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Principles of Thermodynamics

In this introductory textbook, thermodynamics is presented as a natural extension of
mechanics so that the laws and concepts learned in mechanics serve to get acquainted with
the theory. The foundations of thermodynamics are presented in the first part. The second
part covers a wide range of applications, which are of central importance in the fields of
physics, chemistry and engineering, including calorimetry, phase transitions, heat engines
and chemical reactions. In the third part, devoted to continuous media, Fourier and Fick’s
laws, diffusion equations and many transport effects are derived using a unified approach.
Each chapter concludes with a selection of worked examples and several exercises to
reinforce key concepts under discussion. A full solutions manual is available at the end
of the book. It contains more than 150 problems based on contemporary issues faced by
scientists and engineers that are solved in detail for undergraduate and graduate students.
Jean-Philippe Ansermet is a professor of physics at École Polytechnique Fédérale de Lausanne
(EPFL), a fellow of the American Physical Society and a past president of the Swiss
Physical Society. He coordinated the teaching of physics at EPFL for 12 years. His course
on mechanics, taught for 25 years, was based on his textbook and a massive open online
course (MOOC) that has generated over half a million views. For more than 15 years, he
has taught thermodynamics to engineering and physics students. An expert in spintronics,
he applies thermodynamics to analyse his pioneering experiments on giant magnetoresistance, or heat–driven spin torques and predict novel effects.
Sylvain D. Brechet completed his PhD studies in theoretical cosmology at the Cavendish
Laboratory of the University of Cambridge as an Isaac Newton fellow. He is lecturer
at the Institute of Physics at EPFL. He teaches mechanics, thermodynamics and electromagnetism to first-year students. His current research focuses on theoretical modelling in
condensed matter physics and more particularly in spintronics. Merging the fields of nonequilibrium thermodynamics, continuum mechanics and electromagnetism, he brought
new insight to spintronics and fluid mechanics. In particular, he predicted in 2013 the
existence of a fundamental irreversible thermodynamic effect now called the Magnetic
Seebeck effect.

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Principles of Thermodynamics
JEAN-PHILIPPE ANSERMET
École Polytechnique Fédérale de Lausanne

SYLVAIN D. BRECHET
École Polytechnique Fédérale de Lausanne

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University Printing House, Cambridge CB2 8BS, United Kingdom
One Liberty Plaza, 20th Floor, New York, NY 10006, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India
79 Anson Road, #06–04/06, Singapore 079906
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781108426091
DOI: 10.1017/9781108620932
© Jean-Philippe Ansermet and Sylvain D. Brechet 2019
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2019
First edition in French published by Presses Polytechniques et Universitaires Romandes, 2016.

Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Ansermet, Jean-Philippe, 1957- author. | Brechet, Sylvain D., 1981– author.
Title: Principles of thermodynamics / Jean-Philippe Ansermet
(École Polytechnique Fédérale de Lausanne), Sylvain D. Brechet
(École Polytechnique Fédérale de Lausanne).
Other titles: Thermodynamique. English
Description: Cambridge ; New York, NY : Cambridge University Press, 2018. |
Originally published in French: Thermodynamique (Lausanne : EPFL, 2013). |
Includes bibliographical references and index.
Identifiers: LCCN 2018030098 | ISBN 9781108426091 (hardback : alk. paper)
Subjects: LCSH: Thermodynamics–Textbooks. | Thermodynamics–Problems,
exercises, etc.
Classification: LCC QC311.28 .A5713 2018 | DDC 536/.7–dc23
LC record available at />ISBN 978-1-108-42609-1 Hardback
Additional resources for this publication at www.cambridge.org/9781108426091
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.

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Contents

Preface
Acknowledgments


page xiii
xv

Part I Foundations
1 Thermodynamic System and First Law
1.1
Historical Introduction
1.2
Thermodynamic System
1.3
State, Variables and State Functions
1.4
Processes and Change of State
1.5
Extensive and Intensive Quantities
1.6
First Law of Thermodynamics
1.7
Thermodynamics and Mechanics
1.8
Internal Energy
1.9
Damped Harmonic Oscillator
1.10 Worked Solutions
Exercises

2 Entropy and Second Law
2.1
Historical Introduction
2.2

Temperature
2.3
Heat and Entropy
2.4
Second Law of Thermodynamics
2.5
Simple System
2.6
Closed and Rigid Simple System
2.7
Adiabatic and Closed Mechanical System
2.8
Open, Rigid and Adiabatic System
2.9
Closed Simple System
2.10 Open, Rigid and Diathermal System
2.11 Worked Solutions
Exercises

