Energy Minimum Principle
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On Entropy, Information, and Conservation of Information
entropy Article On Entropy, Information, and Conservation of Information Yunus A. Çengel Department of Mechanical Engineering, University of Nevada, Reno, NV 89557, USA; [email protected] Abstract: The term entropy is used in different meanings in different contexts, sometimes in contradic- tory ways, resulting in misunderstandings and confusion. The root cause of the problem is the close resemblance of the defining mathematical expressions of entropy in statistical thermodynamics and information in the communications field, also called entropy, differing only by a constant factor with the unit ‘J/K’ in thermodynamics and ‘bits’ in the information theory. The thermodynamic property entropy is closely associated with the physical quantities of thermal energy and temperature, while the entropy used in the communications field is a mathematical abstraction based on probabilities of messages. The terms information and entropy are often used interchangeably in several branches of sciences. This practice gives rise to the phrase conservation of entropy in the sense of conservation of information, which is in contradiction to the fundamental increase of entropy principle in thermody- namics as an expression of the second law. The aim of this paper is to clarify matters and eliminate confusion by putting things into their rightful places within their domains. The notion of conservation of information is also put into a proper perspective. Keywords: entropy; information; conservation of information; creation of information; destruction of information; Boltzmann relation Citation: Çengel, Y.A. On Entropy, 1. Introduction Information, and Conservation of One needs to be cautious when dealing with information since it is defined differently Information. Entropy 2021, 23, 779. -
Practice Problems from Chapter 1-3 Problem 1 One Mole of a Monatomic Ideal Gas Goes Through a Quasistatic Three-Stage Cycle (1-2, 2-3, 3-1) Shown in V 3 the Figure
Practice Problems from Chapter 1-3 Problem 1 One mole of a monatomic ideal gas goes through a quasistatic three-stage cycle (1-2, 2-3, 3-1) shown in V 3 the Figure. T1 and T2 are given. V 2 2 (a) (10) Calculate the work done by the gas. Is it positive or negative? V 1 1 (b) (20) Using two methods (Sackur-Tetrode eq. and dQ/T), calculate the entropy change for each stage and ∆ for the whole cycle, Stotal. Did you get the expected ∆ result for Stotal? Explain. T1 T2 T (c) (5) What is the heat capacity (in units R) for each stage? Problem 1 (cont.) ∝ → (a) 1 – 2 V T P = const (isobaric process) δW 12=P ( V 2−V 1 )=R (T 2−T 1)>0 V = const (isochoric process) 2 – 3 δW 23 =0 V 1 V 1 dV V 1 T1 3 – 1 T = const (isothermal process) δW 31=∫ PdV =R T1 ∫ =R T 1 ln =R T1 ln ¿ 0 V V V 2 T 2 V 2 2 T1 T 2 T 2 δW total=δW 12+δW 31=R (T 2−T 1)+R T 1 ln =R T 1 −1−ln >0 T 2 [ T 1 T 1 ] Problem 1 (cont.) Sackur-Tetrode equation: V (b) 3 V 3 U S ( U ,V ,N )=R ln + R ln +kB ln f ( N ,m ) V2 2 N 2 N V f 3 T f V f 3 T f ΔS =R ln + R ln =R ln + ln V V 2 T V 2 T 1 1 i i ( i i ) 1 – 2 V ∝ T → P = const (isobaric process) T1 T2 T 5 T 2 ΔS12 = R ln 2 T 1 T T V = const (isochoric process) 3 1 3 2 2 – 3 ΔS 23 = R ln =− R ln 2 T 2 2 T 1 V 1 T 2 3 – 1 T = const (isothermal process) ΔS 31 =R ln =−R ln V 2 T 1 5 T 2 T 2 3 T 2 as it should be for a quasistatic cyclic process ΔS = R ln −R ln − R ln =0 cycle 2 T T 2 T (quasistatic – reversible), 1 1 1 because S is a state function. -
Lecture 4: 09.16.05 Temperature, Heat, and Entropy
3.012 Fundamentals of Materials Science Fall 2005 Lecture 4: 09.16.05 Temperature, heat, and entropy Today: LAST TIME .........................................................................................................................................................................................2� State functions ..............................................................................................................................................................................2� Path dependent variables: heat and work..................................................................................................................................2� DEFINING TEMPERATURE ...................................................................................................................................................................4� The zeroth law of thermodynamics .............................................................................................................................................4� The absolute temperature scale ..................................................................................................................................................5� CONSEQUENCES OF THE RELATION BETWEEN TEMPERATURE, HEAT, AND ENTROPY: HEAT CAPACITY .......................................6� The difference between heat and temperature ...........................................................................................................................6� Defining heat capacity.................................................................................................................................................................6� -
