1. The Thermodynamic Definition. The idea of
entropy as a measure of the mechanical unavailability
of energy—that is, of that part of a given amount of
energy which cannot be transformed into useful me-
chanical work and is therefore lost or dissipated for
all practical purposes—originated in early nine-
teenth-century studies on the efficiency of steam en-
gines. The first to raise the question of maximum effi-
ciency in steam-power engineering was Nicolas
Léonard Sadi Carnot (1769-1832). In a famous memoir
he asked: “How can we know that the steam is used
in the most advantageous way possible to produce
motive power?” (Carnot, 1824). Carnot studied heat
engines whose thermal interaction with their sur-
roundings consists only in the exchange (absorption or
rejection) of heat with appropriate reservoirs of fixed
temperatures, and he showed that the reversibly oper-
ating engine is more efficient than its irreversible
counterpart when working between the same temper-
atures. He derived this conclusion, which became
known as “Carnot's Theorem,” on the basis of the
caloric theory according to which heat is regarded as
a fluid. As scattered statements in the literature of that
period indicate, Carnot and some of his contemporaries
have been fully aware that in any practical trans-
formation of heat into mechanical work part of the
stored energy is always dissipated. The exact answer
as to precisely what part is necessarily lost could not
be given before the Second Law of Thermodynamics
was explicitly stated.
During the years 1840-48 J. P. Joule, J. R. Mayer,
and H. von Helmholtz discovered the equivalence
between heat and work and thus established the First
Law of Thermodynamics (conservation of energy in
a closed system). It invalidated Carnot's assumptions
but not his conclusions. Referring to the conflict be-
tween Carnot and Joule, William Thomson (Lord
Kelvin, 1824-1907) declared that further experiments
were needed to resolve this dilemma (Thomson, 1849).
But only one year later Rudolf Julius Emmanuel
Clausius (1822-88) showed without the benefit of fur-
ther experimentation that the issue can be resolved by
either taking “Carnot's Theorem” as an independent
principle or by deriving it from the First Law of
Thermodynamics in conjunction with the premiss that
“heat always shows a tendency to equalize temperature
differences and therefore to pass from hotter to colder
bodies” (Italics in the original; Clausius, 1850).
With these words Clausius introduced rather casually
the Second Law of Thermodynamics, namely, that it
is impossible for a self-acting cyclic machine, unaided
by any external agency, to convey heat from one body
at a given temperature to another at a higher tempera-
ture, a statement which Max Planck later called “the
Clausius Formulation of the Second Law.” Gibbs
rightly remarked that with Clausius' memoir of 1850
“the science of thermodynamics came into existence”
(Gibbs, 1889). In fact, it was the first paper to contain
the two principal laws of thermodynamics. Thomson
derived Carnot's Theorem from the First Law and the
premiss that “it is impossible by means of inanimate
material agency to derive mechanical effect from any
portion of matter by cooling it below the temperature
of the coldest of the surrounding objects” (Thomson,
1851). It is easy to show that the Kelvin Formulation
of the Second Law, as Planck called the preceding
statement (impossibility of perpetual motion of the
second kind), is fully equivalent to the Clausius formu-
lation (Huang, 1963). In a second paper Thomson
discussed the cosmological implications of the Second
Law and concluded that “within a finite period of time
past, the earth must have been, and within a finite
period of time to come the earth must again be, unfit
for the habitation of man as at present constituted,
unless operations have been, or are to be performed,
which are impossible under the laws to which the
known operations going on at present in the material
world are subject” (Thomson, 1852).
Although the preceding formulations of the Second
Law as well as Thomson's sweeping generalization
expressed essentially what subsequently became known
as “the entropy principle,” the concept of entropy as
such was still unknown. Its definition was made possible
only after Clausius demonstrated the following
theorem: if in a cyclic transformation qi denotes the
quantity of heat drawn from (positive), or rejected by
(negative), a heat reservoir at the (absolute) tempera-
ture Ti, then the expression ∑qi/Ti is equal to zero
for reversible cycles and negative for irreversible ones.
The first part of this statement was found inde-
pendently also by Thomson. In fact, as Planck showed
in a critical analysis of Clausius' paper (Planck, 1879),
Thomson's derivation of the so-called “Clausius Equal-
ity” ∑qi/
Ti = 0, or in the limit of infinitesimal
quanti-
ties of heat
ϕ ʃ δq / T = 0 (1)
for reversible (more precisely, quasi-static) closed-cycle
processes, was logically superior. Nevertheless, it was
Clausius who first realized in the same paper (Clausius,
1854) that for reversible processes δq/T is a total (or
exact) differential, or equivalently that T-1 is an inte-
grating factor. The line integral of a total differential,
as shown in the calculus, depends only on the limits
of integration and not on the particular path chosen
for the integration. In other words, it defines a point
function or, in thermodynamics, a state function; that
is, a function which depends on the thermodynamic
variables, like volume or temperature, of the state
under consideration.
It took another eleven years for Clausius to realize
the importance of the state function defined by the
above-mentioned total differential. Writing
dS = δ
q/
T
and integrating, he obtained
where the path of integration corresponds to a reversi-
ble transformation from the thermodynamic state
A
to the thermodynamic state
B. By combining an irre-
versible transformation from
A to
B with a reversible
one from
B to
A and taking notion of the “Clausius
Theorem,” he concluded that
Looking for an appropriate name for the state function
S, Clausius remarked that just as the (inner) energy
U signifies the heat and work content (
Wärme- und
Werkinhalt) of the system, so
S, in view of the pre-
ceding results, denotes its “transformation content”
(
Verwandlungsinhalt). “But as I hold it to be better,”
he continued “to borrow terms for important magni-
tudes from the ancient languages, so that they may
be adopted unchanged in all modern languages, I pro-
pose to call the magnitude
S the
entropy of the body,
from the Greek word τροπή, transformation. I have
intentionally formed the word
entropy so as to be as
similar as possible to the word
energy, for the two
magnitudes to be denoted by these words are so nearly
allied in their physical meanings, that a certain simi-
larity in designation appears to be desirable” (Clausius,
1865).
Clausius' thermodynamic definition of entropy, based
as it is ultimately on a certain existence theorem in
the theory of differentials, is obviously rather abstract
and far removed from visualizability, in spite of the
fact that the differential expression under discussion
reflects an operational result in steam-power engineer-
ing. As the preceding equations (1) and (2) show, the
entropy of a closed (adiabatically isolated—change of
state without transfer of heat) system can never de-
crease, for δq/T = 0 implies SB ≥ SA. Extrapolating
this result for the universe as a whole, Clausius con-
cluded his paper with the famous words: “The energy
of the universe is constant—the entropy of the uni-
verse tends toward a maximum.”
That irreversibility indeed entails increase of en-
tropy—the so-called “entropy principle”—follows
logically from the two statements, (1) that the entropy
of the universe never decreases, and (2) that a process,
accompanied by entropy increase, is irreversible
(Gatlin, 1966).
The fundamental importance of the entropy concept
was soon understood to lie in the fact that it makes
it possible to predict whether an energy transformation
is reversible (dS = δq/T), irreversible (dS > δq/T), or
impossible (dS < δq/T), even if the total energy in-
volved is conserved. Moreover, with the help of the
entropy concept other thermodynamic state functions
could be defined, such as the Helmholtz free energy
or the Gibbs thermodynamic potential, which proved
extremely useful for the calculation of the maximum
attainable work under conditions of constant tempera-
ture or constant temperature and constant pressure,
respectively.
