Chapter-6: Entropy. 1 Clausius Inequality. 2 Entropy - A Property

Transcription

1 hater-6: Entroy When the first law of thermodynamics was stated, the existence of roerty, the internal energy, was found. imilarly, econd law also leads to definition of another roerty, known as entroy. If first law is said to be law of internal energy, then second law may be stated to be the law of entroy. lausius Inequality Whenever a system undergoes a cycle, ( is zero if the cycle is reversible and negative if the cycle is eversible, i.e. in general, ( 0 roof: onsider a reversible heat engine cycle oerating between reservoirs at temeratures H and L. For this cycle, the cyclic integral of the heat transfer, δ, is greater than 0. δ = H ince, H and L are constant, from the definition of absolute temerature scale, and for a reversible cycle, H L = = 0 H L = 0 rev Now, let s consider an eversible cyclic heat engine oerating between the same H and L and receiving the same quantity of heat H. omaring the eversible cycle with the reversible one, W < W rev ince H L = W; for both the reversible and eversible cycles, < L > 0 H L, H L, rev L, > L, rev onsequently, = H L, > 0 H L, = < 0 H L < 0 hus, we conclude that ( 0 for all heat engine cycle and similarly it can be roved that ( 0 for refrigerator cycles. Entroy - roerty here exists a roerty of a closed system such that a change in its value is equal to reversible rocess undergone by the system between state and state. for any [R Gnyawali / imilsina] age

2 Let a system undergo a reversible rocess from state to state along a ath, and let the cycle be comleted along ath, which is also reversible. For reversible cycle, we can write, = 0 ( + ( = 0... eqn( Now consider another reversible cycle, which roceeds first along ath and is then comleted along ath. For this cycle we can write, ( ( 0... eqn( = δ + = δ From these equations ( and (, we can write, ( = ( ince the quantity is same for all reversible aths between states and, we conclude that this quantity is indeendent of ath and it is a function of the end states only. herefore, it is a roerty. his roerty is called entroy and is denoted by. o, ( δ rev = d nd for eversible rocess, Let the cycle is made of eversible rocess and reversible rocess. hus this is an eversible cycle. o, ( ( 0... eqn(3 = δ + < δ From above equations ( and (3 ( > ( ince ath is reversible, and it is a roerty ( d = d = [R Gnyawali / imilsina] age

3 hus, d > o Entroy change in an eversible rocess, ( δ < d hus in general, Entroy change d ( Entroy and econd Law of hermodynamics for an Isolated ystem he microscoic disorder of a system is rescribed by a system roerty is called entroy. he entroy, an extensive equilibrium roerty, must always increase or remain constant for an isolated system. his is exressed mathematically as, Or, d 0..eq( isolated ( 0 eq( Final Initial Isolated Entroy, like our other thermodynamic roerties, is defined only at equilibrium states or for quasiequilibrium rocesses. Equation ( shows that the entroy of the final state is never less than that of the initial state for any rocess which an isolated system undergoes. Entroy is a measure of the molecular disorder of the substance. Larger values of entroy imly larger disorder or uncertainty and lower values imly more microscoically organized states. he term entroy roduction or entroy generation gen is considered in eq ( to eliminate the inequality sign. Or ( d δ = 0 gen isolated ( = 0 Final gen Initial Isolated Here, δ gen is the entroy generated during a change in system state and is always ositive or zero. rincial of increase of entroy (Entroy generation he entroy of an isolated system increases in all real rocesses and is conserved in ideal rocesses. s a result of natural rocesses the entroy of the universe steadily increases. δ = 0 for reversible rocess. gen δ > 0 for eversible rocess. Entroy gets increasing in eversible rocess. gen [R Gnyawali / imilsina] age 3

4 It is thus roved that the entroy of an isolated system can never decrease. It always increases and remains constant only when the rocess is reversible. his is known as the rincile of increase of entroy (entroy generation or simly entroy rincile. 3 Entroy he entroy of a system is a thermodynamic roerty which is a measure of the degree of molecular disorder existing in the system. It describes the randomness or uncertainty of the system. It is a function of a quantity of heat which shows the ossibility of conversion of heat into work. hus, for maximum entroy, there is minimum availability for conversion into work and for minimum entroy there is a maximum availability for conversion into work. haracteristics:. It increases when heat is sulied esective of the fact whether temerature changes or not.. It decreases when heat is removed whether the temerature changes or not. 3. It remains unchanged in all adiabatic reversible rocesses. 4. he increase in entroy is small when heat is added at a high temerature and is greater when heat addition is made at a lower temerature. 4 Lost Work For an infinitesimal reversible rocess by a closed system, dr = du R + d eq( In reversible rocess only, dw = d, ut in eversible rocess dw d If the rocess is eversible, di = du I + dw eq( ince U is roerty, du R = du I From equation ( and (, dr d = di dw dr = di + d dw Dividing on both sides by d d d dw = + R I he difference ( d dw indicates the work that is lost due to eversibility, and is called the lost work. he lost work aroach zero as the rocess aroaches reversibility as a limit. 5 Entroy-roerty and Relation for an ideal gas and Incomressible substances he Gibbs equation, an imortant relation in thermodynamics, is given by: du = d d eq( his relation relates the equilibrium thermodynamics roerties. H = U + dh = du + d + d.eq( From eq( and eq( yields, dh = d + d..eq(3 [R Gnyawali / imilsina] age 4

