Thermodynamics - Fundamental Principles and Applications

444 Pages·2019·6.075 MB·English

by  Antonio Saggion, Rossella Faraldo, Matteo Pierno

UNITEXT for Physics
Antonio Saggion
Rossella Faraldo
Matteo Pierno
Thermodynamics
Fundamental Principles and Applications
UNITEXT for Physics
Series Editors
Michele Cini, University of Rome Tor Vergata, Roma, Italy
Attilio Ferrari, University of Turin, Turin, Italy
Stefano Forte, University of Milan, Milan, Italy
Guido Montagna, University of Pavia, Pavia, Italy
Oreste Nicrosini, University of Pavia, Pavia, Italy
Luca Peliti, University of Napoli, Naples, Italy
Alberto Rotondi, Pavia, Italy
Paolo Biscari, Politecnico di Milano, Milan, Italy
Nicola Manini, University of Milan, Milan, Italy
Morten Hjorth-Jensen, University of Oslo, Oslo, Norway
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Antonio Saggion Rossella Faraldo
(cid:129) (cid:129)
Matteo Pierno
Thermodynamics
Fundamental Principles and Applications
123
AntonioSaggion Rossella Faraldo
Dipartimento di Fisica eAstronomia LiceoStatale “Celio-Roccati”
“Galileo Galilei” ViaG.Carducci, 8
Universitàdi Padova 45100Rovigo, Italy
Padova,Italy
Matteo Pierno
Dipartimento di Fisica eAstronomia
“Galileo Galilei”
Universitàdi Padova
Padova,Italy
ISSN 2198-7882 ISSN 2198-7890 (electronic)
UNITEXTfor Physics
ISBN978-3-030-26975-3 ISBN978-3-030-26976-0 (eBook)
https://doi.org/10.1007/978-3-030-26976-0
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Antonio Saggion dedicates his contribution to
the memory of Nicolò Dallaporta, former
Professor of Theoretical Physics of the
University of Padova and renowned scientist.
Preface
In order to explain what pushed us to write this book on thermodynamics, and to
help the reader understand how to approach it, it seems appropriate to begin with
the following quote:
In November 1919, the London Times published an article by A. Einstein
entitled “My Theory.” We have copied here a section of the article:
Thereareseveralkindsoftheoryinphysics.Mostofthemareconstructive.Theseattempt
to build a picture of complex phenomena out of some relatively simple proposition. The
kinetic theory of gases, for instance, attempts to refer to molecular movement the
mechanical,thermal,anddiffusionalpropertiesofgases.Whenwesaythatweunderstand
agroupofnaturalphenomena,wemeanthatwehavefoundaconstructivetheorywhich
embracesthem.
Butinadditiontothismostweightygroupoftheories,thereisanothergroupconsistingof
whatIcalltheoriesofprinciple.Theseemploytheanalytic,notthesyntheticmethod.Their
starting point and foundation are not hypothetical constituents, but empirically observed
general properties of phenomena, principles from which mathematical formulae are
deduced of such a kind that they apply to every case which presents itself.
Thermodynamics,forinstance,startingfromthefactthatperpetualmotionneveroccursin
ordinaryexperience,attempts todeducefrom this,byanalyticprocesses, atheorywhich
will apply in every case. The merit of constructive theories is their comprehensiveness,
adaptability,andclarity;thatofthetheoriesofprinciple,theirlogicalperfection,andthe
securityoftheirfoundation.
Alongwithspecialrelativity,thermodynamicsisatheoryofprinciple,accordingto
Einstein’s categorization. This means that it is a theory based upon a universally
accepted assumption, starting from which the analytic method is used to deduce
constraints or correlations within all natural phenomena. As with relativity, its
fundamental property consists in generality.
Other macroscopic theories such as mechanics or electromagnetism bring
together many experimental observations in an abstract mathematical formulation.
