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Thermal physics in the introductory physics course: Why and how to teach it from a unified atomic perspective
1.F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw–Hill, New York, 1965).
2.F. Reif, Statistical Physics, Berkeley Physics Course, Vol. 5 (McGraw–Hill, New York, 1967).
3.D. Halliday and R. Resnick, Physics (Wiley, New York, 1978), 3rd ed.
4.There are, of course, other ways that thermal physics can be taught from a microscopic point of view in an introductory physics course. For example, another such recent attempt is described in T. A. Moore and D. V. Schroeder, “A different approach to introducing statistical mechanics,” Am. J. Phys. 65, 26–36 (1997).
4.Also T. A. Moore, Six Ideas That Shaped Physics (Unit T: Some Processes are Irreversible) (McGraw–Hill, New York, 1997).
5.It is wise to avoid use of the word “microscopic” because students are likely to confuse it with the similarly sounding word “macroscopic.”
6.Negative work done on the system corresponds to positive work done by the system. Similarly, negative heat absorbed by the system corresponds to positive heat given off by the system.
7.The form (3) of the law ensures consistency with mechanics where W is used to denote the work done on a system. It also clearly exhibits the similarity between the work done on a system and the heat absorbed by it. (These advantages are not shared by a historical convention that uses W to denote the work done by a system.)
8.The analogy is due to H. B. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, 1985), 2nd ed.
9.Negative flow into the lake corresponds to positive flow out of the lake. Similarly, negative condensation into the lake corresponds to positive evaporation from the lake.
10.One can check that condensation has been eliminated by also eliminating the flow and checking that the water level then remains unchanged.
11.One can check that heat transfer has been eliminated by also eliminating all macroscopic work and checking that the thermometer indication then remains unchanged.
12.The work W in Eq. (3) has been defined (as in mechanics) as work done on a system. Hence the infinitesimal work done on a system by a pressure p in a small volume change is (that is, it is negative if the volume increases).
13.All the preceding basic ideas can be illustrated by giving students some simple problems dealing with tossed coins or dice. The aim should merely be to let students exemplify the definition of probability in some very simple cases. However, there is no need to introduce students to more complex knowledge about probabilities.
14.The term “basic state,” rather than “microstate,” is used here to avoid confusion with the similarly sounding word “macrostate.”
15.It is advisable to warn students that this specification of a basic state, although adequate for present purposes, must ultimately be replaced by a more correct quantum-mechanical specification.
16.There is no need to distinguish this specification in terms of velocities from the more formal specification in terms of momenta.
17.Many of today’s students already know the utility of digitizing information. For example, computers use such digitizing to specify possible positions on a display by discrete positions (“pixels”) or to specify possible sound intensities by discrete possible values.
18.Such graphs are sufficient to show what happens when the number N of molecules becomes larger. Thus there is no real need to teach students the binomial distribution.
19.Instead of observing fluctuations in an assembly of N systems, one may observe a single system at a large number N of successive instants (for example, by taking movie pictures every second). In a time-independent equilibrium situation the resultant pictures then constitute a statistical assembly equivalent to N different systems observed at any one time. Equivalent probability statements can then be made about these pictures, that is, about the fluctuations in a single system observed in the course of time.
20.It is instructive to show students successive computer-generated movie frames (or actual running movies) displaying simulations of gas molecules spreading out throughout a box (or the reverse process where they concentrate themselves into one half of the box). In particular, this reverse process looks strikingly different for 40 molecules or for 4 molecules. [Such movie frames are shown on pp. 22–25 of volume 5 of the Berkeley Physics Course (see the reference in Note 2).]
21.The absolute temperature can be negative if one considers special systems that have no kinetic energy. For example, as the internal energy of a system of spins in a magnetic field increases, the system’s entropy first increases and then decreases. In this case β, as well as T, can be negative.
22.Some examples or exercises with small numbers, describing unrealistically small systems, can help to clarify these considerations.
23.For very simple systems, such as ideal gases, Eq. (20) is equivalent to the condition that the average energy per particle is the same in each system.
24.More generally, heat flows from the system with lower β to the system with higher β. This is true even in the case of a system (of the kind mentioned in Ref. 21) where β or T can be negative. Unlike β, the absolute temperature T is then an inconvenient concept because a negative absolute temperature describes a very warm system whose temperature goes through infinity as the system cools down.
25.The topics discussed in Sec. III G can be accompanied by many of the exercises and problems commonly used in conventional approaches to thermal physics (for example, problems examining applications of the ideal gas laws, the temperature change caused by the free expansion of a gas, the dependence of molecular speeds on temperature or masses of the molecules, etc.).
26.The notation (with a prime ornamenting the d) is here used to denote an infinitesimal quantity which is not necessarily a small difference.
27.It is instructive to let students contrast a sudden adiabatic expansion of a gas (where its entropy increases) with a quasi-static adiabatic expansion (where this entropy remains unchanged). It can be pointed out that the latter case involves two competing effects: (a) an increase of the entropy because of the increased gas volume and (b) a decrease of the entropy because of the decreased internal energy (caused by the negative work done on the gas during its slow expansion).
28.There is no need to express these relations in terms of partial derivatives [for example, to write ] because beginning students may still be unfamiliar with this notation.
29.Similar entropy arguments apply to refrigerators designed to transfer heat from a reservoir at a lower absolute temperature to another one at a higher temperature.
30.To deal also with pressure effects, one may consider a system in contact with an environment at constant temperature and pressure so that In this case phase transformations may again be discussed by simple entropy considerations without resorting to the Gibbs free energy.
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