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≫k, tempergy is the energy transfer required to increase the dimensionless entropy by unity.

1.
J. S. Dugdale, Entropy and its Physical Meaning (Taylor & Francis, Bristol, PA, 1996), pp. 101–102.
2.
T. V.
Marcella
, “
Entropy production and the second law of thermodynamics: An introduction to second law analysis
,”
Am. J. Phys.
60
,
888
895
(
1992
).
3.
H. S.
Leff
, “
Thermodynamic entropy: The spreading and sharing of energy
,”
Am. J. Phys.
64
,
1261
1271
(
1996
).
4.
R.
Baierlein
, “
Entropy and the second law: A pedagogical alternative
,”
Am. J. Phys.
62
,
15
26
(
1994
).
5.
H. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, 1985), pp. 46–47, 331.
6.
L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison–Wesley, Reading, MA, 1958), p. 34.
7.
J. E. Mayer and M. G. Mayer, Statistical Mechanics (Wiley, New York, 1977), 2nd ed., p. 90.
8.
C. Kittel and H. Kroemer, Thermal Physics (Freeman, New York, 1980), 2nd ed.
See also the first edition (Wiley, New York, 1969), pp. 29, 132–133,
and C. Kittel, Elementary Statistical Physics (Wiley, New York, 1958), p. 27.
9.
B. K. Agrawal and M. Eisner, Statistical Mechanics (Wiley, New York, 1988), pp. 40–47.
10.
See also B. H. Lavenda, 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.”
11.
Strictly speaking the reversibility implied by dS=δQrev/T is sufficient, but not necessary. It is necessary only that the system in question undergo a quasistatic process, which can occur even if the system interacts irreversibly with its environment (see Ref. 2).
12.
Planck calculated NA in 1900 (see Ref. 19) using data on radiation.
A “direct” determination of NA, using Einstein’s theory of Brownian motion, was published in
J. B.
Perrin
,
Ann. Chim. (Paris)
18
,
1
(
1909
);
an English translation is in F. Soddy, Brownian Motion and Reality (Taylor & Francis, London, 1910).
See also A. Einstein, Investigations on the Theory of the Brownian Movement (Dover, New York, 1956), especially pp. 60–62.
13.
A student laboratory experiment for measuring Boltzmann’s constant was developed by
M.
Horne
,
P.
Farago
, and
J.
Oliver
, “
An experiment to measure Boltzmann’s constant
,”
Am. J. Phys.
41
,
344
348
(
1973
). This paper also contains interesting historical information.
14.
The modern value of NA is known from measurements of the lattice constant, density, and atomic mass of silicon. See R. D. Deslattes, “Recent estimates of the Avogadro constant,” in Atomic Masses and Fundamental Constants, edited by J. H. Sanders and A. H. Wapstra (Plenum, New York, 1975), Vol. 5.
15.
For a review of modern determinations of the gas constant, see B. W. Petley, 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.
16.
See, for example, M. W. Zemansky and R. H. Dittman, Heat and Thermodynamics (McGraw–Hill, New York, 1997), 7th ed., pp. 186–191.
17.
Clausius used a similar, but less well known, method to obtain Eq. (1). His method also required the definition of absolute temperature given in the text. See R. J. E. Clausius, The Mechanical Theory of Heat (John Van Voorst, London, 1867), pp. 134–135.
Interesting discussions of Clausius’s method can be found in
W. H.
Cropper
, “
Rudolf Clausius and the road to entropy
,”
Am. J. Phys.
54
,
1068
1074
(
1986
) and
P. M. C.
Dias
,
S. P.
Pinto
, and
D. H.
Cassiano
, “
The conceptual import of Carnot’s theorem to the discovery of entropy
,”
Arch. Hist. Exact Sci.
49
,
135
161
(
1995
).
18.
See Ref. 5, pp. 27–32.
19.
See M. Planck, 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.
20.
See, for example, R. Baierlein, Atoms and Information Theory (Freeman, San Francisco, 1971), Chap. 3.
21.