This allows for precise predictions, which, however, are constrained to the field of
phenomena within which they were synthesized.
vii
viii Preface
Thermodynamicsisalsobasedonanabstractmathematicalformulation,butthis
producestwotypes ofresults: inequalities andgeneral relations between quantities
which appear independent from each other at first sight. This is why thermody-
namics is so often carried out by engineers and is so important for biologists.
Another very important aspect, which symbolizes the cultural choices made in
this work, regards the connection with statistical mechanics. It is often believed
that, for example, the concept of temperature is, essentially, absorbed within
“preexisting”conceptsinmechanics,likethatofkineticenergyinitsvariousforms.
In these cases, there may be the conviction that, after the “discovery of atoms,”
mechanics, developing an opportune probabilistic language, can make thermody-
namics obsolete, as happened for optics, which had its “own laws” and ended up
becoming a chapter of electromagnetism.
In the previous quote, Einstein brings the problem clearly into focus: Kinetic
theory (and statistical mechanics) represents an attempt to attribute some so-called
thermal phenomena to the movement of molecules. This reductionist attitude can
provide results which may agree, more or less, with observations but cannot rep-
resent a “true” law in the sense to which Einstein refers:
TheexampleIsawbeforemewasthermodynamics.Thegeneralprinciplewastheregiven
inthestatement:thelawsofnaturearesuchthatitisimpossibletoconstructaperpetuum
mobile.
For these reasons, thinking about our students, it seemed necessary to us to high-
light the distinction between the two areas: that of thermodynamics and that of
statisticalmechanics.Differentlyfromthechoicesmadeinmanyofthemorerecent
treatises, in which the two aspects are mixed, partly for reasons of didactical
completeness and partly to show how the two aspects support each other recipro-
cally, in this work we have avoided referring, even if only to make analogies, to
kinetic theories (or of statistical mechanics in general). The work on thermome-
chanical effects in Part III is no exception to this. The relations obtained with the
general methods of thermodynamics of irreversible processes have been covered
again intheKnudsengascase usingaclassicmechanical-statisticalmodelingtool.
This is a very informative comparison which provides a “hands-on approach” to
understandingtheamountofparticularhypothesesthatneedtobesatisfiedinorder
to obtain the same result.
Thermodynamicsnowadaysisbaseduponasetofconceptsandonthechoiceof
a mathematical structure which hasundergone significant evolutions, starting from
the first theorizations of the second half of the nineteenth century but despite this,
the conceptual and formal original structure is often maintained. Examples of this
are the persistence of the calorimeter definition of “heat” and the establishment
of the second law according to the original formulation of Lord Kelvin or
R.Clausius. Thefirstexample(whichriskskeepingalivetheoldideaof“caloric,”
Preface ix
meaning a fluid that flows from one body to another) faces the student with two
substantial difficulties bordering on real conceptual errors. On the one hand, the
concept of heat is persistently associated with variations in temperature1; on the
other hand, it creates the necessity to “demonstrate” the heat–work equivalence
(which further reinforces the idea of caloric) in order to then “demonstrate” that
energy is a function of state.
Commentingonthesecondexample(theSecondPrinciple)ismoredemanding.
One arrives at the concept of entropy, which is the theorization, meaning the
translation into a mathematical form, of the fundamental postulate on the impos-
sibilityofperpetualmotion,afteralong,strenuousandmathematicallyquestionable
process.
Whatever the solution of the first and second principles, their formulation is
always dependent on the definition of the amount of heat transferred in an inter-
action. This represents a limitation that has to be overcome, and this can only be
obtained through further postulation. The problem lies in the fact that the concept
of the amount of heat transferred only makes sense for closed systems, meaning
systems which do not exchange mass, therefore leaving out all of those systems
(open systems) which are of primary interest to biology and to applications in
engineering. This further step is carried out by generalizing the relation between
energy and entropy, that the first and second principles indicated as for closed
systems.
Thefinalpostulation,whichbringseverythingtogether,consistsinestablishing,
foreverymacroscopicsystem,theexistenceoftheso-calledFundamentalRelation.
