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                                       Jérôme VIARD
                                 Université Cl. Bernard-Lyon1

ABSTRACT: It is well known that, for many students, the concept of entropy remains
difficult because the word itself is obscure. A teaching strategy, using the metaphor of
disorder, has proved to be ineffective. To solve the difficulty created by the term entropy, for
the last ten years, we have used an alternative strategy with third year university students,
which consist of getting the students to read excerpts from Carnot and Clausius. Groups of ten
to fifteen students read and interpret these texts during their regular course time. It is clear
that students involved in this experimental teaching welcome this new way of introducing the
concept of entropy. This teaching was introduced in the context of history of science lectures.
We hope that it may be used more widely.

KEY WORDS: teaching thermodynamics; entropy and disorder; using the history of the
entropy concept; Carnot and Clausius.


Very often physics teachers think that the use of an adequate name borrowed from the current
language to explain physical concepts is sufficient to enable students to use these concepts.
We have shown elsewhere (Gréa & Viard 1995, Viard & Langlois 2001) that even in the case
of basic concepts such as, for example, that of electrical resistance this hope was illusory,
owing to the fact that the way physicists use it differs from that of everyday use. In the case of
entropy the previous teaching strategy consisting of a name followed by a concept, does not
work. The word entropy does not belong to current language and it is clear that for many
students ’the concept of entropy remains difficult because of the obscurity of the word used
to name it‘.
As a result, ’How to introduce the concept of entropy to students?‘ is a real question for
physics teachers.
We shall first recall, the usual teaching strategy commonly followed by physics teachers faced
with the previous difficulty, based on the use of the metaphor of disorder ; and the unfortunate
effect this has on the students’ understanding of the subject. We shall then propose an
alternative teaching strategy based on the work of Carnot and Clausius, as they themselves
tried to answer, the following questions many years before the word existed:

       -’What is this unknown thing which appears in the field of science, which analyses
       “the relationships between the thermal and mechanical properties of bodies”,
       according to Maxwell when he defines thermodynamics?‘ (1872/1891, p. 1),
       -’Why is it necessary to introduce a new physical quantity?‘,
       -’Can we ascribe a physical meaning to this physical quantity?‘.

We begin by examining the use of a metaphorical language made by some teachers in their
lectures on thermodynamics and its possible repercussions on students’ understanding.


Aware of the difficulty caused by the obscurity of the name given by Clausius to his concept,
some teachers try to give a new meaning to this term by way of a metaphor. We can read for
example in a French dictionary of physics:

       Entropy characterizes the state of disorder of a system. (Sarmand, 1988).

Similar introductions to entropy are supplied in many French textbooks but with a concern for
greater accuracy. See for example:

       the entropy of a system […] is a measure of its state of disorder, (Jancovici 1969, p. 95),
or :

       Entropy appears then as a quantitative measure of the degree of disorder of a given state.
       (Faroux, Renault, 1997, p.158),

or :

       In statistical thermodynamics, entropy is interpreted as a measure of the state of disorder of
       this system. (Queyrel, Mesplède 1996, p.123).

Other physicists have however expressed criticisms of these statements over the years:

       “Entropy is a name given to a quantitative measure of disorder” represents, not the received
       doctrine of physical science, but (to say the least) a highly contentious opinion.(Wright 1970),

while others discuss the Wright’s point of view, such as Baierlein in the American Journal of
Physics who states in a paper entitled: ’Entropy and the second law: A pedagogical
alternative‘ :

       There is generally nothing wrong with referring to entropy as a measure of disorder. (1994,
       p. 25),

and takes then an example borrowed from the everyday life:

       Imagine a bedroom with the usual complements of shoes, socks and T-shirts. Suppose, further,
       that the room is one that we intuitively characterize as “orderly”. […] There are strong spatial
       correlations between the shoes in a pair or the T-shirts on the dresser. Those correlations limit
       severely the ways in which shoes and T-shirts can be distributed in the room, and so the
       objects exhibit a small multiplicity and a low entropy. (Idem).

Should we believe Wright or Baierlein? An inquiry about the understanding of the concept of
entropy with fifth year university or Ph. D. physics students (Viard 1987 ; Brosseau & Viard
1992 ; Gréa & Viard 1995), will allow us to answer the previous question. This enquiry was
carried out through semi-directive interviews. During the interviews, a sample group of ten
students had to answer three questions and solve a short problem. All the students had
attended two semester courses on thermodynamics and statistical physics one or two years
previously. These interviews were given outside the university context and questions were
formulated in a language as close as possible to current language.
Two of the questions were based on the relationships between entropy, heat and temperature.
The third question and the associated problem directly concern the subject of this paper.
Question 1 ask students above their personal conception of entropy:

       1.“What is entropy for you? What do you imagine when you think of entropy?”

