Lecture 6 The Second Law Of Thermodynamics
The Second Law ofThermodynamics(Lecture 6)1st semester, 2021Advanced Thermodynamics (M2794.007900)Song, Han Ho(*) Some materials in this lecture note are borrowed from the textbook of Ashley H. Carter.
The Second Law of ThermodynamicsIntroductionèThe first law of thermodynamics can’t explain the followings.GasVacuumPossible!GasWork à Heat100% conversionPossible!Heat à Work100% conversionNot Possible!Not Possible!GasVacuumSomething is missing! We need a second fundamental lawfor a complete description of our world!2
The Second Law of ThermodynamicsThe Mathematical Concept of EntropyèLet’s define a new state variable, entropy(S).The first law of thermodynamics for a reversible process (subscript ‘r’ represents‘reversible’),dU dQr - dWrwhere δQr and δWr are inexact differential.Here,dWr PdV ordWrP dVHere, dV is exact differential, and 1/P is called an integrating factor for δWr, aninexact differential. Then, how about δQr?dQrT3 dSClausius’s definitionof the entropy S
The Second Law of ThermodynamicsThe Mathematical Concept of EntropyèContinue on.dU dQr - dWr TdS - PdVèGibbs Equation!Gibbs equation is one of the most important equations inthermodynamics.èHere are two questions regarding the Gibbs equation:1. Is dS an exact differential or is S (entropy) a state variable?2. Does Gibbs equation apply for irreversible process, as well as forreversible process?à We will learn the answers throughout this lecture!4
The Second Law of ThermodynamicsIrreversible Processes (Examples)Dissipative workFree expansion5Mixing of two gasesThermal equilibrium process (T2 T1)
The Second Law of ThermodynamicsClassical Statements of Second LawèHistorically, the impossibility of certain process was first introduced in thefollowing two famous statements of the second law:lClausius statement“It is impossible to construct a devicethat operates in a cycle and whose soleeffect is to transfer heat from a cooler bodyto hotter body.”lClausius statementKelvin-Planck statement“It is impossible to construct a device thatoperates in a cycle and produces no othereffect than the performance of work andthe exchange of heat with a single reservoir.”6Kelvin-Planck statement
The Second Law of ThermodynamicsCarnot’s TheoremèFirst Proposition“It is impossible to construct an engine that operates between two givenreservoirs and is more efficient than a reversible engine (or Carnotengine) operating between the same two reservoirs.”hirr h revif hirr h rev,QL' QL7
The Second Law of ThermodynamicsCarnot’s TheoremèSecond Proposition“All engines that operate on the Carnot cycle between two givenreservoirs have the same efficiency, independent of working substance.”h rev f (TH , TL )orw qH - qLqLTLh 1 1qHqHqHTHReversibleengine 18Reversibleengine 2
The Second Law of ThermodynamicsThe Clausius Inequality and the Second LawèThe second law of thermodynamics leads to the inequality of Clausiusfor both reversible and irreversible heat engines (or refrigerators).èFor Carnot (reversible) engines,qHqLòæ QHçç orè QLö TH ø TLdQQH QL 0TTH TLCarnot cycle in P-v diagram9
The Second Law of ThermodynamicsThe Clausius Inequality and the Second LawèFor irreversible heat engines operating between the same T reservoirsas for the Carnot (reversible) engine,Then,Finally,èFor both reversible and irreversible heat engines,where equality is for reversible engines. Similarly, the inequality ofClausius can be demonstrated for both rev. and irrev. refrigerators.10
The Second Law of ThermodynamicsThe Clausius Inequality and the Second LawèèConsider the reversible processes A, B, and C.Applying the inequality of Clausius,AB1¾¾ 2¾¾ 1CB1¾¾ 2¾¾ 1è11Then,
The Second Law of ThermodynamicsThe Clausius Inequality and the Second Law12èContinue on.èæ dQ ö is independent of the path, or a point function.So, çè T ø revèDefine this point function as entropy, or S.èThen, the change in entropy can be evaluated as,