3 Thermodynamics of Subsystems
3.1
3.2

Historical Introduction
Rigid and Impermeable Diathermal Wall

v

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1
3
3
5
5
7
8
9
11
13
16
18
23
26
26
28
29
31
32
35
36
37
37
39
41
46
49
49
50



vi

Contents

3.3
Moving, Impermeable and Diathermal Wall
3.4
Rigid and Permeable Diathermal Wall
3.5
Movable and Permeable Diathermal Wall
3.6
Worked Solutions
Exercises

4 Thermodynamic Potentials
4.1
Historical Introduction
4.2
Fundamental Relations
4.3
Legendre Transformation
4.4
Thermodynamic Potentials
4.5
Equilibrium of Subsystems Coupled to a Reservoir
4.6
Heat and Work of Systems Coupled to Reservoirs
4.7
Maxwell Relations

4.8
Worked Solutions
Exercises

Part II Phenomenology
5 Calorimetry
5.1
Historical Introduction
5.2
Thermal Response Coefficients
5.3
Third Law of Thermodynamics
5.4
Mayer Relations
5.5
Specific Heat of Solids
5.6
Ideal Gas
5.7
Thermal Response Coefficients of the Ideal Gas
5.8
Entropy of the Ideal Gas
5.9
Worked Solutions
Exercises

6 Phase Transitions
6.1
Historical Introduction
6.2

The Concavity of Entropy
6.3
The Convexity of Internal Energy
6.4
Stability and Entropy
6.5
Stability and Thermodynamic Potentials
6.6
Phase Transitions
6.7
Latent Heat
6.8
The Clausius–Clapeyron Equation
6.9
Gibbs Phase Rule
6.10 Van der Waals Gas
6.11 Worked Solutions
Exercises

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53
55
58
59
67
70
70
71
74

76
79
82
84
87
96
101
103
103
105
108
109
110
111
112
114
118
127
129
129
131
133
135
137
139
143
144
145
147
151

160


vii

Contents

7 Heat Engines
7.1
Historical Introduction
7.2
Thermal Machine
7.3
Carnot Cycle
7.4
Reversible Processes on an Ideal Gas
7.5
Carnot Cycle for an Ideal Gas
7.6
Efficiency and Coefficients of Performance
7.7
Endoreversible Carnot Cycle
7.8
Stirling Engine
7.9
Heat Pump and Refrigerator
7.10 Worked Solutions
Exercises

8 Chemistry and Electrochemistry

8.1
Historical Introduction
8.2
Chemical Reactions
8.3
Matter Balance and Chemical Dissipation
8.4
Molar Volume, Entropy and Enthalpy
8.5
Mixture of Ideal Gases
8.6
Osmosis
8.7
Electrochemistry
8.8
Worked Solutions
Exercises

Part III Continuous Media
9 Matter and Electromagnetic Fields
9.1
Historical Introduction
9.2
Insulators and Electromagnetic Fields
9.3
Conductors and Electromagnetic Fields
9.4
Conductor and External Electromagnetic Fields
9.5
Adiabatic Demagnetisation

9.6
Worked Solutions
Exercises

10 Thermodynamics of Continuous Media
10.1 Historical Introduction
10.2 Continuity Equations
10.3 Evolution Equations
10.4 Worked Solutions
Exercises

11 Thermodynamics of Irreversible Processes
11.1

Historical Introduction

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166
166
168
171
174
176
179
181
183
185
187
196

203
203
204
208
209
212
216
218
223
232
239
241
241
243
250
259
262
266
273
277
277
278
287
298
304
308
308


viii


Contents

11.2 Linear Empirical Relations
11.3 Chemical Reactions and Viscous Friction
11.4 Transport
11.5 Fluid Dynamics
11.6 Worked Solutions
Exercises

Part IV Exercises and Solutions
1 Thermodynamic System and First Law
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9

State Function: Mathematics
State Function: Ideal Gas
State Function: Rubber Cord
State Function: Volume
Cyclic Rule for the Ideal Gas
Evolution of Salt Concentration
Capilarity: Contact Angle
Energy: Thermodynamics Versus Mechanics

