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Teaching the in introductory Harvey S. Leffa) Department of Physics, California State Polytechnic University, Pomona, 3801 West Temple Avenue, Pomona, California 91768 ͑Received 19 October 2001; accepted 28 March 2002͒ The is often the only for which equations of state are studied in introductory physics. The can be a rich supplement to the ideal gas, and a vehicle for introducing 20th century physics concepts. © 2002 American Association of Physics Teachers. ͓DOI: 10.1119/1.1479743͔

I. INTRODUCTION emission rates are equal. Thus, an apparently empty con- tainer actually is filled with a photon gas, a fact that can The monatomic classical ideal gas, which is called the intrigue students. ideal gas in this paper, is a staple of introductory physics. It Unlike the ideal gas, for which there are three independent is understandable in the context of a system of noninteracting variables, N, T, and V, the photon gas has just two indepen- point ,1 and its equations of state are tractable math- dent, controllable variables, T and V. We can envisage build- ematically. These virtues of the ideal gas are summarized in ing a photon gas from stored in the container walls. Table I. With them come several less desirable characteris- Consider the container in Fig. 1. Imagine purging it of all tics. First, it is based on classical physics or the semiclassical with a vacuum pump, and then moving the piston to limit of a quantum ideal gas, and hence does not provide the left until it touches the left wall. The is then zero insights on quantum or relativistic phenomena. Second, the and the walls ͑including the piston͒ have T. is independent of volume, a property that Now slowly move the piston to the right, keeping the wall holds only in the low limit for real . Third, the temperature constant using a reservoir. will pour out ideal gas gives no insight into changes of state such as the of the walls as the volume increases, until the dynamic equi- librium described above occurs. In this way, we mentally to liquid transition. Finally, because it is usually the construct a photon gas of volume V and temperature T, with only system for which equations of state are encountered, students tend to come away with a sense that every system average photon number N(T,V). Building the photon gas behaves as an ideal gas. using this thought experiment can help develop an under- standing of the nature of the variable photon gas. In contrast, the photon gas is a quantum mechanical sys- The most straightforward approach for introducing the tem of particles ͑quanta of the electromagnetic field͒ called 2 photon gas in introductory physics is to define it in a way photons. The thermal behavior of photons in blackbody ra- similar to that in the preceding paragraphs, and to then dis- diation has played a pivotal role in the development of quan- play the relevant of state and ex- tum mechanics. In addition, because photons move with the amine their implications. The extent to which the equations speed of , the photon gas is a relativistic system. Thus it of state are used can vary. At minimum, the equations can be reflects two major developments of 20th century physics: presented and interpreted. If time allows and the interest and relativity. Over a decade ago, the level is sufficient, they also can be used to analyze isother- Introductory University Physics Project called for more 20th mal and adiabatic processes for the photon gas, as is done in century physics in introductory courses.3 Inclusion of the the body of this paper. photon gas would toward this goal. A more ambitious approach is to use kinetic to The quantum mechanical amplitudes for pho- establish the connection between internal energy and pres- tons can interfere constructively or destructively with one sure for the photon gas, and to use calculus to derive the another, but photons do not ordinarily affect one another’s equations of state. This procedure is presented in the Appen- , momenta, or polarizations, which simplifies their dix as a resource for teachers. thermodynamic behavior. Unlike the ideal gas, the internal A rich literature, mainly related to blackbody radiation, 4 5 energy function for the photon gas is volume dependent. Re- exists in books on modern physics, quantum physics, 6 7–13 markably, despite its nonatomic nature, the photon gas can optics, and classical and statistical . Nu- provide insights into the liquid–vapor phase transition. Fi- merous citations to the literature are given in Ref. 14. Nev- nally the very notion of the photon gas disabuses students of ertheless, the photon gas has not found its way into introduc- the thought that every thermodynamic system behaves as an tory physics textbooks. The main purpose of this paper is to ideal classical gas. encourage teachers of introductory physics and textbook au- thors to adopt the photon gas as a supplement to the ideal A key feature of the photon gas is that it has a variable gas. Some of the ideas here might also be useful to teachers , N. Consider a container of volume V, of modern physics and junior-senior level . whose walls are maintained at temperature T. Suppose it has been emptied of matter by a vacuum pump. It cannot be II. PHOTON GAS EQUATIONS OF STATE AND entirely ‘‘empty’’ because the walls radiate photons into the PROCESSES container. Some photons scatter off the walls, with some be- ing absorbed and new ones being emitted continually. A dy- The equations of state for the internal energy U(T,V) and namic equilibrium exists when the average absorption and P(T) are

