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The Laws of Thermodynamics - Revisited Arthur Shavit, Professor Emeritus Department of Mechanical Engineering Technion – Israel Institue of Technology Haifa, ISRAEL Krakow 9/12/2011 The second Law Usual statement A PMM2 is impossible A PMM2 is cyclic device that produces positive work while interacting with a single reservoir Questions: Is that a law of thermodynamics? What is a law of thermodynamics? To answer these question some introduction is needed Krakow The Structure of Thermodynamics • Definitions • Experimental facts • Laws (axioms) • Theorems and Corollaries • Applications 9/12/2011 Definitions • Body • Environment • Primitive property • State • Identical states • Change of state • Allowed states • System • Path • Interaction • Process • Cycle 9/12/2011 Definitions – some examples Body A body is a part of space enclosed by a well defined boundary. The boundary may be physical or mathematical, fixed or changing in time, closed or open to passage of matter. Environment Everything outside the boundary of the body Primitive Property Primitive property of a body is specified by subjecting the body to an operation or a test, that requires no previous knowledge of the body, the result of which at a specific time is the value of the primitive property at that time. A primitive property may be determined without the need to change the conditions of the body. Krakow Definitions – examples cont. State The condition of the body identified by all its primitive properties Identical states States that have the same values of the corresponding primitive properties. Change of state Occurs when the value of at least one primitive property is changed. Allowed states Allowed states of a body are all the states which the body may inherently attain consistent with the definition of the body. Krakow System An idealization of a body that may includes only part of the allowed states of the body. It is also required that the system could be isolated from its environment The allowed states may be given as an explicit list. or implicitly, by describing one state and all possible variations of state. These variations must be consistent with: 1. the laws of matter, 2. the constraints. 3. the passive resistances. Closed System A system where matter may not cross its boundary. Open System A system where matter may cross its boundary. Krakow General Property An observable characteristic of the system, whose change between two end states is independent of the path. Derived property: A property that is not primitive Examples: – Ampere-hour on a battery. – Life time of an incandescent lamp. Krakow 9/12/2011 Classifications of properties 9/12/2011 • Primitive – Derived • Extensive – Intensive • Independent – Dependent • Conservative – Non conservative Equilibrium An equilibrium state is one that can not be changed without a corresponding change in the environment. 4 types according to the changes required in the environment Stable Unstable Neutral Metastable Mutual equilibrium Krakow 9/12/2011 Types of Equilibria The type of equilibrium is characterized by the required change in the environment for a finite change in the system For a Stable Unstable Neutral Change of state in system Permanent Same order Temporary Smaller order Temporary Smaller order Metastable Stable up to a limit Unstable above the limit Krakow 9/12/2011 Rate of change of state in system Permanent Same order Permanent Same order Permanent Same order Stable up to a limit Unstable above the limit Neutral vs Themodynamic property Neutral Property A property of neutral equilibrium that can change in both directions by only temporary changes in the environment. Example: The horizontal position of the system in a gravity field. Substate A state different from others only by neutral properties. Thermodynamic Property Any property that is not neutral. Note: A thermodynamic property may have several substates. Thermodynamic State A state that includes only thermodynamic properties Krakow Work Interaction Work is an interaction between two systems such that whatever happened in each system and its boundary could be repeated exactly while the sole external effect is a change of level of a weight. Measure of work The work of a system equals the number of weights, in the test, that underwent a unit change of level. Adiabatic process A process having no interactions other than work. Modes of quasistatic work Dislacement of a wire pulled