The Second Law of ThermodynamicsThe Clausius Inequality and the Second LawèConsider reversible processes A & Band irreversible process C. From theinequality of Clausius,AB1¾¾ 2¾¾ 1CB1¾¾ 2¾¾ 1è13Because entropy is a point function,
The Second Law of ThermodynamicsThe Clausius Inequality and the Second LawèAs path C was an arbitrary irreversible process, in general,èFor a reversible process, the second law isèFor an irreversible process, the second law isIf any irreversible effects occur while the amount of heat(δQ) is transferred,the change of entropy will be greater than for the reversible process.14
The Second Law of ThermodynamicsThe Clausius Inequality and the Second LawèInteresting fact around the second law for our universe:dS ³dQT 0 (for isolated system)“The entropy of an isolated system increases in any irreversibleprocess and is unaltered in any reversible process. This is theprinciple of increasing entropy.”èThis indicates that there is a direction for the sequence of naturalevents.The law of increasing entropy The arrow of time15
The Second Law of ThermodynamicsCombined First and Second LawsèHere are two questions regarding the Gibbs equation again:dU dQr - dWr TdS - PdV1. Is dS an exact differential or is S (entropy) a state variable?2. Gibbs equation applies for irreversible process, as well as forreversible process?èLet’s consider the second question.The first law in most general from,dU dQ - dWFrom the second law for an irreversible process,TdS dQr dQ or dQr dQ e (e 0 )Then, substitute this into the first law for reversible process,dU dQr - dWr dQ e - dWr16
The Second Law of ThermodynamicsCombined First and Second LawsèContinue on.Comparing these two equations,dU dQ - dW and dU dQ e - dWr dW dWr - eHere, Ɛ is called lost work, associated with irreversibility.Finally,dU dQ - dW (dQr - e ) - (dWr - e ) dQr - dWr TdS - PdVThis indicates that the Gibbs equation is applied to ANY process.èTwo interesting examples:ll17dQ, dW 0 but PdV , TdS finiteAdiabatic stirring: dQ 0 but TdS finite, PdV 0 but dW ¹ 0Free expansion:
The Second Law of ThermodynamicsDetermining Entropy in Real System (extra topic)èFor simple substances, Q.M. & S.M. can be used to directly enumeratemicrostates(statistical thermodynamics). S k ln WèFor complex substances, the Gibbs equation is used.Gibbs Equation (EOS for Entropy)µi1PdS dU dV - å dN iTTT18
The Second Law of Thermodynamics2nd Law of Thermodynamics (extra topic)[ S ] : dS system dSin - dS out dS produced - dS destroyedaccumulationtransfers 0 (2nd Law) 0 (2nd Law)Transfers of Entropy:Work 0Heat dQTMatter sdN19(reversible work modes only)(Irreversible work modes as heat)
The Second Law of Thermodynamics2nd Law of Thermodynamics (extra topic)[ S ] : dS system dSin - dS out dS produced - dS destroyedaccumulationtransfersdQTdSdS produced 0 (2nd Law) 0 (2nd Law)(Closed System)Differential:dS dQT dS producedWhole process 1 à 2:State 1State 2(Compression stroke)20S 2 - S1 ò21dQT2 ò dS produced1
The Second Law of Thermodynamics2nd Law of Thermodynamics (extra topic)[ S ] : dS system dSin - dS out dS produced - dS destroyedaccumulationsdNtransfersdQ 0 (2nd Law) 0 (2nd Law)(Open System)TDifferential:dSdS produceddS dQT sdN dS producedWhole process 1 à 2:State 1State 2(Intake stroke)21S 2 - S1 ò21dQT22 ò sdN ò dS produced11
10 The ClausiusInequality and the Second Law The Second Law of Thermodynamics è For irreversible heat engines operating between the same T reservoirs as for the Carnot (reversible) engine, Then, Finally, è For both reversible and irreversible heat engines, where equality is for reversible engines.Similarly, the inequality of
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