792 Am. J. Phys. 70 ͑8͒, August 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers 792 Table I. Summary of monatomic classical ideal gas properties. Given the foregoing, it is straightforward to analyze iso- thermal processes for the photon gas. Consider a quasistatic Property type Description volume change from V to Vϩ⌬V. From Eq. ͑1͒, if T is Independent variables N, V, T constant, System Collection of N noninteracting 4 point particles, each with m, ⌬UϭbT ⌬V. ͑4͒ described by Temperature-independent result 3 From the first law of thermodynamics, ⌬UϭQϩW, where Uϭ 2PV from kinetic theory, relating the WϭϪ͐P(T)dV is the work done on the photon gas. When internal energy U and pressure P T is constant, so is P(T), and Eq. ͑2͒ implies that Internal energy 3 Uϭ 2NkT Pressure PϭNkT/V 1 4 WϭϪ 3 bT ⌬V. ͑5͒ Entropya SϭNk͓ln(T3/2V/N) 3/2 5 ϩln(2␲mk/h) ϩ 2͔ Thus the energy gained by the photon gas from the concomi- Work W on the gas for isothermal WϭϪNkT ln(1ϩ⌬V/V) tant increase or decrease in the number of photons, which volume change by ⌬V constitutes a radiative process, is Energy Q added to gas by heat QϭNkT ln(1ϩ⌬V/V) process for isothermal volume Qϭ 4 bT4⌬V. ͑6͒ change by ⌬V 3 change of gas for isothermal ⌬SϭNk ln(1ϩ⌬V/V) The slow isothermal volume change under consideration is volume change by ⌬V Reversible adiabatic condition, PV5/3ϭconstant reversible, and the entropy change, Q/T, of the photon gas is where the ratio ⌬Sϭ 4 bT3⌬V. ͑7͒ C p /CVϭ5/3 3 aThe expression for the entropy is the Sackur–Tetrode equation, the classical This entropy change, which is linear in ⌬V, is very different limit for Bose–Einstein and Fermi–Dirac quantum ideal gases with atoms from the logarithmic volume dependence of the correspond- of mass m. Planck’s constant h connotes the Sackur–Tetrode equation’s ing entropy change for an ideal gas, which is shown in Table quantum origin, and k reflects the thermodynamic nature of the gas. I. Now suppose we build the photon gas as described earlier by choosing the initial volume in Eq. ͑7͒ to be zero and the final volume to be V. We then allow the piston to move to ϭ 4 ͑ ͒ U͑T,V͒ bVT 1 the right slowly to volume V, creating the photon gas. Be- and cause at zero volume, the photon number Nϭ0, evidently Sϭ0; that is, there can be no entropy if there are no photons. 1 4 P͑T͒ϭ 3 bT . ͑2͒ Equation ͑7͒ then implies that at volume V,

4 3 The constant b cannot be determined from thermodynamics, Sϭ 3 bVT . ͑8͒ but its value can be borrowed from or ex- perimental results. It is given by Notice that Eqs. ͑1͒ and ͑2͒ imply that the enthalpy15 H ϭUϩPV is 8␲5k4 Ϫ16 Ϫ4 Ϫ3 bϭ 3 3 ϭ7.56ϫ10 J K m . ͑3͒ 4 4 15h c Hϭ 3 bVT . ͑9͒ Note that b depends on Planck’s constant h, which reflects Clearly for an isothermal volume change, ⌬HϭQ and for an the quantum mechanical nature of the photon gas, the speed expansion from zero volume to volume V, HϭQϭTS. This of light c, which reflects its relativistic nature, and Boltz- example makes clear that is the energy needed to mann’s constant k, which reflects its thermodynamic nature. form the photon gas and to do the work needed to make At the introductory level, we can introduce Eqs. ͑1͒ and ͑2͒ available the volume V it occupies.16 and give the numerical value of b, without broaching the Next, consider a slow adiabatic volume change. Adiabatic formula in Eq. ͑3͒. means that no photons are emitted or absorbed by the con- tainer walls. For a photon gas, the only possible type of heat process is via radiation; that is, energy can be exchanged with the container only by the emission and absorption of photons. An adiabatic volume change requires that the con- tainer walls be perfectly reflecting mirrors. Under such a process the photon number cannot change because a perfect reflector is also a nonemitter of photons. Because the number of photons cannot change, N(T,V)ϭconstant. Furthermore, a slow, reversible leaves the entropy of the photon gas unchanged, so S(T,V)ϭconstant. The constancy of both N(T,V) and S(T,V) implies that N(T,V)ϭconstant ϫS(T,V), which along with Eq. ͑8͒, leads to the conclusion Fig. 1. Photon gas in a container with wall temperature T and volume V. that The right wall is movable and its quasistatic movement can alter V revers- ibly by ⌬V. N͑T,V͒ϭrVT3, ͑10͒