Students were supposed to take their answers to the question into consideration when they
answered the problem:

       A system expands under the effect of internal pressure (of a gas compressed by a frictionless
       piston). The transformation should be reversible. The system is thermally isolated. What
       happens to the system’s entropy? Does it increase and if yes or not why?

To summarize the results of the enquiry we can say that nine of the ten interviewed students
referred to entropy as disorder or the measure of a system disorder following the
interpretation given by the previous textbooks’ authors.
Six students answered that entropy increased; two that the entropy decreased; one supplied
the expected answer that the entropy remains constant, and one did not understand the
The reasoning of the six students who answered that entropy increased goes as follows:

             Increase of volume ==> Increase of disorder I.E. Increase of entropy

These firsts results obtained with a few students were confirmed by asking the same
questions, every year, to third year university students during a ten year period , from 1995 to
The fact that for the great majority of students, disorder is understood exclusively as a spatial
disorder, had escaped Baierlen and many textbooks’ authors. Students forget the kinetic part
of entropy but they are not alone. Textbooks very often limit their examples of disorder to
spatial disorder, as is the case for Baierlen just above, and then reinforce the natural tendency
of students to ignore the other part of entropy.
The result of these considerations is that the cure - the metaphor of disorder - is worse than
the disease – the obscurity of the term entropy.
We need clearly an alternative teaching strategy, it will be borrowed from the history of


The previous unfruitful strategy was still based on the fallacious idea that the choice of an
accurate word allow us to understand what it is about. This strategy is not applicable when
the thing we are speaking about is not clearly identified. The strategy followed by the
researchers who founded thermodynamics is the opposite of the above one and may be
summarized as follows: the thing or the concept precedes the word.
To allow students to make sense of the word entropy we propose that they follow themselves:

                                        The road to entropy

as says Croppers (1986), the one followed by Clausius himself, before he creates the word,
many years later.
From 1995 to 2005 we have proposed, to small groups of up to fifteen of third year university
students, the reading of some chapters of Carnot and Clausius.
The reading and the interpretation of these texts by students were made during the lecture.
The lecture started with shorts texts from Carnot’s Réflexions sur la puissance motrice du
feu. (1824). These texts are basis on which the theory of Clausius is built.
We present below an analysis of the content of the most important part of the texts proposed
to students during the lecture.
As said above in the introduction, the first job that the two researchers gave themselves was to
identify the characteristic elements of the new physics discipline they are investigating.
The heuristic strategy of Carnot in the chosen text is grounded on an analogy between falling
water in water engines and falling heat in heat engines. Carnot establishes that the amount of
motive power produced in a steam engine is a function of two variables in the same way that
the motive power produced in a water engine is a function of the height of the waterfall and
the amount of falling water. He indicates however that the function, which expresses the
relationship, between motive power and the variables on which it depends on is not the same
in the two cases.

       […] we may compare with enough exactness the motive power of heat to the one of a
       waterfall […]. The motive power of a waterfall depends on its height and on the quantity of
       liquid ; the motive power of heat depends also on the quantity of caloric used, and on what, we
       could name, on what we actually shall call the height of its fall i.e. on the difference of
       temperature between the bodies between which the exchange of caloric is made. In the
       waterfall, the motive power is rigorously proportional to the difference of level between the
       upper reservoir and the lower reservoir. In the fall of the caloric, the motive power increases,
       doubtless with the difference of temperature but we don’t know if it is proportional to this
       difference. (1824, p. 28-29).

Starting from this comparison, Carnot draws a general conclusion about steam engines:

       The production of movement in steam engines is always associated with a circumstance to
       which we have to pay attention. This circumstance is the restoration of the equilibrium in the
       caloric, I.E. its passage from a body where temperature is more or less high to another one
       where it is lower. (1824, p. 9),

elsewhere, the circumstance becomes a cause:

       all restoration of the equilibrium in the caloric may be the cause of the production of motive
       power. (Idem, p. 23).

