Entropy – the story of lost opportunities...

1. Entropy – the story of lost opportunities... Gas expands without doing any mechanical work vs. There was an opportunity for a spontaneous process – heat flow from Th to Tc. Heat transfer between a hot and cold object without mechanical work done.

2. What do these two processes have in common? Spontaneous (NOT quasi-static) expansion of a gas and heat transfer between two objects with different temperatures are both irreversible processes – lost opportunities. What kind of simple reversible processes do we have in stock? Thermal reservoir with constant temperature No heat transfer at all. isothermal adiabatic

3. How do we convert internal energy or heat into work? We build a heat engine. Isothermal engine 100% of the heat transferred to the system is converted to work…. In principle one can get an unlimited amount of work… BUT it will require an infinitely large expansion! What are we going to do after the gas expands? Run it back?

4. Isothermal engine As the system expands all the heat transferred to the system is converted to work…. As the system contracts back, though, the same amount of work is done by the surroundings and all the energy is returned to the reservoir. W < 0 W > 0

5. Adiabatic engine The positive work is now limited by the internal energy of the insulated system. But again, no net work is done if you go back and forth along the same adiabat. W < 0 W > 0

6. We need an engine working in cycles and converting heat supplied from the outside into mechanical work, possibly, with a high efficiency… How efficient can it be? The isothermal engine could convert 100% heat into work, but did not work cyclically. Can we match this performance with an engine operating in cycles? Any fundamental law prohibiting it? The second law of thermodynamics (Kelvin-Plank statement): It is impossible to construct a heat engine operating in a cycle that extracts heat from a reservoir and delivers and equal amount of work.

7. It is impossible to construct a heat engine operating in a cycle that extracts heat from a reservoir and delivers and equal amount of work That would be an ideal heat engine… What is a real heat engine doing? • Works between two temperatures, - a hot reservoir and a cold reservoir. (Hot side and cold side). • Gets some heat Qh (obtained from, say, burning a fuel) from the hot side • Rejects some heat Qc to the cold side. • Works in a cycle, so that the internal energy does not change, DU=0. • Does work W = Qh - Qc • Has an efficiency e = W/Qh

8. Carnot cycle http://www.ntu.edu.sg/home2000/S7231633I/

9. A - B Carnot cycle A B - C B C - D D C D - A What is the total work by the gas?

10. Carnot cycle What is the total work done by the gas? A B D C After some calculations we get Efficiency

11. Carnot cycle Efficiency • For highest efficiency we would want to run our engine between a very hot and a very cold reservoirs. • Large temperature difference, Th -Tc, and low temperature of the cold reservoir, Tc, are very helpful. • Efficiency can in principle reach 100% for Tc = 0, but we normally do not have such reservoirs available…