793 Am. J. Phys., Vol. 70, No. 8, August 2002 Harvey S. Leff 793 Table II. Comparison of equations for classical ideal and photon gases.

Classical ideal gas Photon gas

N is specified and fixed NϭrVT3 3 UϭbVT4ϭ2.7NkT Uϭ 2NkT PϭNkT/V 1 4 Pϭ 3bT ϭ0.9NkT/V 3/2 3/2 5 4 3 SϭNk͓ln(T V/N)ϩln(2␲mk/h) ϩ 2͔ Sϭ 3bVT ϭ3.6Nk

Fig. 2. A plot of pressure P versus volume V for a reversible using a photon gas as the working fluid. The horizontal segments 1–2 and compare it with the ideal gas. One comparison examines 3–4 are isothermals at Th and Tc , respectively. Segment 2–3 corresponding equations of state of the two gases. Such a is a reversible adiabatic expansion. Segment 4–1 occurs at zero volume and comparison is shown in Table II, using the fact that N therefore entails only the container walls and not the photon gas. ϭ2.03ϫ107 VT3. In this view, the pressure and internal en- ergy functions are remarkably similar if the dependent vari- able N is displayed explicitly for the photon gas. Thus for where r is a constant. As was the case for b, r cannot be example, the average energy per photon in a photon gas is evaluated within the domain of thermodynamics, but its 2.7 kT, compared with 1.5 kT for the ideal gas. Similarly, value can be borrowed from statistical physics,17 the pressure P of the photon gas is 0.9 NkT/V compared 3 k with NkT/V for the ideal gas. Keep in mind, however, that N rϭ60.4 ϭ2.03ϫ107 mϪ3 KϪ3. ͑11͒ ͩ hcͪ is not an independent variable for the photon gas, so the similarities are strictly formal. The entropy functions are As before, the constants h, c, and k illustrate the quantum very different looking for the photon and ideal gases. Nota- mechanical, relativistic, and thermodynamic nature of the bly, the entropy per photon is 3.6k, independent of tempera- photon gas. Also as before, to keep the discussion elemen- ture. tary, the value of r can be stated without broaching the for- A numerical comparison is shown in Table III. The ideal mula in Eq. ͑11͒. gas is taken to have the mass of monatomic argon and is at From Eqs. ͑8͒ and ͑2͒, it is clear that the condition of 300 K and normal atmospheric pressure. The numerical val- constant entropy is ues of N, U, P, and S for the photon gas are all approxi- T3Vϭconstant or PV4/3ϭconstant. ͑12͒ mately 10 or more orders of magnitude smaller than for the ideal gas, which is why we can ignore the photon gas when Interestingly, the second form in Eq. ͑12͒ is similar to the discussing the thermodynamics of an ideal gas in the vicinity corresponding condition in Table I for an ideal gas, for which ␥ of room temperature and atmospheric pressure. PV ϭconstant, where ␥ϭC p /CVϭ5/3. However, the simi- On the other hand, for sufficiently high temperatures, the larity is only formal because for the photon gas, C p does not number of photons can exceed the number of ideal gas atoms even exist, because one cannot vary the temperature at con- in an equal volume V. Indeed, it is straightforward to show stant pressure for a photon gas. that if 1.00 mol of argon ideal gas is at standard atmospheric Having found Eqs. ͑2͒ and ͑12͒, we can sketch a reversible pressure, 1.01ϫ105 Pa, the corresponding average number Carnot cycle on a pressure–volume diagram. We take advan- of photons exceeds the number of atoms for any V if T tage of the fact that the photon gas can be brought to zero Ͼ1.38ϫ105 K. The equations in Table II imply that for T volume, so that the Carnot cycle looks as shown in Fig. 2. 5 The horizontal isotherms 1–2 and 3–4 come from the fact Ϸ1.41ϫ10 K, the ideal and photon gases have nearly the that PϭP(T). The adiabatic segment 2–3 is qualitatively same internal energies and , while the ideal gas en- tropy is still significantly larger than the photon gas entropy. similar to reversible adiabatic curves for the ideal gas. The 3 4 vertical segment 4–1 corresponds to heating the container Of course, because of the T behavior of N and S, and the T behavior of P and U, the photon gas will dominate the ideal walls from Tc to Th , with zero photons in the zero volume container. Along this segment, the walls undergo a heat pro- gas in all respects for sufficiently high T. In this discussion cess, but there is no photon gas present and thus Qϭ0 for a we have ignored the ionization of the ideal gas atoms that photon gas working fluid. would occur at such high temperatures. Figure 2 makes it clear that the pressure–volume represen- Other comparisons are possible. For example, the ideal tation of a Carnot cycle does not necessarily appear as it does gas entropy becomes negative for sufficiently small T and for an ideal gas. The purpose here is not to exhibit a real diverges to negative infinity in the limit T 0. This inad- → working fluid, but rather to illustrate that a Carnot cycle’s P – V plot can differ from that obtained for the ideal gas. It is a good exercise for students to use the photon gas equations Table III. Numerical comparison of classical ideal and photon gas functions. 5 to show that, as expected, the thermal efficiency is ␩ϭ1 Here the ideal gas is 1.00 mol of monatomic argon at Pϭ1.01ϫ10 Pa, Vϭ2.47ϫ10Ϫ2 m3, and Tϭ300 K. ϪTc /Th , the reversible Carnot cycle efficiency for any working fluid. Function Classical ideal gas Photon gas

