Physics 301 25-Oct-2002 19-1Heat and WorkNow we want to discuss the material covered in chapter 8 of K&K. This material mightbe considered to have a more classical thermodynamics rather than statistical mechanicsflavor. We’ve already discussed a lot of this material in bits and pieces throughout theterm, so we will try to focus on the material not yet covered and just hit the highlights ofthe remaining material.Heat and work occur during processes. They are energy transfers. Work is an energytransfer by macroscopic means and heat is an energy transfer by microscopic means. We’vediscussed reversible processes several times and we’ll assume reversible processes unlesswe explicitly state otherwise. When work is done to or by a system, the macroscopicparameters of the system are changed—for example changing the volume causes pdVwork to be performed. Performing work changes the energy, U,ofasystem. Butworkdoes not change the entropy. Heat transfer changes the entropy as well as the energy:dU =¯dQ = τdσ.A very important activity in any modern society is the conversion of heat to work.This is why we have power plants and engines, etc. Basically all forms of mechanical orelectrical energy that we use involve heat to work conversion. Not all of them involve fossilfuels, and in some cases it may be hard to see where the heat enters. For example, whatabout hydro-electric power? This is the storage of water behind a dam and then releasingthe gravitational potential energy of the water to run an electric generator. Where is theheat supplied? Heat is supplied in the form of sunlight which keeps the weather goingwhich provides water in the atmosphere to make rain to fill the lake behind the dam. Ofcourse, the economics of this process are quite different from the economics of an oil firedelectrical generating plant.It was the steam engine (conversion of heat, obtained by burning coal, into work)that allowed the industrial revolution to proceed. The desire to make better steam enginesproduced thermodynamics!With an irreversible process you can turn work completely into heat. Actually, thisstatement is not well defined. What we really mean to say is that with an irreversibleprocess we can use work to increase the internal energy of a system and leave that systemin a final configuration that would be exactly the same as if we had reversible heated thesystem. For example, consider a viscous fluid in an insulating container. Immersed inthe fluid is a paddle wheel which is connected by a string running over various pulleysand whatever to a weight. The weight is allowed to fall under the influence of gravity.Because the fluid is so viscous, the weight drops at a constant slow speed. Once the weightreaches the end of its travel, we wait for the fluid to stop sloshing and the temperature andpressure in the fluid to become uniform. Thus essentially all of the mechanical gravitationalCopyrightc 2002, Princeton University Physics Department, Edward J. GrothPhysics 301 25-Oct-2002 19-2potential energy is converted to internal energy of the fluid. We can take the fluid fromthe same initial state to the same final state by heating slowly (reversibly!) until we havethe same temperature rise.There is no known way to convert (reversibly or non-reversibly) heat (more properly,internal energy, U) entirely into work with no other change. This is one of the ways ofstating the second law of thermodynamics.It is certainly possible to convert heat into work. (I’m getting tired of trying to say itexactly correctly, so I’ll just use the vernacular and you know what I mean, right?) Theconstraints are that you can’t convert all of it to work or there must be some permanentchange in the system or both. For example, suppose we reversibly add heat to an ideal gaswhile we keep the volume constant. Then we insulate the gas and allow it to reversiblyexpand until its temperature is the same as when we started. Then the internal energyof the gas is the same as when we started, so we have completely converted the heat intowork, but the system is not the same as when we started. The gas now occupies a biggervolume and has a lower pressure.The problem is that when we reversibly add heat to a system we add internal energydU =¯dQ and we also add entropy dσ =¯dQ/τ , but when we use the system to perform work,we remove only the energy dU =¯dW and leave the entropy! If we want to continue usingthe system to convert heat to work, we have to remove the entropy as well as the energy,so there is no accumulation of entropy. The only way to remove entropy (reversibly) is toremove heat. We want to remove less heat than we added (so we have some energy leftover for work) so we must remove the heat at a lower temperature than it was added inorder to transfer the same amount of entropy.To make this a little more quantitative, consider some time interval (perhaps a com-plete cycle of a cyclic engine) during which heat Qhis transfered into the system attemperature τh,heat−Qlis transfered into the system at temperature τl, and energy −Win the form of work is transfered into the system. (So heat Ql> 0 leaves the system andwork W>0 is performed on the outside world.) At the end of this time interval we wantthe system to be in the same state it was when we started. This means∆U =0=Qh− Ql− W,and∆σ =0=Qhτh−Qlτl.We findQhQl=τhτl,andηC=WQh=1−τlτh.Copyrightc 2002, Princeton University Physics Department, Edward J. GrothPhysics 301 25-Oct-2002 19-3The ratio of the heat input and output is the same as the ratio of the temperatures ofthe input and output reservoirs. The energy conversion efficiency or just efficiency, η isdefined as the work output over the heat input, W/Qinput. For the ideal engine we’vebeen considering, the efficiency is ηC,theCarnot efficiency, and is the upper limit to theefficiency of any real (i.e. non-reversible) engine operating between temperature extremesτhand τl. Carnot might be called the father of thermodynamics. He worked in the early1800’s and understood the second law. This was before heat was recognized as a form ofenergy!Of course, this definition of efficiency is motivated by the fact that if you’re an electricpower company, you can charge your customers based on W but you have to pay yoursuppliers based on Qhand you want to maximize profits!We live on the surface of the Earth and any engine must dump its waste heat, Ql,at what amounts to