Simple Nature - Light and Matter

Carnot engines operating between a given pair of temperatures T Hand T L have the same efficiency. (This can be proved by the methodsof section 5.4.) Thus a Carnot engine is the most efficient possibleheat engine.h / Entropy can be understoodusing the metaphor of a waterwheel. Letting the water levelsequalize is like letting the entropymaximize. Taking water from thehigh side and putting it into thelow side increases the entropy.Water levels in this metaphorcorrespond to temperatures inthe actual definition of entropy.5.3.3 EntropyWe would like to have some numerical way of measuring thegrade of energy in a system. We want this quantity, called entropy,to have the following two properties:(1) Entropy is additive. When we combine two systems andconsider them as one, the entropy of the combined system equalsthe sum of the entropies of the two original systems. (Quantitieslike mass and energy also have this property.)(2) The entropy of a system is not changed by operating a Carnotengine within it.It turns out to be simpler and more useful to define changesin entropy than absolute entropies. Suppose as an example that asystem contains some hot matter and some cold matter. It has arelatively high grade of energy because a heat engine could be usedto extract mechanical work from it. But if we allow the hot andcold parts to equilibrate at some lukewarm temperature, the gradeof energy has gotten worse. Thus putting heat into a hotter areais more useful than putting it into a cold area. Motivated by theseconsiderations, we define a change in entropy as follows:∆S =QT[change in entropy when addingheat Q to matter at temperature T ;∆S is negative if heat is taken out]A system with a higher grade of energy has a lower entropy.Entropy is additive. example 10Since changes in entropy are defined by an additive quantity (heat)divided by a non-additive one (temperature), entropy is additive.Entropy isn’t changed by a Carnot engine. example 11The efficiency of a heat engine is defined byefficiency = 1 − Q L /Q H ,and the efficiency of a Carnot engine isefficiency = 1 − T L /T H ,so for a Carnot engine we have Q L /Q H = T L /T H , which can berewritten as Q L /T L = Q H /T H . The entropy lost by the hot reservoiris therefore the same as the entropy gained by the cold one.312 Chapter 5 Thermodynamics