N 6.02ϫ1023 atoms 1.35ϫ1013 photons III. COMPARISON: IDEAL AND PHOTON GASES U 3.74ϫ103 J 1.51ϫ10Ϫ7 J P 1.01ϫ105 Pa 2.04ϫ10Ϫ6 Pa After introducing the photon gas and examining its behav- S 155 J/K 6.71ϫ10Ϫ10 J/K ior under isothermal and adiabatic processes, it is useful to

794 Am. J. Phys., Vol. 70, No. 8, August 2002 Harvey S. Leff 794 equacy of the ideal gas model comes from its classical char- W mϪ2 KϪ4, is the Stefan–. In appropri- acter. In contrast, the photon gas entropy approaches zero in ate contexts, one can move on to study Planck’s radiation the Tϭ0 limit. Similarly, the ideal volume heat law and, ultimately, quantum mechanics. capacity is constant and remains so in the zero temperature limit, unlike real gases, whose heat capacities approach zero, consistent with the third law of thermodynamics. The photon V. CONCLUSIONS 3 gas heat capacity at constant volume, CVϭ4bVT , ap- The photon gas can enrich the introduction to thermody- proaches zero in this limit, as the number of photons ap- namics. Its basic equations lead students into new territory proaches zero. The behavior of the photon gas reflects its involving creation and annihilation of photons, which pro- quantum mechanical nature. vides a thought provoking introduction to modern physics The ideal gas gives no indication of the condensation phe- ideas. nomenon that a real gas experiences at temperatures below An important related point is that photons are everywhere. its critical temperature. As a real gas is compressed isother- That is, because all matter radiates, it is literally impossible mally at a temperature below its critical temperature, part of to have a region of space that is free of photons. In this it begins to condense into the liquid state, keeping the pres- sense, the photon gas has the distinction of being ubiquitous, sure constant. Although it is a very different kind of system, another point that can pique the intellectual curiosity of stu- with very different physics, the photon gas can shed light on dents. this phenomenon because under isothermal compression, Despite the evident richness of the photon gas, its equa- photons get absorbed, becoming part of the energy of the tions of state are tractable and have straightforward interpre- walls, providing an analogy to the vapor-to-liquid transition: tations. In addition to its potential for enriching the study of the photons play the role of the vapor , and the thermal physics, the photon gas serves as a good foundation absorbed energy of the walls are the analog of the liquid for subsequent introduction to cavity radiation. molecules. The energy of compression to zero volume for the In summary, the photon gas has much to offer teachers and photon gas is the rough analog of the heat of condensation students. Its study can supplement the ideal gas or can be for the real gas. initiated in a course on modern physics. A more in-depth For the photon gas, the pressure remains constant as the treatment is appropriate for junior or senior level thermal volume decreases at constant temperature because the num- physics. ber of photons decreases while the energy of the walls in- creases. For a real gas, the pressure remains constant during isothermal compression because the number of gas mol- ACKNOWLEDGMENT ecules decreases as gas molecules become liquid molecules. Condensation phenomena occur because of attractive I thank John Mallinckrodt and Ron Brown for very helpful between molecules, while photon absorption occurs because comments on a first draft of this paper. atoms and molecules continually absorb and emit radiation, as their electronic energies increase and decrease. Although APPENDIX the physics differs for the two phenomena, both are charac- terized by constant temperature, constant pressure, and a The objective of the Appendix is to derive Eqs. ͑1͒ and variable number of gas particles. Discussion of these matters ͑2͒. Kinetic theory enables us to deduce a simple relationship can shed light not only on radiation, but on the phenomenon between the internal energy U and pressure P of a photon of condensation. gas, 1 U͑T,V͒ 1 P͑T͒ϭ ϭ u͑T͒. ͑A1͒ 3 V 3