Entropy: Ideal Gas Processes
Chapter 19: The Kinec Theory of Gases Thermodynamics = macroscopic picture Gases micro -> macro picture One mole is the number of atoms in 12 g sample Avogadro’s Number of carbon-12 23 -1 C(12)—6 protrons, 6 neutrons and 6 electrons NA=6.02 x 10 mol 12 atomic units of mass assuming mP=mn Another way to do this is to know the mass of one molecule: then So the number of moles n is given by M n=N/N sample A N = N A mmole−mass € Ideal Gas Law Ideal Gases, Ideal Gas Law It was found experimentally that if 1 mole of any gas is placed in containers that have the same volume V and are kept at the same temperature T, approximately all have the same pressure p. The small differences in pressure disappear if lower gas densities are used. Further experiments showed that all low-density gases obey the equation pV = nRT. Here R = 8.31 K/mol ⋅ K and is known as the "gas constant." The equation itself is known as the "ideal gas law." The constant R can be expressed -23 as R = kNA . Here k is called the Boltzmann constant and is equal to 1.38 × 10 J/K. N If we substitute R as well as n = in the ideal gas law we get the equivalent form: NA pV = NkT. Here N is the number of molecules in the gas. The behavior of all real gases approaches that of an ideal gas at low enough densities. Low densitiens m= enumberans tha oft t hemoles gas molecul es are fa Nr e=nough number apa ofr tparticles that the y do not interact with one another, but only with the walls of the gas container. -
Lecture 6: Entropy
Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 6: Entropy 1 Introduction In this lecture, we discuss many ways to think about entropy. The most important and most famous property of entropy is that it never decreases Stot > 0 (1) Here, Stot means the change in entropy of a system plus the change in entropy of the surroundings. This is the second law of thermodynamics that we met in the previous lecture. There's a great quote from Sir Arthur Eddington from 1927 summarizing the importance of the second law: If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equationsthen so much the worse for Maxwell's equations. If it is found to be contradicted by observationwell these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of ther- modynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. Another possibly relevant quote, from the introduction to the statistical mechanics book by David Goodstein: Ludwig Boltzmann who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. There are many ways to dene entropy. All of them are equivalent, although it can be hard to see. In this lecture we will compare and contrast dierent denitions, building up intuition for how to think about entropy in dierent contexts. The original denition of entropy, due to Clausius, was thermodynamic. -
Internal Energy in an Electric Field
Internal energy in an electric field In an electric field, if the dipole moment is changed, the change of the energy is, U E P Using Einstein notation dU Ek dP k This is part of the total derivative of U dU TdSij d ij E kK dP H l dM l Make a Legendre transformation to the Gibbs potential G(T, H, E, ) GUTSijij EP kK HM l l SGTE data for pure elements http://www.sciencedirect.com/science/article/pii/036459169190030N Gibbs free energy GUTSijij EP kK HM l l dG dU TdS SdTijij d ij d ij E kk dP P kk dE H l dM l M l dH l dU TdSij d ij E kK dP H l dM l dG SdTij d ij P k dE k M l dH l G G G G total derivative: dG dT d dE dH ij k l T ij EH k l G G ij Pk E ij k G G M l S Hl T Direct and reciprocal effects (Maxwell relations) Useful to check for errors in experiments or calculations Maxwell relations Useful to check for errors in experiments or calculations Point Groups Crystals can have symmetries: rotation, reflection, inversion,... x 1 0 0 x y 0 cos sin y z 0 sin cos z Symmetries can be represented by matrices. All such matrices that bring the crystal into itself form the group of the crystal. AB G for A, B G 32 point groups (one point remains fixed during transformation) 230 space groups Cyclic groups C2 C4 http://en.wikipedia.org/wiki/Cyclic_group 2G Pyroelectricity i Ei T Pyroelectricity is described by a rank 1 tensor Pi i T 1 0 0 x x x 0 1 0 y y y 0 0 1 z z 0 1 0 0 x x 0 0 1 0 y y 0 0 0 1 z z 0 Pyroelectricity example Turmalin: point group 3m Quartz, ZnO, LaTaO 3 for T = 1°C, E ~ 7 ·104 V/m Pyroelectrics have a spontaneous polarization. -
Thermodynamic Processes: the Limits of Possible
