Solved Problems in Thermodynamics and Statistical Physics

292 Pages·2019·5.441 MB·English

by  Gregor Skačej, Primož Ziherl

Gregor Skačej
Primož Ziherl
Solved Problems
in Thermodynamics
and Statistical
Physics
Solved Problems in Thermodynamics and Statistical
Physics
č ž
Gregor Ska ej Primo Ziherl
(cid:129)
Solved Problems
in Thermodynamics
and Statistical Physics
123
Gregor Skačej PrimožZiherl
Faculty of Mathematics andPhysics Jožef StefanInstitute
University of Ljubljana Faculty of Mathematics andPhysics
Ljubljana, Slovenia University of Ljubljana
Ljubljana, Slovenia
ISBN978-3-030-27659-1 ISBN978-3-030-27661-4 (eBook)
https://doi.org/10.1007/978-3-030-27661-4
The authors have obtained the copyright to publish in English from the Slovenian publishers. The
contents of this book were originally published in Slovenian in 2005, in two volumes in Slovenian
language:
P.ZiherlandG.Skačej,Rešenenalogeiztermodinamike(DMFA-založništvo,Ljubljana,2005).
G.SkačejandP.Ziherl,Rešenenalogeizstatističnefizike(DMFA-založništvo,Ljubljana,2005).
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To Mira and Katja
Preface
Mastering a topic in physics generally includes solving a suitable set of problems,
either in tutorials and homework assignments or while preparing for an examina-
tion. Thermodynamics and statistical physics are no exception, although their
conceptual framework is seemingly simple, at least at the level of equilibrium
phenomena, and they are usually technically undemanding compared to, e.g.,
theory of elasticity or electromagnetism. Most standard textbooks on thermody-
namicsandstatisticalphysicsdocontainsomeproblemsthatthereaderscanuseto
consolidate their knowledge, but the dedicated solved-problems volumes address
the development of these specific skills in a more focused, hands-on, and com-
prehensive fashion. This book belongs to the latter category.
The material included is or has been used in the undergraduate course on sta-
tisticalthermodynamics for students of physicsat the University of Ljubljana. The
topicsdiscussedcoverthestandardsyllabusofmostsuchcoursesfromtheequation
of state to the kinetic theory of gases, and elementary knowledge of classical and
quantum physics is sufficient to tackle most problems. With the selection of
problems, we wish to emphasize that the theoretical apparatus of thermodynamics
and statistical physics is quite universal and that it does not apply solely to pVT
systems best known from the typical general-physics course for freshmen. This is
why we often discuss electric, magnetic, and other non-pVT systems, equations of
state, etc., so as to offer readers the opportunity to recognize the universality by
themselves. At the same time, we put emphasis on examples from soft condensed
matter physics, touching upon the many instances where excluded-volume inter-
actions lead to interesting effects. (Here we must refer the interested reader to the
excellentEntropyBeyondtheSecondLawbyAttard.)Inseveralcases,weresortto
seemingly artificial problems, such as when discussing the two-dimensional Fermi
gas, because there exists an exact closed-form solution, which is well worth
deriving and examining. Some of the problems are actually case studies dealing
with selected special topics from different fields of physics. These include the
isothermal–isobaric ensemble, Tonks gas, Debye–Hückel theory of plasma,
Onsager theory of the nematic–isotropic transition, rubber elasticity, Flory
mean-field theory of polymers, transfer-matrix formalism in spin and polymer
vii
viii Preface
chains,renormalizationgroup,latticegasmodel,etc.Afewsystemsareconsidered
repeatedly but from diverse perspectives: For example, the one-dimensional Ising
chain is analyzed using as many as four different approaches, and thermionic
emission is studied both within thermodynamics and statistical mechanics.
The book may be of interest primarily to undergraduate students of physics as
wellastotheirinstructors,butitmayalsobehelpfulasareference foracourseof
chemicalorengineeringthermodynamics.Someofthe234problemsexaminedare
discussedintutorialsandothersweredevelopedfortheopen-notes/open-booksstyle
examinations at theUniversity ofLjubljana.Many wereoriginally proposedinthe
Slovenian-language textbook Toplota by Kuščer and Žumer, albeit without solu-
tions,andsome areadaptedfromorinspired byother textbooks. Quite afew were
conceivedbyourselves.Thematerialcontainedinthisbookisintendedtobeusedin
tutorials and for self-study, and the book itself is a companion text rather than a
self-contained,standalonevolumeaswedidnotincludeanyintroductorymaterialto
review the main concepts in a given topic. When organizing the material into
chapters,wearrangedtheproblemsaccordingtothemaintheme;insomecases,this
cannot bedone unambiguouslyand hence some arbitrariness in thedivision.