IV. TWO BRIEF EXAMPLES The right-hand side defines the u(T). To obtain Eq. ͑A1͒, we make several assumptions. First, Perhaps the most exciting example of a photon gas is the we assume isotropy, namely, the average number of right- cosmic microwave background radiation.18 The latter is in ward moving photons within a specified range of velocity is essence a gas of ‘‘old’’ photons that was created in the early, the same as the corresponding average number of leftward hot Universe approximately 13 billion years ago, and which moving photons. Denote the number of photons per unit vol- has cooled to 2.7 K. Inserting the latter temperature and the ume with x components of velocity between cx and cx value of r in Eq. ͑11͒ into Eq. ͑10͒ gives N/V ϩdcx by n(cx)dcx . The assumed isotropy implies n(cx) ϭ416 photons/cm3. Such photons, which make even dino- ϭn(Ϫcx). ͑Because photons all have speed c, cx varies saur bones seem rather young, are in our vicinity all the time. solely because of differing velocity directions.͒ Integration An awareness of the photon gas opens the door to an under- over cx gives standing of this remarkable phenomenon. c c N A second example uses a well-known result from kinetic n͑cx͒dcxϭ2 n͑cx͒dcxϭ , ͑A2͒ theory19 together with Eq. ͑10͒. Suppose a photon gas exists ͵Ϫc ͵0 V in a cavity within a solid that is at temperature T, and that where N is the ͑average͒ number of photons. The function photons can leak out through a small opening in the walls. 1 Vn(cx)/N is a function that can be The kinetic theory result for the particle flux is 4(N/V)c used to calculate averages such as 1 3 2 ϭ 4rcT , and the energy flux, measured in watts/m , is c c 1 1 4 Vn͑cx͒ Vn͑cx͒ 4(N/V)(U/N)ϭ 4cbT . This result is the well-known energy 2ϭ 2 ϭ 2 8 cx cx dcx 2 cx dcx . ͑A3͒ flux from a blackbody, where cb/4ϭ␴ϭ5.67ϫ10 ͵Ϫc N ͵0 N