What is new in the previous remarks of Carnot? It is not the production of motive force,
which has been a feature of the field of mechanics for some time. It is not the restoration of
equilibrium in the caloric, which has belonged to the field of the science of heat for as
similarly long period. No, the novelty is entirely contained in the necessary relationship
established by Carnot between these two phenomena and which is what constitutes the new
discipline, as stated by Maxwell in the reference above. Doing that, Carnot points out the
similarity that exists between the two cases he considers: the water engine and the steam
engine. In the two situations the production of motive power is associated with the restoration
of the equilibrium between the levels of water in the case of the water engine or between the
levels of temperature in the caloric in the case of the steam engine.
These points have not escaped Clausius, who analyses the work of Carnot very carefully.
The lecture continues with the proposition to students of a chapter of Clausius referring to the
previous texts of Carnot. This chapter is entitled:


Clausius proposes a new reading and a new formulation of Carnot’s remarks:

       The theorem of Carnot, integrated with the first theorem, expresses a relationship between two
       kinds of transformations: the transformation of heat into work, and the passage of heat from a
       warm body to a cooler body, a passage which we can treat as a transformation of a quantity of
       heat at high temperature into heat at a lower temperature. (1854/1868, p.137).

There are two basic contributions of Clausius in this short text which relates to the initial
work of Carnot.
First, Clausius introduces the conversion of heat into work, an idea that is absent from
Carnot’s memoir, to take into account the first theorem, as he says himself, and he also
introduces the term of transformation, which does not belong to Carnot’s terminology.
According to Kim (1983), the term, transformation was first used by William Thomson.
Reporting a statement of Carnot: ’that there is a an absolute waste of mechanical energy
available to man when heat is allowed to pass from a body to another at a lower temperature‘,
Thomson remarks:

       As it is most certain that Creative Power alone can either call into existence or annihilate
       mechanical energy, the “waste” referred to cannot be annihilation but must be some
       transformation of energy. (Thomson 1852, quoted by Kim (1983) ).

Secondly Clausius modifies the nature of the relationship existing between the two kinds of
With Carnot the relationship between the two kinds of transformations have a causal
Falling heat is the cause of the production of work.
With Clausius the identity of the transformations is his centre of interest. He notes first that:

       Each one of the two transformations occurring in a reversible cycle, taken in inverted sense,
       can replace the other one ; in such a way that, if a transformation of any kind has taken place,
       it can be annihilated and replaced by a transformation of the other kind, without introducing
       any other permanent modification. (1854, 1868, p.144)

and concludes:

       We can see therefore that these two kinds of transformations can be considered as phenomena
       the nature of which is the same ; we shall name equivalent two transformations which can
       replace themselves mutually in this way. (Idem).

His reasoning goes as follows: after an entire reversible cycle, the state of the system is the
same as it was at the beginning of the cycle.
As a result, the two transformations, which occur in the cycle, have to balance or compensate
themselves during the cycle.
A condition of this compensation is that their nature must be the same. They are said to be
equivalent to each other.
Clausius constructs then a quantitative transformation theory:

       The question is now to find the law according to which we shall have to represent
       mathematically these transformations, in order that their equivalence results from the equality
       of their values. We shall name the mathematical value of a transformation its equivalence
       value. (Idem).

It is important to note that in the former quotation the equivalence refers to the nature of the
transformations, whilst in the latter the equivalence refers to the amount of each
transformation. The algebraic value of each transformation has to be equal and opposite to the
value of the other one in such a way as to make their sum equal to zero.
To fix the quantity of each transformation Clausius makes basic assumptions according to
which the equivalence value of the first kind of transformation is proportional to the quantity
of heat Q converted in work and to a function of the temperature t at which this conversion
occurs. As a result the equivalence value is equal to:

                                         Q.f (t) (Idem, p.145).

In the same way the equivalence value of the second kind of transformation will be
proportional to the amount of heat Q, which passes from the temperature t1 to the temperature
t2 and to a function of these two temperatures, as a result the equivalence value is equal to:

                                           Q.F(t1, t2), (Idem).

A negative value is given to the conversion of heat into work and a positive value to the
passage of heat from a given temperature to a lower one. Thus the conversion of a quantity Q
of heat at the temperature t, associated to the passage of quantity Q1 from the temperature t1 at
the lower temperature t2 supplies the equation:

                               -Q. f (t) + Q1 . F(t2, t1) = 0. (Idem, p. 146).

Applying this equality to different cycles Clausius arrives at the fact that the second function
can be expressed by a linear combination of the first one:

                                F(t, t’) = f (t’) - f (t’) (Idem, p.148)

By identification of the function f (t) to the inverse of the absolute temperature T, the two
equivalence values can both be expressed, by a linear combination of the ratio:


Third year university students of can follow all the steps of the reasoning of Clausius reported
above without difficulties.
So far we have answered two of the three questions asked in our introduction referring to the
identification of a new object in the field of physics: the thermo dynamical transformations
and the necessity of the formulation of a quantitative theory of these transformations.
One question has still no answer: that referring to the physical meaning of the quantity
The formal likeness of the value of equivalence of the two transformations expresses their
identical nature in a symbolic way. This mathematical treatment is only a new formulation of
the previous reasoning expressed in current language but it does not supply new information
about the physical meaning of the quantities introduced. One question in particular remains
unanswered, ’How do we interpret the fact that the quantity of transformation has for measure
the ratio Q/T?‘. What does it mean for a physicist to divide a quantity of heat by temperature?
Clausius is aware of this difficulty:

       Although the necessity of this theorem [the theorem of the equivalence of transformations]
       admits of strict mathematical proof if we start from the fundamental proposition quoted above,
       it thereby nevertheless retains an abstract form the mind finds hard to grasp, and we feel
       compelled to seek the precise physical cause of which this theorem is a consequence.
       (1862/1868, p. 256).