by a force Change of volume under pressure. Change of magnetization in magnetic field. Change of surface area with surface tension . Etc. Krakow 9/12/2011 First law / Energy Krakow First law The work of a system undergoing an adiabatic process depends only on the end states. Energy A property whose change between two end states is determined by the adiabatic work. 9/12/2011 The Laws of Thermodynamics A law is a generalization of all known experimental facts. Zeroth Law Krakow Maxwell, 1891 First Law Clausius, 1850 (Joule, 1848) Second Law Clausius, 1850 (Carnot, 1824) State principle Kline & Koenig, 1957 Third Law Nernst, 1906 9/12/2011 Work of a system in a stable state Theorem: A system, in a stable equilibrium state, cannot change its state while the only external effect is the rise of the level of weight. Proof Krakow Assume that the theorem is incorrect then there should be at least one case where a the stable state changes while the only external effect is a rise in a level of a weight. It possible to lower the weight and impart a velocity to the system. In this case the net external effect is zero while the state of the system changed from a stable state to another. That violates the definition of a stable state. 9/12/2011 Quasi-stable State Some non equilibrium states can be made stable by eliminating some of the allowed stated, while retaining others. This can be achieved by altering passive resistances and constraints. A state made stable by altering passive resistances and/or constraints is called a quasi-stable state. The stable state so produced is called corresponding stable state. Krakow 9/12/2011 Heat Interaction Heat is an interaction between two systems each in a stable state with no change in the constraints and the passive resistances. Heat Interaction between systems not in stable states. Interaction during which the system vary only through the corresponding stable states. Krakow 9/12/2011 Zeroth Law If two systems, A and B, are each in mutual equilibrium with a system C then they are in mutual equilibrium with each other. Is that trivial??? 9/12/2011 Temperature T T T T Thermometer Temperature is a property that is common to all systems in mutual equilibrium. Krakow 9/12/2011 State Principle The stable state of a system bounded by a fixed boundary and subjected to prescribed force fields is fully determined by its energy. The state principle fixes the number of independent properties of a system in a stable state. These are the parameters of the boundary and the force fields and the energy. Krakow 9/12/2011 Heat Machines A closed system that undergoes a cycle while having interactions Heat Machine dW Heat Engine dW dQ 0 Heat Pump - Refrigerator dW dQ 0 Krakow dQ 0 The Second Law Reservoir A system in a stable state whose temperature stays constant under finite interactions. PMM2 A heat engine that communicates with a single reservoir The Second Law (Two statements) • A PMM2 is not possible • It is not possible to transfer heat from a reservoir at a low temperature to one at a higher with no other effects Are these really Laws (axioms)??? Krakow 9/12/2011 Clausius Inequality dQ 0 T (a corollary of the second law) For a reversible process Leads to define a property Krakow dQ T rev 0 dQ dS T rev The Law of Stable Equilibrium A system having specified allowed states can reach, from any given state, one and only one stable state and leave no effect on the environment. Krakow 9/12/2011 Gibbs Principle of General Inertia A finite rate of change (or a finite rate of a rate of change) cannot be stopped by means of infinitesimal alteration in the circumstances. (J.W. Gibbs, Collected Works, Yale University. Press, Vol. 1 p.56,1948) Krakow 9/12/2011 The Unified Laws First law Second law Law of stable equilibrium Zeroth Law Gibbs Principle Krakow Prague 14.04.2003 Structure of Thermodynamics State principle Pressure weight Theorem Krakow Pressure is a thermodynamic property. A weight is an idealized body whose only independent property is its level in a gravitational field. A process involving no effects except the lowering of weights is impossible. Work Interaction Work is an interaction between two systems such that whatever happened in each system and its boundary could be repeated exactly while the sole external effect is a change of level of a weight. 