Physics 207: Lecture 27, Pg 1 Lecture 26Goals: Chapters 18, entropy and second law of thermodynamics Chapters 18, entropy and second law of thermodynamics Chapter 19, heat engines and refrigerators Chapter 19, heat engines and refrigerators No lab this week.

Physics 207: Lecture 27, Pg 2 Equipartition theorem l 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

Physics 207: Lecture 27, Pg 3 Equipartition theorem l The thermal energy is equally divided among all possible energy modes (degrees of freedom). The average thermal energy is (1/2)k B T for each degree of freedom. ε avg =(3/2) k B T (monatomic gas) ε avg =(6/2) k B T (solids) ε avg =(5/2) k B T (diatomic molecules) l Note that if we have N particles: E th =(3/2)N k B T =(3/2)nRT (monatomic gas) E th =(6/2)N k B T =(6/2)nRT (solids) E th =(5/2)N k B T =(5/2)nRT (diatomic molecules)

Physics 207: Lecture 27, Pg 4 Specific heat l Molar specific heats can be directly inferred from the thermal energy. E th =(6/2)N k B T =(6/2)nRT (solid) ΔE th =(6/2)nRΔT=nCΔT C=3R (solid) l The specific heat for a diatomic gas will be larger than the specific heat of a monatomic gas: C diatomic =C monatomic +R

Physics 207: Lecture 27, Pg 5 Second Law and Entropy l A perfume bottle breaks in the corner of a room. After some time, what would you expect? A) B)

Physics 207: Lecture 27, Pg 6 very unlikely probability=(1/2) N l The probability for each particle to be on the left half is ½.

Physics 207: Lecture 27, Pg 7 Second Law of thermodynamics l The entropy of an isolated system never decreases. It can only increase, or in equilibrium, remain constant. l The laws of probability dictate that a system will evolve towards the most probable and most random macroscopic state l Thermal energy is spontaneously transferred from a hotter system to a colder system.

Physics 207: Lecture 27, Pg 8 Reversible vs Irreversible l 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.)

Physics 207: Lecture 27, Pg 9 Reversible vs Irreversible l 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.

Physics 207: Lecture 27, Pg 10 Heat Engines l Turning heat into work: Industrial revolution. Volume Pressure i f

Physics 207: Lecture 27, Pg 11 Key concepts l Work done by the system: W system =-W external l Energy reservoir: An object that interacts with the system that is sufficiently large such that its temperature is almost constant. Q H : The amount of heat transferred to/from hot reservoir Q C : The amount of heat transferred to/from cold reservoir

Physics 207: Lecture 27, Pg 12 Energy-transfer diagram Hot reservoir Cold reservoir QHQH QCQC W out cyclic system ΔE system =0 W out =Q H -Q C

Physics 207: Lecture 27, Pg 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 efficiency η (lowercase Greek eta), defined as We can also write the thermal efficiency as

Physics 207: Lecture 27, Pg 14 l What is the largest thermal efficiency that a heat engine can have? A)  =2B)  =1 C)  =1/2 D)  =0 l What is the lowest thermal efficiency that a heat engine can have? A)  =1/2B)  =0 C)  =-1/2 D)  =-1

Physics 207: Lecture 27, Pg 15 Refrigerators l Devices that uses work to transfer heat from a colder object to a hotter object. Hot reservoir Cold reservoir QHQH W in W in +Q C =Q H K=Q C /W in QCQC

Physics 207: Lecture 27, Pg 16 Is perfect engine possible? Hot reservoir Cold reservoir Q H1 W out W in Q H2 QCQC = QCQC QHQH

Physics 207: Lecture 27, Pg 17 Turbines: Brayton Cycle

Physics 207: Lecture 27, Pg 18 l Which of the following processes would have the largest work output per cycle? V P V V P P A) B)C)

Physics 207: Lecture 27, Pg 19 Internal combustion engine: gasoline engine (Adiabats) l 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

Physics 207: Lecture 27, Pg 20 The best thermal engine ever, the Carnot engine l 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.

Physics 207: Lecture 27, Pg 21 The Carnot Engine l 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. l Carnot showed that the thermal efficiency of a Carnot engine is: