Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*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

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 δ*q* / *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 δ*q*/*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

*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

Werkinhalt

*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 δ*q*/*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* = δ*q*/*T*), irreversible (*dS* > δ*q*/*T*), or

impossible (*dS* < δ*q*/*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.

Dictionary of the History of Ideas | ||