795 Am. J. Phys., Vol. 70, No. 8, August 2002 Harvey S. Leff 795 2 The last step follows because n(cx)cx is an even function of The individual entropy changes of the photon gas along ϭ 4 ϭ cx . Equation ͑A3͒ will be used in our derivation of Eq. ͑A1͒. the Carnot cycle are ⌬S12 3u(Th)V/Th , ⌬S23 0, ⌬S34 4 Our second assumption is that there is a well-defined av- ϭϪ 3u(Tc)(VϩdV)/Tc , and ⌬S41ϭ0, respectively. Be- erage photon energy ͗e͘, which depends solely on the wall cause S is a , these changes also must add to temperature and not on the system volume V. This assump- zero along the cycle. Using dVӶV and dTϵThϪTcӶTc , tion is motivated by the expectation that the energy distribu- we have 1/ThϷ(1ϪdT/Tc)/Tc , and the condition for zero tion of the emitted photons depends on the wall temperature. entropy change along the cyclic path becomes In contrast, the average number of photons N must depend Ϫ on both temperature T and volume V because in equilibrium, 4 ͑u͑Th͒ u͑Tc͒͒V u͑Tc͒dV u͑Th͒V dT Ϫ Ϫ 2 ϭ0. 3 ͫ Tc Tc T ͬ the absorption and emission rates can be equal only if N c achieves a sufficiently large value, which increases with V. ͑A6͒ Our last assumption is that N is proportional to the system Replacing Tc with T and Th with TϩdT, and retaining only volume; that is, N(T,V)ϭn(T)V. We now proceed with the first order terms, Eq. ͑A6͒ becomes kinetic theory derivation. Fix the right wall in Fig. 1 and denote the container’s uV dT V duϪu dVϪ ϭ0. ͑A7͒ horizontal length by L and its cross sectional area by A. T Choose a small time interval ⌬tӶL/c, and consider right- ward moving photons with x-components of velocity be- We can eliminate dV by combining Eqs. ͑A5͒ and ͑A7͒ to obtain tween cx and cxϩdcx , located within distance cx⌬t from the right wall. The latter region has spatial volume Acx⌬t, 1 u and an average number of photons, (Ac ⌬t)n(c )dc , duϭ dT. ͑A8͒ x x x 4 T within it will collide with the right wall in time interval ⌬t. The magnitude for a photon with energy e is e/c Finally, integration of Eq. ͑A8͒ gives u(T)ϭbT4, which is and the x component of its momentum is (e/c)(cx /c). In an equivalent to Eq. ͑1͒. As mentioned, the numerical value of b elastic with the right wall a photon’s momentum is obtained from . The combination of change is 2(e/c)(cx /c), and the average it exerts on Eqs. ͑1͒ and ͑A1͒ gives Eq. ͑2͒. 2 20 the wall during time ⌬t is 2ecx /(c ⌬t). The average force for all such by photons with average energy ͗e͘ a͒Electronic mail: [email protected] 1Weak interactions must exist in order for the gas to achieve thermody- and x component of velocity between cx and cxϩdcx is 2 namic equilibrium. It is assumed that these interactions have negligible (2͗e͘cx /c )(Acx)n(cx)dcx . effects on the equations of state. 2 If we integrate over cx from 0 to c and divide by area A, The term ‘‘photon’’ was coined by Gilbert N. Lewis, ‘‘The conservation of the average pressure on the wall from photons impinging at photons,’’ Nature ͑London͒ 118, 874-875 ͑1926͒. 3 all angles is See Lawrence A. Coleman, Donald F. Holcomb, and John S. Rigden, ‘‘The introductory university physics project 1987–1995: What has it accom- 2͗e͘ c ͗e͘N 1 ͗e͘N plished,?’’ Am. J. Phys. 66, 124–137 ͑1998͒, and references therein. 2 2 4 Pϭ 2 n͑cx͒cx dcxϭ 2 cx ϭ . ͑A4͒ Stephen T. Thornton and Andrew F. Rex, Modern Physics, 2nd ed. ͑Saun- c ͵0 Vc 3 V ders College Publishing, New York, 2000͒, pp. 91–96, 296–298. 5David J. Griffiths, Introduction to Quantum Mechanics ͑Prentice-Hall, The penultimate step follows from Eq. ͑A3͒ and the last step Englewood Cliffs, NJ, 1995͒, pp. 216–218, 311–312. 2 2 6 follows from the isotropy condition, cx ϭc /3. In Eq. ͑A4͒, Eugene Hecht, Optics ͑Addison-Wesley Longman, Reading, MA, 1998͒, N͗e͘ϭU, the internal energy of the photon gas. With our pp. 50–57, 575–583. 7C. J. Adkins, Equilibrium Thermodynamics ͑McGraw-Hill, New York, assumptions that ͗e͘ is solely a function of T and N 1968͒, pp. 146–159. 1 8 ϭn(T)V, Eq. ͑A4͒ reduces to Eq. ͑A1͒, P(T)ϭ 3u(T), Ralph Baierlein, Thermal Physics ͑Cambridge University Press, Cam- where u(T)ϭU/V. It is clear from Eq. ͑A1͒ that pressure is bridge, 1999͒, pp. 116–130. 9 solely a function of temperature and thus, the pressure– Ashley H. Carter, Classical and Statistical Thermodynamics ͑Prentice- volume isotherms for a photon gas form a family of Hall, Upper Saddle River, NJ, 2001͒, pp. 186–190. 10Daniel V. Schroeder, An Introduction to Thermal Physics ͑Addison- constant–pressure curves. We now use the forms PϭP(T) Wesley, New York, 2000͒, pp. 288–307. and Uϭu(T)V, along with the Carnot cycle in Fig. 2 to 11Keith Stowe, Introduction to Statistical Mechanics and Thermodynamics derive Eq. ͑1͒. Along segments 1–2, 2–3, 3–4, and 4–1, the ͑Wiley, New York, 1984͒, Chap. 26. 12Raj K. Pathria, Statistical Mechanics, 2nd ed. ͑Butterworth-Heineman, internal energy changes of the photon gas are ⌬U12 1 Vϩ⌬V Oxford, 1996͒, Appendixes B, D, E. ϭu(Th)V, ⌬U23ϭϪ 3͐V u(T)dV, ⌬U34ϭϪu(Tc)(V 13Daniel J. Amit and Yosef Verbin, Statistical Physics: An Introductory ϩ⌬V), and ⌬U41ϭ0, respectively. Because U is a state Course ͑World Scientific, Singapore, 1995͒. Relevant kinetic theory is function, these changes must add to zero along the cycle. discussed in Part I, Chap. 1, and the photon gas is discussed in Part IV, Simplification occurs when T ϪT ӶT and ⌬VӶV, in Chap. 4. h c c 14Harvey S. Leff ͑in preparation͒. which case we replace ⌬V by dV. Then the addition of the 15Students of introductory physics do not typically encounter enthalpy in 1 four internal energy changes gives u(Th)VϪ 3u(T)dV their courses, and the discussion in the text can be skipped. On the other hand, students are introduced to enthalpy in their courses, and Ϫu(Tc)(VϩdV)ϭ0, where TcрTрTh . Replacement of the discussion in the text might help them appreciate, if not fully under- u(T) with u(Tc) induces an additive error рduϵu(Th) stand, enthalpy in a wider context. Ϫu(Tc). Thus to first order in du and dV the sum of the 16See Ref. 10, p. 33. internal energy changes around the cycle reduces to 17In Eq. ͑11͒, the prefactor 60.4 in r arises from an integral, which gives 8␲␨(3)⌫(3)ϭ60.4. Here ␨(3) and ⌫(3) are zeta and gamma functions, 4 V duϪ 3 u dVϭ0. ͑A5͒ respectively.