The information theory development of statistical mechanics was pioneered by
E. T.
Jaynes
, “
Information theory and statistical mechanics
,”
Phys. Rev.
106
,
620
630
(
1957
);
E. T.
Jaynes
, “
Information theory and statistical mechanics. II
,”
Phys. Rev.
108
,
171
190
(
1957
).
A good summary is in K. Ford, Editor, Statistical Physics—1962 Brandeis Summer Institute Lectures in Theoretical Physics (Benjamin, New York, 1963), Vol. 3, pp. 181–218.
22.
One exception is Ref. 8, in which τ is used throughout.
23.
Callen (Ref. 5) and Landau and Lifshitz (Ref. 6) adopt this view. A contrasting view is taken in F. Reif, 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.”
24.
In Refs. 6,7,8,9, the extensive dimensionless entropy is labeled σ, which is related to the Clausius (extensive) entropy S by S=kσ. In contrast, we use the lower case σ for the intensive dimensionless entropy per particle and Sd for the extensive dimensionless entropy.
25.
σ can be expressed in terms of the entropy per mole, smol=S/n and the entropy per unit mass, smass=S/M as follows: σ=Msmass/(Nk)=nsmol/(Nk)=smol/R. The latter expression shows that if one knows the molar entropy in J K−1 mol−1, then σ=smol/8.314. If smol is in calories/mol, then σ=smol/2 to a good approximation.
26.
The third law of thermodynamics implies that entropy, and thus dimensionless entropy, approaches the same value for all paths leading toward absolute zero temperature. By convention, this value is defined to be zero.
27.
R. P. Bauman, Modern Thermodynamics with Statistical Mechanics (Macmillan, New York, 1992), p. 345. In this book, the Sackur–Tetrode equation is written S=R[1.5 ln M+2.5 ln T−ln P]+172.298, where M is the mass, in kg, of one mole (6.02×1023 particles). For example, for neon, M=0.020 18 kg. In contrast, in our Eq. (13), M=20.18 mass units.
28.
If an exact expression for 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 σ=ln q+(g/N)ln N 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.
29.
Icosane (also called eicosane), with molecular formula C20H42, has 62 atoms per molecule. Although it is a solid at room temperature, Lange’s Handbook of Chemistry lists an entropy value that implies σ=112 for icosane gas under standard conditions.
30.
R. C. Tolman, 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−u2/c2)1/2]average=3kT. Equipartition holds for the kinetic energy per particle only in the nonrelativistic limit.
31.
R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, Oxford, 1996), 2nd ed., Chap. 7.
32.
K. Huang, Statistical Mechanics (Wiley, New York, 1987), 2nd ed., pp. 283–286.
33.
R.
Baierlein
, “
The meaning of temperature
,”
Phys. Teach.
28
,
94
96
(
1990
).
34.
R. Baierlein, Thermal Physics (Cambridge U.P., New York, 1999), pp. 6–8, 177–178, 347–349.
35.
This is known as Trouton’s rule. See, for example, M. Sprackling, Thermal Physics (American Institute of Physics, New York, 1991), p. 242. The value Δsmol≈85 J mol−1K−1 cited by Sprackling implies Δσvap≈10.
36.
The second term in (34) is much less than 1 if (T/TD)3≫1/(2bN). Using the heat capacity, C(T)=bNk(T/TD)3, this implies C(T)≫k/2. The latter agrees (in an order of magnitude sense) with the condition C̄≫k for Q=kT to induce the change ΔSd=1.
37.
See Ref. 16, p. 355, Table 13.4.
38.
In “bit language,” the dimensionless entropy is Sd[bits]=Sd/ln 2=Sd/0.693, and Eq. (30) becomes ΔSd[bits]=1.44 bits upon addition of energy Q=τ. This reflects the additional missing information associated with 2.718… times as many accessible states.
39.
See Ref. 5, pp. 203–207.
40.
A unique solution would not exist if (∂2Sd/∂U2)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.
41.
See Ref. 32, pp. 138–140.  
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.