Lecture 10 ENTROPY – 2nd law of thermodynamics

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ENTROPY – LAW OF THERMODYNAMICS ND OUTLINE • Reversible Process vs Irreversible Process • Quasi-Static vs Quick Process • Carnot’s theorem • Clausius’s Integration • Entropy • The Principle of Increase of Entropy • The Change in Entropy of an Ideal Gas REVERSIBLE – IRREVERSIBLE PROCESS In a reversible process, the system can be returned to its initial conditions along the same path on a PV diagram, and every point along this path is an equilibrium state A process that does not satisfy these requirements is irreversible P P 1 Quasi–static process irreversible Quick (sudden) process 2 reversible reversible V V QUASI-STATIC vs QUICK PROCES P P Quasi–static process Quick (sudden) process 2 irreversible reversible V V Carnot's theorem Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle that specifies limits on the maximum efficiency any heat engine can obtain Carnot's theorem states: • All heat engines between two heat reservoirs are less efficient than a Carnot heat engine operating between the same reservoirs • Every Carnot heat engine between a pair of heat reservoirs is equally efficient, regardless of the working substance employed or the operation details e max  eCarnot Tc  1 Th CARNOT ENGINE Two reservoirs, temperature Th, Tc CLAUSIUS’S INTEGRATION Qj, Tj P e  e carnot 1  Divide any reversible cycle into a series of thin Carnot cycles, where the isotherms are infinitesimally short: Q'c T  1 c Qh Th Q'c T  c Qh Th Qc Q  h Tc Th Qh Q  c 0 Th Tc Qi, Ti V   reversible cycle Qi   T   irriversible cycle i i     reversible cycle  0 T   irriversible cycle Q  ENTROPY   reversible cycle  0 T   irriversible cycle Q Consider a reversible cycle 1a2b1 The Clasius integration has sign “=“ P Q  a b  1a V Definition: We define a state variable S that the change in entropy dS is equal to this amount of energy for the reversible process divided by the absolute temperature of the system: 1a b1  1a T Q T Q   T b1 Q T 0   Q b1 Q  1a _ rever T T   1b _ rever  S  0 1 _ reversible dS  Q rev T Q T Q T ENTROPY (Cont.) Consider an irreversible cycle 1a2: irreversible 2b1: reversible The Clasius integration has sign “ the change in entropy during a process depends only on the endpoints => the change in entropy is independent of the actual path followed Consequently, the entropy change for an irreversible process can be determined by calculating the entropy change for a reversible process that connects the same initial and final states The principle of Increase of Entropy S12 Q rev  T 12 S  S  S  For an isolated system dQ=0 => S12  0  irreversible _ process  reversible _ process S > 0, for irreversible processes S = 0, for reversible processes S < 0, the process is impossible The entropy of the Universe increases in all real processes The Change in Entropy of an Ideal Gas dS  Q rev P2 V2 S  nC v ln  nC p ln P1 V1 T dU  Q  PdV Q  dU  PdV dS  dU  PdV i nRdT nR   dV T T V T2 ISOTHERMAL: V i nRdT nR S     dV T V T V 1  nC v ln T2 V  nR ln T1 V1  nC v ln P2 V2 V  nR ln P1V1 V1  nC v ln P2 V V  nC v ln  nR ln P1 V1 V1 i nRT; PV  nRT i Cv  R i2 Cp  R  Cv  R U The Change in Entropy of an Ideal Gas dQ Q12   T T nRT ln S   Isothermal Process V2 V1 T  nR ln V2 V1 Isovolumetric Process nC v dT T2 S    nC v ln T T1 Isobaric Process S   Adiabatic Process S  nC p dT T  nC p ln S  const T2 T1 Iso_entropy Process Example 22.6 Change in Entropy: Melting A solid that has a latent heat of fusion Lf melts at a temperature Tm Calculate the change in entropy of this substance when a mass m of the substance melts dQ T S   T  Tmelt  Const S  Q mLf  T Tmel ... • Carnot’s theorem • Clausius’s Integration • Entropy • The Principle of Increase of Entropy • The Change in Entropy of an Ideal Gas REVERSIBLE – IRREVERSIBLE PROCESS In a reversible process,... Iso _entropy Process Example 22.6 Change in Entropy: Melting A solid that has a latent heat of fusion Lf melts at a temperature Tm Calculate the change in entropy of this substance when a mass m of. .. can be determined by calculating the entropy change for a reversible process that connects the same initial and final states The principle of Increase of Entropy S12 Q rev  T 12 S  S 