1. C. Truesdell,The efficiency of a homogeneous heat engine, J. Math. and Phys. Sciences (Madras),7 (1973), pp. 349–371;Corrigenda, ibidem,9 (1975), pp. 193–194.
2. R. Fosdick -J. Serrin,Global properties of continuum thermodynamic processes, Arch. Rational Mech. Anal.,59 (1975), pp. 97–109.
3. IfP is the power of the applied forces andK is the kinetic energy of the body, then W=P−
$$\dot K$$
.
4. The « paradox » was raised byA. Sommerfeld, Exercise 1.6 of hisThermodynamik und Statistik, ed.F. Bopp andJ. Meixner, Wiesbaden, 1952. The « solution » given there asserts that there are no adiabats connecting the isotherms above and below the inversion temperature for a given pressure. Such a claim could be just only if the inversion temperature were independent of pressure; experiment shows that it is not, nor need we assume it is in order to explain the « paradox », which merely reflects misunderstanding of classical thermodynamics. Only reversible cycles according to classical thermodynamics need be considered. Sketch (A) in fig. 2 represents the particular case in whichH has the same value on each of the adiabatic processes of the cycle. The cycle itself does not correspond either to a Carnot process or to aC-process. However, according to the classical thermodynamics of reversible processes. Net Work of (A)+Net Work of (C)=Net Work of (B); both (B) and (C) representC-processes in which γ+=γ−, and (20) applies to both. Hence Net Work of (A)=0. In sketch (A) the value ofH on the two adiabatic parts of the cycle is the same. A cycle of similar form but using adiabats with different values ofH can be obtained from sektch (A) by adjoining a Carnot cycle on the left-hand side or a Carnot refrigerator on the right-hand side. In the former case the total work done is that of the Carnot cycle; in the latter case, the total work consumed is that of the Carnot refrigerator. These same results may be obtained easily from the equations. IfH=H
1 on the left-hand adiabat andH=H
2 on the right-hand one, from (7) we see thatC
+−C
−=(θ
max−θ
min)(H
2−H
1) and hence from (2) thatU=(θ
max−θ
min)(H
2−H
1). IfH
2≦H
1, the classical estimate (18) holds trivially, while ifH
2 >H
1, then (7) and (1)2 show thatC
+>θ
max(H
2−H
1) and so again (18) follows. The « paradox » was explained correctly byJ. S. Thomsen - T. J. Hartka,Strange Carnot cycles, Amer. Journal of Physics,30 (1962), pp. 26–33, 388–389. The cycles themselves had been noticed and analysed correctly earlier byJ. E. Trevor,Carnot cycles of unfamiliar types, Sibley J. Engr.,42 (1928), pp. 274–278. In these papers the cycles look « strange » indeed because the authors begin by describing them in the plane of pressure and volume, which is altogether unsuited to fundamental studies. I mention the matter here partly because the solution does not seem to be well known and partly because it is produced instantly and effortlessly by the present, far more general estimates of efficiency.