Thermodynamic Processes: The Limits of Possible Thermodynamics put severe restrictions on what processes involving a change of the thermodynamic state of a system (U,V,N,…) are possible. In a quasi-static process system moves from one equilibrium state to another via a series of other equilibrium states . All quasi-static processes fall into two main classes: reversible and irreversible processes . Processes with decreasing total entropy of a thermodynamic system and its environment are prohibited by Postulate II Notes Graphic representation of a single thermodynamic system Phase space of extensive coordinates The fundamental relation S(1) =S(U (1) , X (1) ) of a thermodynamic system defines a hypersurface in the coordinate space S(1) S(1) U(1) U(1) X(1) X(1) S(1) – entropy of system 1 (1) (1) (1) (1) (1) U – energy of system 1 X = V , N 1 , …N m – coordinates of system 1 Notes Graphic representation of a composite thermodynamic system Phase space of extensive coordinates The fundamental relation of a composite thermodynamic system S = S (1) (U (1 ), X (1) ) + S (2) (U-U(1) ,X (2) ) (system 1 and system 2). defines a hyper-surface in the coordinate space of the composite system S(1+2) S(1+2) U (1,2) X = V, N 1, …N m – coordinates U of subsystems (1 and 2) X(1,2) (1,2) S – entropy of a composite system X U – energy of a composite system Notes Irreversible and reversible processes If we change constraints on some of the composite system coordinates, (e.g. -
3. Energy, Heat, and Work
3. Energy, Heat, and Work 3.1. Energy 3.2. Potential and Kinetic Energy 3.3. Internal Energy 3.4. Relatively Effects 3.5. Heat 3.6. Work 3.7. Notation and Sign Convention In these Lecture Notes we examine the basis of thermodynamics – fundamental definitions and equations for energy, heat, and work. 3-1. Energy. Two of man's earliest observations was that: 1)useful work could be accomplished by exerting a force through a distance and that the product of force and distance was proportional to the expended effort, and 2)heat could be ‘felt’ in when close or in contact with a warm body. There were many explanations for this second observation including that of invisible particles traveling through space1. It was not until the early beginnings of modern science and molecular theory that scientists discovered a true physical understanding of ‘heat flow’. It was later that a few notable individuals, including James Prescott Joule, discovered through experiment that work and heat were the same phenomenon and that this phenomenon was energy: Energy is the capacity, either latent or apparent, to exert a force through a distance. The presence of energy is indicated by the macroscopic characteristics of the physical or chemical structure of matter such as its pressure, density, or temperature - properties of matter. The concept of hot versus cold arose in the distant past as a consequence of man's sense of touch or feel. Observations show that, when a hot and a cold substance are placed together, the hot substance gets colder as the cold substance gets hotter. -
The Carnot Cycle, Reversibility and Entropy
entropy Article The Carnot Cycle, Reversibility and Entropy David Sands Department of Physics and Mathematics, University of Hull, Hull HU6 7RX, UK; [email protected] Abstract: The Carnot cycle and the attendant notions of reversibility and entropy are examined. It is shown how the modern view of these concepts still corresponds to the ideas Clausius laid down in the nineteenth century. As such, they reflect the outmoded idea, current at the time, that heat is motion. It is shown how this view of heat led Clausius to develop the entropy of a body based on the work that could be performed in a reversible process rather than the work that is actually performed in an irreversible process. In consequence, Clausius built into entropy a conflict with energy conservation, which is concerned with actual changes in energy. In this paper, reversibility and irreversibility are investigated by means of a macroscopic formulation of internal mechanisms of damping based on rate equations for the distribution of energy within a gas. It is shown that work processes involving a step change in external pressure, however small, are intrinsically irreversible. However, under idealised conditions of zero damping the gas inside a piston expands and traces out a trajectory through the space of equilibrium states. Therefore, the entropy change due to heat flow from the reservoir matches the entropy change of the equilibrium states. This trajectory can be traced out in reverse as the piston reverses direction, but if the external conditions are adjusted appropriately, the gas can be made to trace out a Carnot cycle in P-V space. -
Chapter 20 -- Thermodynamics