9/12/2011 Proof of first law X1 Y1 A1 X1 A2 X2 Y2 B X2 Y2 B X0 Y0 X0 Y0 C weights 9/12/2011 X00 Y00 Proof of the State Principle According to the Law of Stable States one and only one stable state is possible for a system of fixed constraints (and passive resistances) that undergoes no interactions (constant energy) It follows that the stable state is determined by the constraints ( E, 1 , 2 ,..., 1 , 2 ,...) Thus Where 9/12/2011 is any property of the system in stable equilibrium are the constraints are the passive resistances Work of a system in combination with a reservoir rev In general (dW dW ) R Define d R dW rev rev 12 W R 1 2 R dW ad,rev d ad R dE 9/12/2011 Called Available Work Entropy Define entropy dSR CR (dE dR ) dSRad,rev 0 (dE d R )ad,rev dSRad 0 9/12/2011 Principle of increase of entropy Criterion of Equilibrium It is necessary and sufficient for equilibrium of an isolated system, not subdivided by adiabatic walls, that all possible variation in state satisfy: (dS)E ≤ 0 (dS)E 0 9/12/2011 for all p.v. Stable Equilibrium (dS)E 0 for all p.v. For every possible variation for which 9/12/2011 (dS)E 0 (DS)E 0 Unstable Equilibrium (dS)E 0 for all p.v. For at least one possible variation for which 9/12/2011 (dS)E 0 (DS)E 0 Metastable Equilibrium (dS)E 0 for all p.v. For all possible variation for which (dS)E 0 And for p.v. smaller than a certain value (DS)E 0 And for some possible variation, larger than that value, at least one p.v. is (DS)E 0 9/12/2011 Neutral Equilibrium (dS)E 0 for all p.v. For at least one possible variation for which and 9/12/2011 (dS)E 0 (DS)E 0 Proof of the Zeroth Law Any property (E, 1, 2 ,...,1,2 ,...) Thus S S ( E, 1 , 2 ,..., 1 , 2 ,...) S [ S ( E )] , Consider two systems A and B in mutual equilibrium S A [S A ( E A )] , S B [S B ( E B )] , 9/12/2011 A dS A dS A dE A dE , B dS B dS B dE B dE , Proof of the Zeroth Law (cont.) A B A B dS dS dS dS A dE B dE dE , dE , A B A For equilibrium B dS A dS dE A 0 B dE A dE , , dS A dS B dE A , dE B , 9/12/2011 B (dS dS )(E A E A ) 0 Temperature T f dS dE , 1 Kelvin scale dS T dE , dE dS , For a system in equilibrium with the reservoir d R 1 dS CR 1 CR TR dE , ,TR dE , ,TR 1 CR 273.16 dS dE d R TR 9/12/2011 Krakow 12.09.2011 For the triple point of water dR dE TRdS The Laws of Thermodynamics Revisited 41 Alternative Criteria of Equilibrium For all possible variations to states of (dS)E 0 equal E (dE)S 0 equal S dE TdS 0 dA SdT 0 } uniform T dA 0 uniform and equal T dE pdV 0 dH Vdp 0 9/12/2011 } uniform T and equal T Alternative Criteria of Equilibrium For all possible variations dE pdV 0 dH Vdp 0 } dH 0 dH TdS 0 dG SdT 0 dG 0 9/12/2011 } From states of to states of uniform T equal S uniform p equal p and S uniform p and T equal p uniform p and T equal p and T Alternative Criteria of Equilibrium dS T f dE , 1 Kelvin scale dS dE T dE , dS , For a system in equilibrium with the reservoir d R 1 dS CR 1 CR TR dE , ,TR dE , ,TR 1 Select CR 273.16 dE d R dS TR 9/12/2011 For the triple point of water dR dE TRdS Many Thanks תודה רבה Dziękuję Bardzo KrakowKrakow 9/12/2011 12.09.2011 The Laws of Thermodynamics Revisited 45 Questions ??? Discussions ??? Krakow Nomenclaure A,a E,e G,g f H,h m n p Q S,s T U,u V,v W m Krakow Helmholzenee energy Energy Gibbs free energy Fugacity Enthalpy Mass Number of moles Pressure Heat Entropy Temperature Internal energy Volume Work constraint Passive resistance Chemical potential Degree of reaction Notations A,a E,e G,g H,h m n p S,s T U,u V,v Krakow Helmholz free energy energy Gibbs free energy enthalpy mass number of moles pressure entropy temperature internal energy volume m degree of reaction passive resistance constraint chemical potential f fugacity W work Q heat Krakow Simple system A system that has only one boundary quasistatic work parameter. If the parameter is the volume the system is called a simple compressible system. Such a system has exactly 2 independent properties: the volume and the energy. (V and E) Krakow 9/12/2011 The volume, v, and energy, e, of a simple system in a stable state are two independent properties. 1 1 (e, v) Namely, any property: Solving for e and v yields: 2 2 (e, v) v v( 1 , 2 ) e e( 1 , 2 ) For example if 1 p and 2 T then v v ( p, T ) e u ( 1 , 2 ) or v ( p, T , v ) 0 for a stable state. In general Krakow Equation of state U = Internal energy Internal energy is the functional relation of two independent properties. U E UE etc. U U (1 , 2 ) Krakow Krakow The Unified Laws Quantum mechanics Thermodynamics Law of stabke equilibrium Gibbs P rinciple First law Zeroth law Second law Third law State principle 9/12/2011 Schrodinger equation