Chapter 13ThermodynamicsChapter 13IntroductionWhen we discussed the ideal gas, we were able to justify theempirical equation of state pV = NkT from the point of viewof Newtonian mechanics. We achieved this by defining certainintuitive ideas, such as temperature, in terms of precisemechanical concepts and by making reasonable assumptionsconcerning the statistical distribution of velocities in thegas. We also showed that our definition of temperature asproportional to the average kinetic energy is consistent withour intuitive notion. Our approach, however, has severalweak points. First, our definition of temperature as propor-tional to the average kinetic energy is appropriate for anideal gas, for which the potential energy is zero by defini-tion. But we would like to define temperature for allsystems, including those (such as solids) whose potentialenergy is significant or even larger in magnitude that thekinetic energy. It is by no means obvious that our definitionof temperature for an ideal gas can be used in other materialsystems. An even more serious problem with our approach isthat we still donÕt know how to explain the simplest phe-nomenon that occurs when two bodies are brought intothermal contact: the fact that their temperatures becomeequal. What is the driving ÒforceÓ for this phenomenon?Why is it that energy flows from the object with the highesttemperature to the object with the lowest temperature? Areversed energy flow would not violate NewtonÕs laws,because energy would still be conserved. However, anspontaneous flow of energy from a cold object to a hotobject is never observed. Why?In principle, the above questions can be answered withNewtonian mechanics. Given the initial positions andvelocities of any system, we can use NewtonÕs laws to predictthe positions and velocities at any later time, so that we cancompute any desired property. (Of course, Newtonianmechanics is superseded by quantum mechanics, but thisdoesnÕt change the argument. If we use quantum mechanics,we know that given the initial wave function of our systemwe can calculate the wave function at all later times byapplying Schr—dingerÕs equation. All measurable quantitiesphy 241 MenŽndez 212Chapter 13can be calculated from a knowledge of the wave function).From a practical point of view, however, our task is com-pletely unrealistic. The number of particles in any macro-scopic system is so fantastically high, that there is absolutelyno hope of computing the evolution of the system by usingNewtonÕs laws. Remember the time it took your spreadsheetto calculate the position as a function of time for twomasses connected by springs. How long would it take to dothe same for 1025 particles? How much computer memorywould you need? In one of the end of chapter problems, youare asked to compute the time needed to print the initialpositions and velocities for the atoms in a small volume ofgas. You will be surprised when you compare this time withthe age of the universe. On the other hand, even if we wereable to apply Newtonian mechanics to a system of very manyparticles, this approach appears as overkill when we try toexplain such simple phenomena as the equalization oftemperatures for objects in thermal contact. This is a verysimple phenomenon known by everybody, even thosewithout any instruction in physics. Simple phenomena areoften the result of simple underlying laws. There must besome simple principle that allows us to solve these problemswithout the full machinery of NewtonÕs laws. The branch ofphysics that deals with this topic is thermodynamics.The suffix ÒdynamicsÓ to the word ÒthermodynamicsÓemphasizes the fact that we are seeking and alternative tothe use of Newtonian dynamics. However, it is somewhatmisleading in that thermodynamics does not provide thetime-dependence of the properties of the system. In otherwords, thermodynamics, as you will soon learn, can explainwhy two bodies in thermal contact reach the same finaltemperature but says nothing about the time needed for thiscommon temperature to be reached. This partial lack ofinformation is one of the prices we will pay for not usingNewtonÕs laws to compute the properties of the system.Thermodynamics deals with equilibrium states: those statesof the system for which the macroscopic properties areconstant as a function of time. For example, if we let a gasphy 241 MenŽndez 213Chapter 13expand in a volume, eventually the gas will occupy the entirevolume. Thermodynamics will be able to predict the proper-ties of the gas once the entire volume is occupied, but willnot be able to make any predictions on the properties of thegas while it is expanding, nor to compute the time neededfor this expansion. Typically, the expansion of a gas in aÒhuman-sizeÓ volume takes a few seconds. In other cases, thetime needed by a system to reach a true ÒequilibriumÓ statemay be extremely long. A diamond, for example, (whichaccording to the TV ads is ÒforeverÓ) is unstable relative tographite: if you wait long enough, all diamonds shouldbecome cheap graphite crystals. However, you donÕt need tobe concerned about the diamonds you may have at home:they will outlast you and whoever you designate as your heir.The thermodynamicsystemWe will deal with systems of particles that are characterizedby a few macroscopic parameters. Examples of such parame-ters are the pressure in a gas, the volume, the electric andmagnetic moments, the temperature, the total number ofparticles, etc. Another set of useful macroscopic parametersare quantities that are conserved in isolated systems ofparticles, such as the internal energy U, the total linearmomentum p, and the total angular momentum L. In ourdiscussions, however, we will only consider systems for whichthe total linear and angular momenta are zero. If we have gasin a container, this insures that the container is fixed inspace an that the gas is not rotating as a whole. Theseconditions must be relaxed when thermodynamic ideas areapplied to the study of the atmosphere or to stellar gases,but we will not deal with these subjects.The macroscopic parameters are usually the only quantitieswe are interested in. For example, when you buy a six-packyou only specify the volume, the relative number of particles(alcohol vs. water, etc.) and the temperature. This is verylittle information for such a complicated system of zillionsof particles. The predictions we will make with thermody-namics concern those macroscopic