Ourbookishardlytheonlyofitskind;some otherreferencesofvariousstyles,
coverages,andlevelsofdifficultyarelistedattheendwherewealsomentionafew
textbooks of interest. The book edited by Lim is a collection of problems from
entranceexaminationsattheuniversitiesintheUS,whicharegenerallydesignedto
testconceptualknowledge.Abbott–VanNess’volumeofSchaum’sOutlineisanice,
engineering-oriented thermodynamics-only collection of solved problems. Kubo’s
classicscombineanoverviewofthemainconceptsineachtopicwithproblemsthat
spanmanylevelsofdifficulty.Althoughnotacollectionofsolvedproblems,Rumer–
Ryvkin’s textbook contains many illuminating case studies or worked examples.
Equallyinterestingastheseestablishedreferencesarethemorerecentonessuchas
the book by Cini, Fucito, and Sbragaglia. In our experience, no single resource
sufficesbyitself,andwestronglysuggestconsultingboththosesingledoutinthelist
ofreferencesaswellasanyrelatedtitlesthatwedonotmentionhere.
The style of Solved Problems in Thermodynamics and Statistical Physics is
intentionally concise and the language is terse. In each solution, we describe all
of the main steps but the intermediate ones are left to the readers as an invitation;
we are convinced that by working them out, they will better understand the
material. Nonetheless, we include an interpretation of the physical phenomenon at
handandreferencestorelatedtopicsandproblemsinthebookasappropriate.Most
problems ask for a numerical answer; still the emphasis is on concepts. Some
figures are included but, in many problems, the readers are expected to sketch one
by themselves. Often it is advantageous to consider a suitable simplification of the
exact result or a limiting case, which usually provides a physical insight. In some
problems, we complement the analytical approximation with the exact numerical
result so as to demonstrate how they compare to each other.
Thetextofproblemsistypesetinitalicwhereasthesolutionsareinuprightfont.
The symbols used are generally consistent with most modern textbooks on ther-
modynamicsandstatisticalphysics;unlessstatedotherwise,“specific”referstothe
Preface ix
quantityinquestionperunitmass.Asusual,theheatispositiveifitisreceivedby
the system and negative if it is emitted, and analogously for work. We use cali-
graphic fonts C to indicate reduced (i.e., dimensionless) quantities, the only
exceptionbeingF whichisusedforthemagnitudeoftheforce.Vectorsandtensors
areinuprightboldfacefontbandinsans-seriffontS,respectively;exceptinoneor
two instances, primed quantities such as U0 denote the initial state and unprimed
ones refer to the final state. Some textbooks use a special symbol such as đ to
indicate improper differentials; here we assume that readers know that heat and
workarenotfunctionsofstateandthatthisneednotbeemphasizedeverytimewe
writethefirstandthesecondlaw.Forconciseness,mathematicalmaterialistypeset
usingtheconventionwhereimpliedmultiplicationhasprecedenceoverdivisionso
that ab/cd stands for (ab)/(cd). We use the Système International (SI) units. For
conciseness,theindexdoesnotincludetheveryfrequententriessuchasenergyand
heat capacity at constant volume.
This book is largely a translation of two volumes that we wrote in Slovenian.
These volumes were reprinted several times before we started working on the
translation and we trust that we managed to remove the majority of errors; at this
point, we thank all students and colleagues who drew our attention to errors and
inconsistencies and helped us to fix them. It is possible that some of them remain
andwewillappreciatethereaderstellingusaboutanythingthatshouldbecorrected
or amended.
We appreciate the unfaltering support of the Faculty of Mathematics and
Physics,UniversityofLjubljana,andwethankourfamiliesfortheirencouragement
andunderstanding.WehopethatthereaderswillliketheSolvedProblemsasmuch
as we do—now that they are solved.
Ljubljana, Slovenia Gregor Skačej
Primož Ziherl
Contents
Part I Thermodynamics
1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The First Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 The Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7 Transport Phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Part II Statistical Physics
8 Classical Canonical Ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10 Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
11 Quantum Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
12 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
13 Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Appendix A: Steam Tables... ..... .... .... .... .... .... ..... .... 279
Appendix B: Metropolis Algorithm. .... .... .... .... .... ..... .... 281
References.... .... .... .... ..... .... .... .... .... .... ..... .... 285
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 287
xi
Part I
Thermodynamics