp (10) p 1 ¢p 1 1 T (p ,T ) 2 2 C (p,T) (p1,T1) A B p Figure 1: Path in p; T plane for thermodynamic process. -
Thermodynamics
ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 6. Thermodynamics January 26, 2011 Contents 1 Laws of thermodynamics 2 1.1 The zeroth law . .3 1.2 The first law . .4 1.3 The second law . .5 1.3.1 Efficiency of Carnot engine . .5 1.3.2 Alternative statements of the second law . .7 1.4 The third law . .8 2 Mathematics of thermodynamics 9 2.1 Equation of state . .9 2.2 Gibbs-Duhem relation . 11 2.2.1 Homogeneous function . 11 2.2.2 Virial theorem / Euler theorem . 12 2.3 Maxwell relations . 13 2.4 Legendre transform . 15 2.5 Thermodynamic potentials . 16 3 Worked examples 21 3.1 Thermodynamic potentials and Maxwell's relation . 21 3.2 Properties of ideal gas . 24 3.3 Gas expansion . 28 4 Irreversible processes 32 4.1 Entropy and irreversibility . 32 4.2 Variational statement of second law . 32 1 In the 1st lecture, we will discuss the concepts of thermodynamics, namely its 4 laws. The most important concepts are the second law and the notion of Entropy. (reading assignment: Reif x 3.10, 3.11) In the 2nd lecture, We will discuss the mathematics of thermodynamics, i.e. the machinery to make quantitative predictions. We will deal with partial derivatives and Legendre transforms. (reading assignment: Reif x 4.1-4.7, 5.1-5.12) 1 Laws of thermodynamics Thermodynamics is a branch of science connected with the nature of heat and its conver- sion to mechanical, electrical and chemical energy. (The Webster pocket dictionary defines, Thermodynamics: physics of heat.) Historically, it grew out of efforts to construct more efficient heat engines | devices for ex- tracting useful work from expanding hot gases (http://www.answers.com/thermodynamics). -
Chapter 11 Liquids, Solids, and Phase Changes
Lecture Presentation Chapter 11 Liquids, Solids, and Phase Changes 11.1, 11.2, 11.3, 11.4, 11.15, 11.17, 11.26, 11.30, 11.32, 11.40, 11.42, 11.60, 11.82, 11.116 John E. McMurry Robert C. Fay Properties of Liquids Viscosity: The measure of a liquid’s resistance to flow (higher intermolecular force, higher viscosity) Surface Tension: The resistance of a liquid to spread out and increase its surface area (forms beads) (higher intermolecular force, higher surface tension) Properties of Liquids low intermolecular force – low viscosity, low surface tension high intermolecular force – high viscosity, high surface tension HW 11.1 Properties of Liquids high intermolecular force – high viscosity, high surface tension For the following molecules which has the higher intermolecular force, viscosity and surface tension ? (LD structure, VSEPRT, dipole of molecule) (if the molecules are in the liquid state) a. CHCl3 vs CH4 b. H2O vs H2 (changed from handout) c. N Cl3 vs N H Cl2 Phase Changes between Solids, Liquids, and Gases Phase Change (State Change): A change in the physical state but not the chemical identity of a substance Fusion (melting): solid to liquid Freezing: liquid to solid Vaporization: liquid to gas Condensation: gas to liquid Sublimation: solid to gas Deposition: gas to solid Phase Changes between Solids, Liquids, and Gases Enthalpy – add heat to system Entropy – add randomness to system Phase Changes between Solids, Liquids, and Gases Heat (Enthalpy) of Fusion (DHfusion ): The amount of energy required to overcome enough intermolecular forces to convert a solid to a liquid Heat (Enthalpy) of Vaporization (DHvap): The amount of energy required to overcome enough intermolecular forces to convert a liquid to a gas HW 11.2: Phase Changes between Solids, Liquids, & Gases DHfusion solid to a liquid DHvap liquid to a gas At phase change (melting, boiling, etc) : DG = DH – TDS & D G = zero (bc 2 phases in equilibrium) DH = TDS o a. -
Thermodynamic Stability: Free Energy and Chemical Equilibrium ©David Ronis Mcgill University
Chemistry 223: Thermodynamic Stability: Free Energy and Chemical Equilibrium ©David Ronis McGill University 1. Spontaneity and Stability Under Various Conditions All the criteria for thermodynamic stability stem from the Clausius inequality,cf. Eq. (8.7.3). In particular,weshowed that for anypossible infinitesimal spontaneous change in nature, d− Q dS ≥ .(1) T Conversely,if d− Q dS < (2) T for every allowed change in state, then the system cannot spontaneously leave the current state NO MATTER WHAT;hence the system is in what is called stable equilibrium. The stability criterion becomes particularly simple if the system is adiabatically insulated from the surroundings. In this case, if all allowed variations lead to a decrease in entropy, then nothing will happen. The system will remain where it is. Said another way,the entropyofan adiabatically insulated stable equilibrium system is a maximum. Notice that the term allowed plays an important role. Forexample, if the system is in a constant volume container,changes in state or variations which lead to a change in the volume need not be considered eveniftheylead to an increase in the entropy. What if the system is not adiabatically insulated from the surroundings? Is there a more convenient test than Eq. (2)? The answer is yes. To see howitcomes about, note we can rewrite the criterion for stable equilibrium by using the first lawas − d Q = dE + Pop dV − µop dN > TdS,(3) which implies that dE + Pop dV − µop dN − TdS >0 (4) for all allowed variations if the system is in equilibrium. Equation (4) is the key stability result.