796 Am. J. Phys., Vol. 70, No. 8, August 2002 Harvey S. Leff 796 18Edward Harrison, Cosmology: The Science of the Universe, 2nd ed. ͑Cam- still holds because the average number of photons coming from a wall bridge University Press, Cambridge, 2000͒, Chap. 17. section in any time interval must equal the number approaching the wall, 19E. H. Kennard, Kinetic Theory of Gases ͑McGraw-Hill, New York, 1938͒, and the average momentum approaching must equal the average momen- Sec. 37. tum leaving. The mathematical treatment used for elastic collisions thus 20The assumption of elastic collisions does not hold for walls that can absorb holds in an average sense when inelastic collisions occur, and isotropy and emit photons. In that case, however, isotropy guarantees that Eq. ͑A1͒ holds.

Lissajous Figure Drawing Device. Lissajous figures are the resultant of two simple harmonic at right angles to each other. In this dedicated device, the SHMs are provided by two pendula swinging at right angles to each other. Horizontal rods coupled to the tops of the pendula drive a stylus P that scrapes the soot from a smoked glass plate held on the stage of the overhead projector. The audience thus sees a black screen with the figure being traced out in white. The illustration is from J. A. Zahm, and Music, second edition ͑A. C. McClurg & Co., Chicago, 1900͒, p. 409 ͑Notes by Thomas B. Greenslade, Jr., Kenyon College͒

797 Am. J. Phys., Vol. 70, No. 8, August 2002 Harvey S. Leff 797