For a closed system that contains an arbitrary pure substance, which can exchange energy as heat and as expansion∕compression work, but no particles, with its surroundings, the inexact differential of the reversibly exchanged heat is a differential in two variables. This inexact differential can be turned into an exact one by an integrating factor that, in general, depends on both variables. We identify the general form of the integrating factor as the reciprocal temperature (Clausius’s well-known $1∕T$), which is guaranteed to be a valid integrating factor by the second law of thermodynamics, multiplied by an arbitrary function of the implicit adiabat equation $ξ(T,V)=constant$ or $ξ(T,P)=constant$. In general, we cannot expect that two different equations of state (corresponding to two different substances) predict identical equations for the adiabats. The requirement of having a universal integrating factor thus eliminates the volume-dependent or pressure-dependent integrating factors and leaves only a function of temperature alone: Clausius’s integrating factor $1∕T$. The existence of other integrating factors is rarely mentioned in textbooks; instead, the integrating factor $1∕T$ is usually taken for granted relying on the second law or, occasionally, one finds it “derived” incorrectly from the first law of thermodynamics alone.

1.

The transfers of heat and of work are two ways in which the system can exchange energy with its surroundings. “Heat” denotes the transfer of energy due to a (possibly infinitesimally small) temperature difference, while “work” is energy exchanged due to a (possibly infinitesimally small) pressure difference between the system and its surroundings. The system itself possesses neither heat nor work, just energy.

2.
See, for example,
Peter
Atkins
and
Julio
de Paula
,
Atkins’ Physical Chemistry
, 7th Ed. (
Oxford University Press
,
Oxford
,
2002
), pp.
73
120
.
3.

The term “reversible” denotes a special way of conducting the transfer of heat or work in a given process. It ensures that, at all times, the differences in temperature and pressure between the system and its surroundings are only infinitesimal; that is, the temperature and pressure are matched in such a way that heat and∕or work are just transferred in the desired direction (to or from the system), but the values of the temperature and pressure are equal for the system and for its surroundings for all practical purposes. The total entropy of the supersystem consisting of the system and its surroundings remains constant in a reversible process, implying that the change in entropy of the system is exactly equal, but opposite in sign, to the change in entropy of the surroundings.

4.
S. M.
Blinder
, “
Carathéodory’s formulation of the second law
,” in
, edited by
H.
Eyring
,
B.
Henderson
, and
W.
Jost
(
,
New York
,
1971
), Vol.
1
, Chap. 10.
5.
Arnold
Münster
,
Chemische Thermodynamik
(
VCH
,
Weinheim
,
1969
), pp.
27
38
;
English translation,
Classical Thermodynamics
(
Wiley-Interscience
,
New York, NY
,
1970
).
6.
Stephen G.
Brush
,
The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century
(
North-Holland
,
Amsterdam
,
1986
), Book 2, pp.
566
583
.
7.
G. Krishna
Vemulapalli
, “
A simple method for showing entropy is a function of state
,”
J. Chem. Educ.
63
,
846
846
(
1986
).
8.
Predrag
Djurdjević
and
Ivan
Gutman
, “
A simple method for showing entropy is a function of state
,”
J. Chem. Educ.
65
,
399
399
(
1988
).
9.
Donald A.
McQuarrie
and
John D.
Simon
,
Physical Chemistry: A Molecular Approach
(
University Science Books
,
Sausalito, CA
,
1997
), pp.
820
and
844
845
, Problem 20–5.
10.
John C.
Wheeler
, “
Entropy does not follow from the first law: Critique of ‘Entropy and the first law of thermodynamics’
,”
Chem. Educator
8
,
171
176
(
2003
).
11.
See, for example,
E. C.
Zachmanoglou
and
Dale W.
Thoe
,
Introduction to Partial Differential Equations with Applications
(
Dover
,
Mineola
, NY,
1986
), pp.
133
137
.
12.
This statement will be true at low and moderate densities as the first nine virial coefficients are positive; it is unknown if $Bn$ becomes negative for any higher value of $n$. See
Stanislav
Labík
,
Jiří
Kolafa
, and
Anatol
Malijevský
, “
Virial coefficients of hard spheres and hard disks up to the ninth
,”
Phys. Rev. E
71
,
021105
1
(
2005
).
13.
See for example,
Terrell L.
Hill
,
An Introduction to Statistical Thermodynamics
(
Dover
,
Mineola
, NY,
1986
), pp.
261
285
.
14.
A. B.
Pippard
,
The Elements of Classical Thermodynamics
(
Cambridge University Press
,
London
,
1957
), pp.
29
42
.
15.
Francis W.
Sears
and
Gerhard L.
Salinger
,
Thermodynamics, Kinetic Theory, and Statistical Thermodynamics
, 3rd Ed. (
,
,
1975
), pp.
138
141
and
168
172
.
16.
H. A.
Buchdahl
, “
On the principle of Carathéodory
,”
Am. J. Phys.
17
,
41
43
(
1949
).
17.
H. A.
Buchdahl
, “
On the theorem of Carathéodory
,”
Am. J. Phys.
17
,
44
46
(
1949
).
18.
H. A.
Buchdahl
, “
Integrability conditions and Carathéodory’s theorem
,”
Am. J. Phys.
22
,
182
183
(
1954
).
19.
Louis A.
Turner
, “
Simplification of Carathéodory’s treatment of thermodynamics
,”
Am. J. Phys.
28
,
781
786
(
1960
).
20.
Louis A.
Turner
, “
Simplification of Carathéodory’s treatment of thermodynamics II
,”
Am. J. Phys.
30
,
506
508
(
1962
).
21.
Francis W.
Sears
, “
A simplified simplification of Carathéodory’s treatment of thermodynamics
,”
Am. J. Phys.
31
,
747
752
(
1963
).
22.
Francis W.
Sears
, “
Modified form of Carathéodory’s second axiom
,”
Am. J. Phys.
34
,
665
666
(
1966
).
23.
Panos
Nikitas
, “
Entropy and the first law of thermodynamics
,”
Chem. Educator
7
,
61
65
(
2002
).
24.
Ben
Widom
, “
Thermodynamics, equilibrium
,” in
Encyclopedia of Applied Physics
, edited by
George L.
Trigg
and
Edmund H.
Immergut
(
Wiley-VCH
,
Weinheim
,
1997
), Vol.
21
, pp.
281
310
.