Physics 301 13-Sep-2002 1-1IntroductionIn this course we will cover selected topics in thermodynamics and statistical mechan-ics. Since we only have twelve weeks, the selection is necessarily limited. You will probablyneed to take a graduate course in thermal physics or do studying on your own in order togain a thorough knowledge of the subject.Classical (or maybe “conventional” is better) thermodynamics is an approach to ther-mal physics “from the large.” Statistical mechanics approaches the subject “from thesmall.” In thermodynamics, one is concerned with things like the pressure, temperature,volume, composition, etc., of various systems. These are macroscopic quantities and inmany cases can be directly observed or felt by our senses. Relations between these quan-tities can be derived without knowing about the microscopic properties of the system.Statistical mechanics takes explicit account of the fact that all systems are madeof large numbers of atoms or molecules (or other particles). The macroscopic properties(pressure, volume, etc.) of the system are found as averages over the microscopic properties(positions, momenta, etc.) of the particles in the system.In this course we will tend to focus more on the statistical mechanics rather thanthe thermodynamics approach. I believe this carries over better to modern subjects likecondensed matter physics. In any case, it surely reflects my personal preference!Some History (mostly taken from Reif)As it turns out, thermodynamics developed some time before statistical mechanics.The fact that heat is a form of energy was becoming apparent in the late 1700’s and early1800’s with Joule pretty much establishing the equivalence in the 1840’s. The second lawof thermodynamics was recognized by Carnot in the 1820’s. Thermodynamics continuedto be developed in the second half of the 19thcentury by, among others, Clausius, Kelvinand Gibbs.Statistical mechanics was developed in the late 19thand early 20thcenturies by Clau-sius, Maxwell, Boltzmann, and Gibbs.I find all of this rather amazing because at the time of the initial development ofthermodynamics, the principle of energy conservation hadn’t been firmly established. Sta-tistical mechanics was developed when the existence of atoms and molecules was still beingdebated. The fact that macroscopic properties of systems can be understood in terms ofthe microscopic properties of atoms and molecules helped convince folks of the reality ofatoms and molecules.Copyrightc 2002, Princeton University Physics Department, Edward J. GrothPhysics 301 13-Sep-2002 1-2Still more amazing is the fact that the foundations of statistical mechanics were de-veloped before quantum mechanics. Incorporating quantum mechanics did make somechanges, especially in the counting of states, but the basic approach and ideas of statisti-cal mechanics remained valid. I suspect that this is a reflection of both the strength andweakness of statistical methods. By averaging over many molecules you derive results thatare independent of the detailed properties of individual molecules. The flip side is thatyou can’t learn very much about these details with statistical methods.Some Thermodynamic ConceptsFrom mechanics, we’re familiar with concepts such as volume, energy, pressure (forceper unit area), mass, etc. Two new quantities that appear in thermodynamics are tem-perature (T )andentropy(S).We will find that temperature is related to the amount of energy in a system. Highertemperature means greater internal energy (usually). When two systems are placed incontact, energy in the form of heat flows from the higher temperature system to the lowertemperature system. When the energy stops flowing the systems are in thermal equilibriumwith each other and we say they are at the same temperature. It turns out if two systemsare in thermal equilibrium with a third system, they are also in thermal equilibrium witheach other. (This is sometimes called the zeroth law of thermodynamics.) So the conceptof temperature is well defined. It’s even more well defined than that as we will see later inthe course.Two systems can exchange energy by macroscopic processes, such as compression orexpansion or by microscopic processes. It is the microscopic process that is called heattransfer. Consider a collision among billiard balls. We think of this as a macroscopicprocess and we can determine the energy transfer involved by making measurements ofa few macroscopic parameters such as the masses and velocity components. If we scaledown by 24 orders of magnitude, we consider a collision between molecules, a microscopicprocess. A very large number of collisions occur in any macroscopic time interval. A typicalmolecule in the atmosphere undergoes ∼ 1010collisions per second. All these collisionsresult in the exchange of energy and it is the net macroscopic transfer of energy resultingfrom all the microscopic energy transfers that we call heat.Recall that the first law of thermodynamics isdU = dQ + dW ,where dU is the change of (internal) energy of a system, dQ is energy added to the systemvia a heat transfer, and dW is energy added by doing work on the system.Aside: you will often see the heat and work written as ¯dQ and ¯dW . This is a reminderthat these quantities are not perfect differentials, just small changes. A system has a wellCopyrightc 2002, Princeton University Physics Department, Edward J. GrothPhysics 301 13-Sep-2002 1-3defined internal energy U(P,V,...) which can be differentiated with respect to P , V , ...,but there is no such thing as the heat or work content of a system. The heat and workrefer to energy transfers during a change to the system.So the first law really boils down to a statement of energy conservation. You canchange the energy of a system by adding energy microscopically (dQ) or macroscopically(dW ).While we’re at it, the second law can be stated in many ways, but one way (withoutworrying too much about rigor) is: it’s impossible to turn heat completely into work withno other change. So for example, if you build a heat engine (like a power plant) you can’tturn all the heat you get (from burning coal) completely into electrical energy. You mustdump some waste heat. From this law, one can derive the existence of entropy and thefact that it must always increase. (Or you can define entropy, and state the second law interms of the increase in entropy).EntropyEarlier, we mentioned