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Lecture 26

Lecture 26. Goals:. Chapters 18, entropy and second law of thermodynamics Chapter 19, heat engines and refrigerators. No lab this week. . Equipartition theorem. Things are more complicated when energy can be stored in other degrees of freedom of the system. . monatomic gas: translation

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Lecture 26

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  1. Lecture 26 Goals: • Chapters 18, entropy and second law of thermodynamics • Chapter 19, heat engines and refrigerators • No lab this week.

  2. Equipartition theorem • Things are more complicated when energy can be stored in other degrees of freedom of the system. monatomic gas: translation solids: translation+potential energy diatomic molecules: translation+vibrations+rotations

  3. Equipartition theorem • The thermal energy is equally divided among all possible energy modes (degrees of freedom). The average thermal energy is (1/2)kBT for each degree of freedom. εavg=(3/2) kBT (monatomic gas) εavg=(6/2) kBT (solids) εavg=(5/2) kBT (diatomic molecules) • Note that if we have N particles: Eth=(3/2)N kBT =(3/2)nRT (monatomic gas) Eth=(6/2)N kBT =(6/2)nRT (solids) Eth=(5/2)N kBT =(5/2)nRT (diatomic molecules)

  4. Specific heat • Molar specific heats can be directly inferred from the thermal energy. Eth=(6/2)N kBT =(6/2)nRT (solid) ΔEth=(6/2)nRΔT=nCΔT C=3R (solid) • The specific heat for a diatomic gas will be larger than the specific heat of a monatomic gas: Cdiatomic=Cmonatomic+R

  5. Second Law and Entropy • A perfume bottle breaks in the corner of a room. After some time, what would you expect? B) A)

  6. very unlikely • The probability for each particle to be on the left half is ½. probability=(1/2)N

  7. Second Law of thermodynamics • The entropy of an isolated system never decreases. It can only increase, or in equilibrium, remain constant. • The laws of probability dictate that a system will evolve towards the most probable and most random macroscopic state • Thermal energy is spontaneously transferred from a hotter system to a colder system.

  8. Reversible vs Irreversible • The following conditions should be met to make a process perfectly reversible: 1. Any mechanical interactions taking place in the process should be frictionless. 2. Any thermal interactions taking place in the process should occur across infinitesimal temperature or pressure gradients (i.e. the system should always be close to equilibrium.)

  9. Reversible vs Irreversible • Based on the above comments, which of the following processes is not reversible? A. Lowering a frictionless piston in a cylinder by placing a bag of sand on top of the piston. B. Lifting the piston described in the previous statement by removing one tiny grain of sand at a time.

  10. Heat Engines • Turning heat into work: Industrial revolution. f Pressure i Volume

  11. Key concepts • Work done by the system: Wsystem=-Wexternal • Energy reservoir: An object that interacts with the system that is sufficiently large such that its temperature is almost constant. QH: The amount of heat transferred to/from hot reservoir QC: The amount of heat transferred to/from cold reservoir

  12. Energy-transfer diagram Hot reservoir QH cyclic system ΔEsystem=0 Wout=QH-QC Wout QC Cold reservoir

  13. Thermal efficiency For practical reasons, we would like an engine to do the maximum amount of work with the minimum amount of fuel. We can measure the performance of a heat engine in terms of its thermal efficiencyη(lowercase Greek eta), defined as We can also write the thermal efficiency as

  14. What is the largest thermal efficiency that a heat engine can have? C) η=1/2 D) η=0 A) η=2 B) η=1 • What is the lowest thermal efficiency that a heat engine can have? C) η=-1/2 D) η=-1 A) η=1/2 B) η=0

  15. Refrigerators • Devices that uses work to transfer heat from a colder object to a hotter object. Hot reservoir QH Win+QC=QH Win K=QC/Win QC Cold reservoir

  16. Is perfect engine possible? Hot reservoir QH1 QH2 QH = Wout Win QC QC Cold reservoir

  17. Turbines: Brayton Cycle

  18. Which of the following processes would have the largest work output per cycle? A) B) C) P P P V V V

  19. Internal combustion engine: gasoline engine • A gasoline engine utilizes the Otto cycle, in which fuel and air are mixed before entering the combustion chamber and are then ignited by a spark plug. Otto Cycle (Adiabats)

  20. The best thermal engine ever, the Carnot engine • A perfectly reversible engine (a Carnot engine) can be operated either as a heat engine or a refrigerator between the same two energy reservoirs, by reversing the cycle and with no other changes.

  21. The Carnot Engine • Carnot showed that the thermal efficiency of a Carnot engine is: • All real engines are less efficient than the Carnot engine because they operate irreversibly due to the path and friction as they complete a cycle in a brief time period.

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