5 hese two equations are also reresented on an intensive basis as du = ds dv dh = ds + vd eq(4 he changes in entroy are obtained directly from these equations as du d = dh d = + d..eq(5 d Or du ds = + dv..eq(6 dh v ds = d Ideal Gas Relations: he internal energy and enthaly can be exressed as dh = m d nd du = m d = mr From above, eq(4 reduces to m d d d = + mr eq(7 m d d d = mr For ommon rocess:. Isochoric rocess = m d d = m d = m d = m ln(. Isobaric rocess = m d d = m d = m d = m ln( 3. Isothermal rocess 4. diabatic rocess = 0 d = = 0 5. olytroic rocess [R Gnyawali / imilsina] age 5

6 = mr ln( d = = mr ln( = mr ln( = mr ln( = mr ln( d = m n γ ( ln( n Entroy hange for Incomressible Fluid or olid ubstances ince volume is constant, change in d is zero. du = md he eq(6 now becomes as md d = d = m = m ln( Isentroic rocess for an Ideal Gas and an Incomressible ubstances n isentroic rocess is a constant-entroy rocess. If a control mass undergoes a rocess which is both reversible and adiabatic, then the second law secifies the entroy change to be zero. lthough the isentroic rocess might be an idealization of an actual rocess, this rocess serves as a limiting rocess, for articular alications. Isentroic rocess for an Incomressible Fluid or olid he entroy change for incomressible Fluid or olid is given as: md d =..eq( For isentroic rocess d = 0, so d = 0. hus, an isentroic rocess is an isothermal rocess for an incomressible fluid or solid. lso, the internal energy is given as; du = md o du = 0 for an isentroic rocess. he change in enthaly is H = U + dh = du + d + d.eq( ut du = 0 for this rocess and d = 0 for incomressible fluids or solids. o, eq( becomes as dh = d In intensive basis, dh = vd.eq(3 his last exression is integrated to yield h ( h = v..eq(4 ince v = constant. his last exression is articularly useful in adiabatic work considerations of liquid ums. [R Gnyawali / imilsina] age 6

7 oncet of reversibility, eversibility and vailability Reversible work: he maximum amount of work that can be develoed by a system undergoing a roblem is called reversible work. vailable Energy: vailable energy is the maximum ortion of energy which could be converted into useful work by ideal rocesses whichh reduces the system to a dead state. Let us take an examle of a cycle heat engine. he maximum amount of work obtainable by this engine is called available energy (E and the minimum energy that has to be rejected to the sink is called unavailable energy. hus, H =.E. + U.E. W max =.E. = H - U.E. For given temerature, H & L For a given H, will increase with decrease of L, he lowest ractical temerature of heat rejection is the temerature of the surroundings. O vailability: When a system is in equilibrium with the environment, no sontaneous change of state will occur, and the system will not be caable of doing any work. herefore, if a system in a given state undergoes a comletely reversible rocess until it reaches a state in whichh it is in equilibrium with the environment, the maximum reversible work will have been done by the system. If a system is in equilibrium with the surroundings, it must certainly be in ressure and temerature equilibrium with the surroundings. hat is at ressure O and temerature O. It must also be in chemical equilibrium with the surroundings. With a given state for any system, the reversible work will be maximumm when the final state of the system is in equilibrium with the surroundings. he theoretical maximumm amount of work which can be obtained from a system at any state and when oerating with a reservoir at the constant ressure and temerature 0 and 0 is called availability. hus, the availability for a system is equal to this maximumm reversible work minus the work done against the surroundings. Irreversibility he actual work which a system does is always less than the idealized reversible work, and the difference between the two is called the rocess. hus, I = W rev W For any rocess, I 0 If real rocesses could take lace in a comletely reversible manner, the reversibility would be zero. [R Gnyawali / imilsina] age 7