One of entropy’s puzzling aspects is its dimensions of energy/temperature. A review of thermodynamics and statistical mechanics leads to six conclusions: (1) Entropy’s dimensions are linked to the definition of the Kelvin temperature scale. (2) Entropy can be defined to be dimensionless when temperature *T* is defined as an energy (dubbed *tempergy*). (3) Dimensionless entropy per particle typically is between 0 and ∼80. Its value facilitates comparisons among materials and estimates of the number of accessible states. (4) Using dimensionless entropy and tempergy, Boltzmann’s constant *k* is unnecessary. (5) Tempergy, *kT*, does not generally represent a stored system energy. (6) When the (extensive) heat capacity $C\u226bk,$ tempergy is the energy *transfer* required to increase the dimensionless entropy by unity.

## REFERENCES

*Entropy and its Physical Meaning*(Taylor & Francis, Bristol, PA, 1996), pp. 101–102.

*Thermodynamics and an Introduction to Thermostatistics*(Wiley, New York, 1985), pp. 46–47, 331.

*Statistical Physics*(Addison–Wesley, Reading, MA, 1958), p. 34.

*Statistical Mechanics*(Wiley, New York, 1977), 2nd ed., p. 90.

*Thermal Physics*(Freeman, New York, 1980), 2nd ed.

*Elementary Statistical Physics*(Wiley, New York, 1958), p. 27.

*Statistical Mechanics*(Wiley, New York, 1988), pp. 40–47.

*Statistical Physics: A Probabilistic Approach*(Wiley, New York, 1991), pp. 14–15. He defines the Boltzmann statistical entropy in the conventional way but then writes, “In most of the subsequent formulas, Boltzmann’s constant will be omitted implying that the temperature will be measured in energy units. This introduces a greater symmetry in the formulas, especially in regard to dual Legendre functions.”

*quasistatic*process, which can occur even if the system interacts irreversibly with its environment (see Ref. 2).

*Brownian Motion and Reality*(Taylor & Francis, London, 1910).

*Investigations on the Theory of the Brownian Movement*(Dover, New York, 1956), especially pp. 60–62.

*Atomic Masses and Fundamental Constants*, edited by J. H. Sanders and A. H. Wapstra (Plenum, New York, 1975), Vol. 5.

*The Fundamental Physical Constants and the Frontier of Measurement*(Hilger, Bristol, 1985), pp. 94–99. The best modern values of Boltzmann’s constant utilize $k=R/NA.$

*Heat and Thermodynamics*(McGraw–Hill, New York, 1997), 7th ed., pp. 186–191.

*The Mechanical Theory of Heat*(John Van Voorst, London, 1867), pp. 134–135.

*The Theory of Heat Radiation*(Dover, New York, 1991). This is an English translation by Morton Masius of Planck’s second edition of

*Vorlesungen über die Theorie der Wärmestrahlung*(1913). Planck wrote (p. 120), “… Boltzmann’s equation lacks the factor

*k*which is due to the fact that Boltzmann always used gram-molecules, not the molecules themselves, in his calculations.” Planck’s numerical calculation of

*k*along with Planck’s constant is given on pp. 172–173. Notably, Planck also calculated Avogadro’s number and the electron charge. References to the relevant original (1901) articles are on p. 216.

*Atoms and Information Theory*(Freeman, San Francisco, 1971), Chap. 3.

*Statistical Physics—1962 Brandeis Summer Institute Lectures in Theoretical Physics*(Benjamin, New York, 1963), Vol. 3, pp. 181–218.

*Fundamentals of Statistical and Thermal Physics*(McGraw–Hill, New York, 1965), pp. 99, 136–137. Here,

*T*is defined to be

*dimensionless*, and the unit

*degree*is taken to be “… a unit in the same sense as the degree of angular measure; it does not involve length, mass, or time.”

^{−1}mol

^{−1}, then $\sigma =smol/8.314.$ If $smol$ is in calories/mol, then $\sigma =smol/2$ to a good approximation.

*Modern Thermodynamics with Statistical Mechanics*(Macmillan, New York, 1992), p. 345. In this book, the Sackur–Tetrode equation is written $S=R[1.5\u200aln\u200aM+2.5\u200aln\u200aT\u2212ln\u200aP]+172.298,$ where

*M*is the mass, in kg, of one mole $(6.02\xd71023$ particles). For example, for neon, $M=0.020\u200a18\u200akg.$ In contrast, in our Eq. (13), $M=20.18$ mass units.

*W*is used to obtain σ using Eq. (14) and small terms are dropped for large

*N*, then use of the inversion formula (15) cannot recover the exact

*W*. For example, if $W=NgqN,$ where

*g*and

*q*are constants, then $\sigma =ln\u200aq+(g/N)ln\u200aN$ and the second term on the right is negligible for sufficiently large

*N*. Taking antilogarithms of the first term, we recover $W=qN,$ but not the factor $Ng.$ Thus, Eq. (15) is useful for obtaining the order of magnitude of

*W*rather than its exact value.

*The Principles of Statistical Mechanics*(Oxford U.P., Oxford, 1938; reprinted by Dover, New York), pp. 95–98. Tolman derives a

*general equipartition principle*and shows that a form of equipartition holds even for a relativistic gas of noninteracting particles, whose energy is

*not*quadratic in momentum. Specifically, if

*u*is particle speed and

*m*rest mass, then $[(mu2)/(1\u2212u2/c2)1/2]average=3kT.$ Equipartition holds for the

*kinetic*energy per particle only in the nonrelativistic limit.

*Statistical Mechanics*(Butterworth-Heinemann, Oxford, 1996), 2nd ed., Chap. 7.

*Statistical Mechanics*(Wiley, New York, 1987), 2nd ed., pp. 283–286.

*Thermal Physics*(Cambridge U.P., New York, 1999), pp. 6–8, 177–178, 347–349.

*Thermal Physics*(American Institute of Physics, New York, 1991), p. 242. The value $\Delta smol\u224885\u200aJ\u200amol\u22121\u200aK\u22121$ cited by Sprackling implies $\Delta \sigma vap\u224810.$

*not*exist if $(\u22022Sd/\u2202U2)V,N$ vanished over a finite

*U*interval, which would imply infinite $Cv$ for this range of

*U*values. Empirically, for a single-phase system, $Cv$ diverges when a critical point is approached.

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