Thermodynamics. Using Affinities to Define Reversible Processes
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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. -
Recitation: 3 9/18/03
Recitation: 3 9/18/03 Second Law ² There exists a function (S) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states. ¡¢ ¡ ² Another way of looking at it: ¡¢ ¡ U1 U2 U=U1+U2 S1 S2 S>S1+S2 ¡¢ ¡ Equilibrium State Figure 1: Second Law ² Entropy is additive. And... � ¶ @S > 0 @U V;N::: ² S is an extensive property: It is a homogeneous, first order function of extensive parameters: S (¸U; ¸V; ¸N) =¸S (U; V; N) ² S is not conserved. ±Q dSSys ¸ TSys For an Isolated System, (i.e. Universe) 4S > 0 Locally, the entropy of the system can decrease. However, this must be compensated by a total increase in the entropy of the universe. (See 2) Q Q ΔS1=¡ ΔS2= + T1³ ´ T2 1 1 ΔST otal =Q ¡ > 0 T2 T1 QuasiStatic Processes A Quasistatic thermodynamic process is defined as the trajectory, in thermodynamic space, along an infinite number of contiguous equilibrium states that connect to equilibrium states, A and B. Universe System T2 T1 Q T1>T2 Figure 2: Local vs. Total Change in Energy Irreversible Process Consider a closed system that can go from state A to state B. The system is induced to go from A to B through the removal of some internal constraint (e.g. removal of adiabatic wall). The systems moves to state B only if B has a maximum entropy with respect to all the other accessible states. -
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. -
Chapter 7 BOUNDARY WORK
Chapter 7 BOUNDARY WORK Frank is struck - as if for the first time - by how much civilization depends on not seeing certain things and pretending others never occurred. − Francine Prose (Amateur Voodoo) In this chapter we will learn to evaluate Win, which is the work term in the first law of thermodynamics. Work can be of different types, such as boundary work, stirring work, electrical work, magnetic work, work of changing the surface area, and work to overcome friction. In this chapter, we will concentrate on the evaluation of boundary work. 110 Chapter 7 7.1 Boundary Work in Real Life Boundary work is done when the boundary of the system moves, causing either compression or expansion of the system. A real-life application of boundary work, for example, is found in the diesel engine which consists of pistons and cylinders as the prime component of the engine. In a diesel engine, air is fed to the cylinder and is compressed by the upward movement of the piston. The boundary work done by the piston in compressing the air is responsible for the increase in air temperature. Onto the hot air, diesel fuel is sprayed, which leads to spontaneous self ignition of the fuel-air mixture. The chemical energy released during this combustion process would heat up the gases produced during combustion. As the gases expand due to heating, they would push the piston downwards with great force. A major portion of the boundary work done by the gases on the piston gets converted into the energy required to rotate the shaft of the engine. -
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. -
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. -
Energy Minimum Principle
ESCI 341 – Atmospheric Thermodynamics Lesson 12 – The Energy Minimum Principle References: Thermodynamics and an Introduction to Thermostatistics, Callen Physical Chemistry, Levine THE ENTROPY MAXIMUM PRINCIPLE We’ve seen that a reversible process achieves maximum thermodynamic efficiency. Imagine that I am adding heat to a system (dQin), converting some of it to work (dW), and discarding the remaining heat (dQout). We’ve already demonstrated that dQout cannot be zero (this would violate the second law). We also have shown that dQout will be a minimum for a reversible process, since a reversible process is the most efficient. This means that the net heat for the system (dQ = dQin dQout) will be a maximum for a reversible process, dQrev dQirrev . The change in entropy of the system, dS, is defined in terms of reversible heat as dQrev/T. We can thus write the following inequality dQ dQ dS rev irrev , (1) T T Notice that for Eq. (1), if the system is reversible and adiabatic, we get dQ dS 0 irrev , T or dQirrev 0 which illustrates the following: If two states can be connected by a reversible adiabatic process, then any irreversible process connecting the two states must involve the removal of heat from the system. Eq. (1) can be written as dS dQ T (2) The equality will hold if all processes in the system are reversible. The inequality will hold if there are irreversible processes in the system. For an isolated system (dQ = 0) the inequality becomes dS 0 isolated system , which is just a restatement of the second law of thermodynamics. -
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 11 ENTROPY
Chapter 11 ENTROPY There is nothing like looking, if you want to find something. You certainly usually find something, if you look, but it is not always quite the something you were after. − J.R.R. Tolkien (The Hobbit) In the preceding chapters, we have learnt many aspects of the first law applications to various thermodynamic systems. We have also learnt to use the thermodynamic properties, such as pressure, temperature, internal energy and enthalpy, when analysing thermodynamic systems. In this chap- ter, we will learn yet another thermodynamic property known as entropy, and its use in the thermodynamic analyses of systems. 250 Chapter 11 11.1 Reversible Process The property entropy is defined for an ideal process known as the re- versible process. Let us therefore first see what a reversible process is all about. If we can execute a process which can be reversed without leaving any trace on the surroundings, then such a process is known as the reversible process. That is, if a reversible process is reversed then both the system and the surroundings are returned to their respective original states at the end of the reverse process. Processes that are not reversible are called irreversible processes. Student: Teacher, what exactly is the difference between a reversible process and a cyclic process? Teacher: In a cyclic process, the system returns to its original state. That’s all. We don’t bother about what happens to its surroundings when the system is returned to its original state. In a reversible process, on the other hand, the system need not return to its original state. -
BASIC THERMODYNAMICS REFERENCES: ENGINEERING THERMODYNAMICS by P.K.NAG 3RD EDITION LAWS of THERMODYNAMICS
Module I BASIC THERMODYNAMICS REFERENCES: ENGINEERING THERMODYNAMICS by P.K.NAG 3RD EDITION LAWS OF THERMODYNAMICS • 0 th law – when a body A is in thermal equilibrium with a body B, and also separately with a body C, then B and C will be in thermal equilibrium with each other. • Significance- measurement of property called temperature. A B C Evacuated tube 100o C Steam point Thermometric property 50o C (physical characteristics of reference body that changes with temperature) – rise of mercury in the evacuated tube 0o C bulb Steam at P =1ice atm T= 30oC REASONS FOR NOT TAKING ICE POINT AND STEAM POINT AS REFERENCE TEMPERATURES • Ice melts fast so there is a difficulty in maintaining equilibrium between pure ice and air saturated water. Pure ice Air saturated water • Extreme sensitiveness of steam point with pressure TRIPLE POINT OF WATER AS NEW REFERENCE TEMPERATURE • State at which ice liquid water and water vapor co-exist in equilibrium and is an easily reproducible state. This point is arbitrarily assigned a value 273.16 K • i.e. T in K = 273.16 X / Xtriple point • X- is any thermomertic property like P,V,R,rise of mercury, thermo emf etc. OTHER TYPES OF THERMOMETERS AND THERMOMETRIC PROPERTIES • Constant volume gas thermometers- pressure of the gas • Constant pressure gas thermometers- volume of the gas • Electrical resistance thermometer- resistance of the wire • Thermocouple- thermo emf CELCIUS AND KELVIN(ABSOLUTE) SCALE H Thermometer 2 Ar T in oC N Pg 2 O2 gas -273 oC (0 K) Absolute pressure P This absolute 0K cannot be obtatined (Pg+Patm) since it violates third law.