Clausius is able to give a physical meaning to the formal expression of the value of
transformation only in the case of the first kind of transformation, the transformation of heat
into work. His physical analysis bears on the work of heat inside the matter he calls the
’interior work‘:

       In the first place, a certain quantity of work is necessary to overcome the natural attraction of
       the particles and to separate them to the distance which they occupy in a state of vapour.
       Secondly, the vapour during its development must, in order to procure room for itself, force
       back an outer pressure. We shall name the former of these interior work, and the latter exterior
       work, […]. (1850/1868, p. 24).

Clausius starts from some basic ideas:

       The action of heat has always […] the effect of increasing the mean distance between
       particles. (1862/1868, p. 258).

Clausius then proposes a quantification of this action and he creates a word to name it:

       We represent the degree of division of the body by a new quantity, which we shall name the
       disgregation of the body, by means of which we can define the action of heat saying simply
       that it tends to increase the disgregation. (Idem).

He defines then the dependence of the work of heat on the ’disgregation‘ saying that:

       We shall fix the measure, until now arbitrary, of this one [the disgregation] so as that, at a
       given temperature, the increasing of disgregation is proportional to the work that the heat can
       do by this means. (1862/1868, p. 266),

He notes then that the force of heat is proportional to absolute temperature T.

       In all cases in which the heat contained in a body does mechanical work by overcoming
       resistances, the magnitude of the resistances that it is capable of overcoming is proportional to
       the absolute temperature. (1862/1868, p. 257),


       The mechanical work, which can be done by heat during any change of the arrangement of a
       body, is proportional to the absolute temperature at which this change occurs, (1862/1868,
       p. 261),

The work of heat is equal to the product of the force of heat by the ’increasing‘ of the mean
distance between particles», I.E. of ’disgregation‘.
If the infinitesimal increase of disgregation is dZ, the amount of heat transformed into work
dQ, by choosing unities we obtain:

                                     dQ= T. dZ or dZ= dQ/T.

In this way the ratio Q/T gets a physical meaning.
Clausius draws attention to his new contribution to his theory of transformations:

       I think that what is essentially new in my equation (II.), is that the quantity Z which is
       involved in, has got by my analysis a definite physical meaning ; in this sense that this
       quantity depends only of the actual arrangement of the particles. (1862/1868, p. 276).

The physical meaning of the second kind of transformation, which depends only on the
quantity of heat contained in the body and not of the arrangement of the particle will be
supplied later by the Maxwell-Boltzmann velocity distribution.
As pointed out by Crooper (1986):

       [Clausius] did not offer a name for the new state function in 1854, nor did he even give it a

All the reasoning was made using expressions such as ’value of equivalence [of
However, there is never any doubt in the reader’s mind what its about. The concept is present
a long time before it is named or has its own symbol.
It is only in 1865 only, that Clausius will say he had chosen to create the word entropy
starting from the ancient Greek word τροπη (transformation) to mean the amount of
transformation contained in the body.
The word entropy now takes on the sense of all the theoretical work contained in the theory of


For a reader of the previous texts, the word entropy loses its erstwhile obscurity and its
mysterious character.
We can see that both students and Clausius, are able to give a physical meaning the spatial
part of entropy only, but that unlike the students, Clausius knows that this spatial part is not
the only part since, he also knows the variables which this entropy depends on.
The reading of some chapters of Boltzmann then allowed the students to make sense of the

kinetic part of entropy, but this point goes beyond the scope of this paper.
To summarize the way followed with students we can say that we started from Carnot’s and
Clausius’ own questions about steam engines, and we the attempted to follow step by step
their attempts to answer these questions. To guide the students’ reflections, the existence of a
precise phenomenological reference was of the most importance.
The choice of the topic of thermodynamics and particularly of entropy was due to a personal
difficulty with the concept of entropy, which has led us to investigate the origins of this
Even in the absence of an investigation of the impact of the lecture, we can say that students
involved in this experimental teaching welcome this new way of introducing the concept of
We hope that this teaching method may be used more widely.


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