This contains all of the properties of a macroscopic system: the conditions of
stability (i.e., of observability) of states of equilibrium and the responses that the
systemprovidetoanyexternalperturbation.Allthethermomechanicalpropertiesof
materials can be expressed with partial derivatives of the Fundamental Relation of
that material. The fact that these relations are verified by experimental observation
allows us to understand that the existence of this relation expresses that value of
generalityasEinsteinseesit,andthereforehasthatcharacteristicof“true,”meaning
not linked to modelings.
Inconclusion,togetherwiththeformulationofthefirstlaw,whichdefinesenergy
and postulates its conservation, the existence and the geometric properties of the
Fundamental Relation constitute the complete and all-inclusive formulation of the
postulate “according to which perpetual motion is impossible.” The Fundamental
Relationanditsdifferentialexpression(whichissometimescalledtheFundamental
Equation) also constitute the starting point for the study of the dynamics of irre-
versibleprocesses.Itisfromthisdifferentialexpressionthatthe“forms”oftheflows
describingthedynamicsoftheprocessesinaction,andoftheforceswhichputthese
processesinto motion,are identified.
1Temperature(absolute)shouldbecorrectlydefinedtogetherwithentropy,i.e.,withintheSecond
Principle.
x Preface
Outline of This work
(cid:129) Part I is dedicated to the fundamental principles that, as in Feynman’s
methodological approach, we can call the “Toolbox.” In conformity with
Einstein’sstatement,thestudentwillfocuson“…theprinciplesthatgiveriseto
mathematically formulated criteria” which will be used to expand their
knowledge of natural phenomena. Knowledge of nature will be reflected in
complete and clear descriptions based on the certainty offoundations.
(cid:129) The First Principle is expressed according to M. Born’s formulation given in
1921 in an article published in Physikalische Zeitschrift 22(21), 224. The most
relevant part of this approach is his definition of amount of heat as an
all-inclusive amount which sums up the energy transferred to the system by all
theinteractionsnotcontrolledbytheobserver.TheSecondPrincipleconsistsin
thedefinitionofentropy.Therearevariousexcellentformulations,butwehave
chosen that adopted by E. A. Guggenheim in his celebrated treatise on ther-
modynamics first published in 1949. In this formulation, entropy and temper-
ature are defined as two fundamental, complementary quantities (as in other
formulations), but the core point consists in the separation of the variation in
entropy,inanyinfinitesimalprocess,intothesumoftwocontributions:onedue
to the processes which occur within the system and one due to the interaction
with theoutside. Thissubdivisionintotwocontributions,whichhastobedone
for the variation of any extensive quantity, is of fundamental importance
because it clearly shows the role of the interactions in the transfer of entropy
fromtheoutside world.Thischoicewillprovetohavebeenveryadvantageous
when studying irreversible processes: The contributions from interactions with
the outside will make up the flows whose dynamics constitute our description
of the processes underway. In Part I, we have also included a chapter on
Maxwell’s relations. The reason for situating this topic, which would appear to
be more technical, in the part dedicated to fundamental principles lies in the
need to show how knowledge of the Fundamental Relation is equivalent to the
knowledge (in general, it will be a knowledge gained from experimental mea-
sures) of the coefficients of compressibility, of thermal expansion, and of a
specific heat of the system under consideration.
(cid:129) InPartII,named“Applications,”weapplythefundamentalprinciplestoaseries
of generalized situations which are frequently to be found. The various topics
are listed in the index, but two of them may be highlighted. One concerns the
particularattentiondevotedtothelawofcorrespondingstates,atopicwhich,in
ouropinion,shouldalwaysformpartofthespecifictraining.Theotherconcerns
the relation between radiation, relativity and thermodynamics; it contains a
summaryoftheroutebywhichPlanckwasabletoextractthecorrectexpression
of spectral density of radiation in thermal equilibrium and which peaked in his
famousarticleattheendof1900.Thereasonforthischoiceisthatweconsider
itnecessarytorememberthefundamentalrolethermodynamicshadinobtaining