Classical Approach to 2 nd Law for CM

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Hugh Russell
5 years ago
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1 Classical Approach to 2 nd aw for CM Start with observations about the ability to build devices (thermodynamic cycles) Clausius Statement of 2 nd aw concerns cycles that cause heat transfer from low temperature body to high temperature body (refrigerators and heat pumps) Kelvin-Planck Statement of 2 nd aw concerns cycles that use heat transfer to produce work/power (heat engines) Both observations can be shown to be equivalent, violation of one equals violation of the other Can make other similar equivalent statements of 2 nd aw 2 nd aw Closed Systems: Classical Approach-1

2 Clausius Statement of the 2 nd aw It is impossible to construct a device that operates in a cycle and produces no effect on surroundings other than heat transfer from a lower temperature body to a higher temperature body (1850) means refrigerators and heat pumps require work input (power) W in Cooled Space Refrigerator Surroundings (hotter) eated Space eat Pump Surroundings (colder) W in 2 nd aw Closed Systems: Classical Approach-2

3 Kelvin-Planck Statement of the 2 nd aw It is impossible to construct a device that operates in a cycle and has no effect on surroundings other than producing a net work output and receiving (an equivalent amount of) heat transfer from a single hermal Energy Reservoir (ER) means heat engines must produce waste heat that must be transferred to surroundings 2 nd aw Closed Systems: Classical Approach-3 ER: fixed volume mass, only energy exchange as, and has uniform and (nearly) constant W out ER eat > eat ER eat <

4 Carnot s Propositions ow is temperature related to Second aw? Answer from examining reversible heat engines (classical example is Carnot cycle) Carnot s Propositions - corollaries of Clausius and Kelvin-Planck statements of 2 nd aw 1.It is impossible to construct a heat engine that operates between two ERs that has a higher thermal efficiency ( ηth Wnet,out ) than a reversible heat engine (η th,irrev < η th,rev ) 2.All reversible heat engines that operate between same two ERs have the same η th,rev 2 nd aw Closed Systems: Classical Approach-4

5 Proof of Carnot s Second Proposition ake two reversible devices, heat engine (E) that runs refrigerator (R) (refrig.heat engine run backwards) using same ERs E η th,e > η th,r (W/,E >W/,R ),R Rev. Refrig.,R W,E Rev. eat,e -,E Refrig.,R -,E <,R (also,e <,R ) But if call both devices one system, it looks like refrigerator with no work input - impossible 2 nd aw Closed Systems: Classical Approach-5

6 hermodynamic emperature Since η th,rev only depends on identity of ERs, only a function of absolute η th,rev η(, ) Rewrite efficiency, use 1 st aw η η th th W Apply this to many reversible engines, implies [ / ] rev f( )/f( ) chose def n. of thermodynamic [ / ] rev /, use in, net,out 1 η th,rev ( ) 1 W out eat Carnot Efficiency - true for totally rev. heat engines 2 nd aw Closed Systems: Classical Approach-6

7 Clausius Inequality, Entropy, and 2 nd aw Consider any reversible heat engine Since cyclic integral of / is zero must be a thermodynamic property, entropy From Carnot s 1 st Proposition, can show rev irrev heat addition + ds heat rejection rev < 0, or in general 0 ds 0 eat Clausius Inequality 2nd aw for Control Mass W 2 nd aw Closed Systems: Classical Approach-7

V. The Second Law of Thermodynamics. V. The Second Law of Thermodynamics

V. The Second Law of Thermodynamics. V. The Second Law of Thermodynamics P V. he Second aw of hermodynamics E. he Carnot Cycle. he reversible heat engine (a piston cylinder device) that operates on a cycle between heat reservoirs at and, as shown below on a P-v diagram, is