We introduced already the Carnot cycle with an ideal gasNow we show:1Energy efficiency of the Carnot cycle is independent of the working substance2Any cyclic process that absorbs heat at one temperature, and rejects heat at one other temperature, and is reversible has the energy efficiency of a Carnot cycle reversibleRemark:Note:P>1Textbook:coefficient of performance

TcThheat engine XLets combine a fictitious heat engine X with with a heat pumprealized by a reversed Carnot cycleTcThheat pumpXCLets calculate with

If X would be a Carnot engine it would produce the work However:

FalseLet X be the heat pump and the Carnot cycle operate like an engineFalse1Energy efficiency of the Carnot cycle is independent of the working substance.2Any cyclic process that absorbs heat at one temperature,and rejects heat at one other temperature, and is reversible has the energy efficiency of a Carnot cycle. WhyBecause: X can be a Carnot engine with arbitrary working substance

Carnots theorem:No engine operating between two heat reservoirs ismore efficient than a Carnot engine. Proof uses similar idea as before:We can design the engine X such that Again we create a composite deviceoperates the Carnot refrigerator

My statementholds man

Lets assume thatNote: this time engine X can be also work irreversible like a real engine does>Heat transferred from the cooler to the hotter reservoir without doing work on the surroundingViolation of the Clausius statement

Any cyclic process that absorbs heat at one temperature, and rejects heat at one other temperature, and is reversible has the energy efficiency of a Carnot cycle. We stated:Why did we calculate energy efficiencies for - gas turbine- Otto cycleBecause:they are not 2-temperature devices, but accept and reject heat at a range of temperaturesEnergy efficiency not given by the Carnot formulaBut:It is interesting to compare the maximum possible efficiency of a Carnot cyclewith the efficiency of engineering cycles with the same maximum and minimum temperatures

Consider the gas turbine again 2341adiabates(Brayton or Joule cycle)EfficiencyPhPlMaximum temperature:@3Minimum temperature:@123Heating the gas (by burning the fuel)41cooling:T3:T1with

Efficiency of corresponding Carnot Cycle WithUnfortunately:Gas turbine useless in the limitBecause:Heat taken per cycle0Work done per cycle0

We showed:Energy efficiency of the Carnot cycle isindependent of the working substance.Definition of temperature independent of any material propertyMeasurement of Temperature ratio

As discussed earlier, unique temperature scale requires fixed point orKelvin-scale:Tfix =Ttripel=273.16KIt turns out:proportional to thermodynamic Temperature Tempirical gas temperatureWhyBecause:Calculation of efficiency of Carnot cycle based on yieldsWitha=1

From definition of thermodynamic temperature If any absolute temperature is positive all other absolute temperatures are positivethere is an absolute zero of thermodynamic temperaturewhen the rejected heat0T=0 can never be reached, because this would violate the Kelvin statementhowever