ChapterChapter 2020 -- ThermodynamicsThermodynamics AA PowerPointPowerPoint PresentationPresentation byby PaulPaul E.E. TippensTippens,, ProfessorProfessor ofof PhysicsPhysics SouthernSouthern PolytechnicPolytechnic StateState UniversityUniversity © 2007 THERMODYNAMICSTHERMODYNAMICS ThermodynamicsThermodynamics isis thethe studystudy ofof energyenergy relationshipsrelationships thatthat involveinvolve heat,heat, mechanicalmechanical work,work, andand otherother aspectsaspects ofof energyenergy andand heatheat transfer.transfer. Central Heating Objectives:Objectives: AfterAfter finishingfinishing thisthis unit,unit, youyou shouldshould bebe ableable to:to: •• StateState andand applyapply thethe first andand second laws ofof thermodynamics. •• DemonstrateDemonstrate youryour understandingunderstanding ofof adiabatic, isochoric, isothermal, and isobaric processes.processes. •• WriteWrite andand applyapply aa relationshiprelationship forfor determiningdetermining thethe ideal efficiency ofof aa heatheat engine.engine. •• WriteWrite andand applyapply aa relationshiprelationship forfor determiningdetermining coefficient of performance forfor aa refrigeratior.refrigeratior. AA THERMODYNAMICTHERMODYNAMIC SYSTEMSYSTEM •• AA systemsystem isis aa closedclosed environmentenvironment inin whichwhich heatheat transfertransfer cancan taketake place.place. (For(For example,example, thethe gas,gas, walls,walls, andand cylindercylinder ofof anan automobileautomobile engine.)engine.) WorkWork donedone onon gasgas oror workwork donedone byby gasgas INTERNALINTERNAL -
10.05.05 Fundamental Equations; Equilibrium and the Second Law
3.012 Fundamentals of Materials Science Fall 2005 Lecture 8: 10.05.05 Fundamental Equations; Equilibrium and the Second Law Today: LAST TIME ................................................................................................................................................................................ 2 THERMODYNAMIC DRIVING FORCES: WRITING A FUNDAMENTAL EQUATION ........................................................................... 3 What goes into internal energy?.......................................................................................................................................... 3 THE FUNDAMENTAL EQUATION FOR THE ENTROPY................................................................................................................... 5 INTRODUCTION TO THE SECOND LAW ....................................................................................................................................... 6 Statements of the second law ............................................................................................................................................... 6 APPLYING THE SECOND LAW .................................................................................................................................................... 8 Heat flows from hot objects to cold objects ......................................................................................................................... 8 Thermal equilibrium ........................................................................................................................................................... -
Forms of Energy: Kinetic Energy, Gravitational Potential Energy, Elastic Potential Energy, Electrical Energy, Chemical Energy, and Thermal Energy
6/3/14 Objectives Forms of • State a practical definition of energy. energy • Provide or identify an example of each of these forms of energy: kinetic energy, gravitational potential energy, elastic potential energy, electrical energy, chemical energy, and thermal energy. Assessment Assessment 1. Which of the following best illustrates the physics definition of 2. Which statement below provides a correct practical definition of energy? energy? A. “I don’t have the energy to get that done today.” A. Energy is a quantity that can be created or destroyed. B. “Our team needs to be at maximum energy for this game.” B. Energy is a measure of how much money it takes to produce a product. C. “The height of her leaps takes more energy than anyone else’s.” C. The energy of an object can never change. It depends on “ D. That performance was so exhilarating you could feel the energy the size and weight of an object. in the audience!” D. Energy is the quantity that causes matter to change and determines how much change occur. Assessment Physics terms 3. Match each event with the correct form of energy. • energy I. kinetic II. gravitational potential III. elastic potential IV. thermal • gravitational potential energy V. electrical VI. chemical • kinetic energy ___ Ice melts when placed in a cup of warm water. • elastic potential energy ___ Campers use a tank of propane gas on their trip. ___ A car travels down a level road at 25 m/s. • thermal energy ___ A bungee cord causes the jumper to bounce upward. ___ The weightlifter raises the barbell above his head.