Damped Harmonic Oscillator

2 Entropy and Second Law
2.1
2.2
2.3
2.4
2.5
2.6
2.7

Entropy as a State Function
Work as a Process-Dependent Quantity
Bicycle Pump
Rubbing Hands
Heating by Stirring
Swiss Clock
Reversible and Irreversible Gas Expansion

3 Thermodynamics of Subsystems
3.1
3.2
3.3
3.4
3.5
3.6
3.7

Thermalisation of Two Separate Gases
Thermalisation of Two Separate Substances

Diffusion of a Gas through a Permeable Wall
Mechanical Damping by Heat Flow
Entropy Production by Thermalisation
Entropy Production by Heat Transfer
Thermalisation by Radiation

4 Thermodynamic Potentials
4.1
4.2
4.3
4.4
4.5

Adiabatic Compression
Irreversible Heat Transfer
Internal Energy as Function of T and V
Grand Potential
Massieu Functions

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310
313
314
331
333
349
361
363
363

363
364
364
366
366
368
369
371
372
372
373
373
374
375
376
378
379
379
380
381
382
385
385
386
388
388
388
389
390
390



ix

Contents

4.6
4.7
4.8
4.9
4.10
4.11

Gibbs–Helmoltz Equations
Pressure in a Soap Bubble
Pressure in a Droplet
Isothermal Heat of Surface Expansion
Thermomechanical Properties of an Elastic Rod
Chemical Power

5 Calorimetry
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8


Heat Transfer as a Function of V and p
Bicycle Pump
Heat Transfer at Constant Pressure
Specific Heat of a Metal
Work in Adiabatic Compression
Slopes of Isothermal and Adiabatic Processes
Adsorption Heating of Nanoparticles
Thermal Response Coefficients

6 Phase Transitions
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13

Melting Ice
Cooling Water with Ice Cubes
Wire through Ice without Cutting
Dupré’s Law
Hydropneumatic Accumulator
Positivity of Thermal Response Coefficients

Heat Pipe
Vapour Pressure of Liquid Droplets
Melting Point of Nanoparticles
Work on a van der Waals Gas
Inversion Temperature of the Joule–Thomson Process
Lever Rule
Eutectic

7 Heat Engines
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9

Refrigerator
Power Plant Cooled by a River
Braking Cycle
Lenoir Cycle
Otto Cycle
Atkinson Cycle
Refrigeration Cycle
Rankine Cycle
Rankine Cycle for a Biphasic Fluid

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391
392
393
394
395
396
399
399
399
400
401
401
402
402
403
405
405
405
406
408
408
410
412
413
414
416
416
417
418

420
420
420
422
424
426
429
432
433
436


x

Contents

8 Chemistry and Electrochemistry
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13

8.14
8.15

Oxidation of Ammonia
Acetylene Lamp
Coupled Chemical Reactions
Variance
Enthalpy of Formation
Work and Heat of a Chemical Reaction
Mass Action Law: Esterification
Mass Action Law: Carbon Monoxide
Entropy of Mixing
Raoult’s Law
Boiling Temperature of Salt Water
Battery Potential
Thermogalvanic Cell
Gas Osmosis
Osmosis Power Plant

9 Matter and Electromagnetic Fields
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8

Vapour Pressure of a Paramagnetic Liquid

Magnetic-Field Induced Adsorption or Desorption
Magnetic Battery
Electrocapilarity
Magnetic Clausius–Clapeyron Equation
Magnetocaloric Effect
Kelvin Probe
Electromechanical Circuit

10 Thermodynamics of Continuous Media
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8

Chemical Substance Balance
Pressure Time Derivative and Gradient
Oil and Water Container
Floating Tub Stopper
Temperature Profile of the Earth’s Atmosphere
Stratospheric Balloon
Velocity Field Inside a Pipe
Divergence of a Velocity Field

11 Thermodynamics of Irreversible Processes
11.1
11.2

11.3
11.4
11.5

Heat Diffusion Equation
Thermal Dephasing
Heat Equation with Heat Source
Joule Heating in a Wire
Thomson Heating in a Wire

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439
439
440
440
442
443
444
445
445
447
448
450
451
452
452
454
456
456

457
458
459
460
461
462
464
467
467
467
469
469
471
473
474
475
478
478
479
480
481
482


xi

Contents

11.6
11.7

11.8
11.9
11.10
11.11
11.12
11.13
11.14
11.15
11.16

Heat Exchanger
Harman Method
Peltier Generator
ZT Coefficient of a Thermoelectric Material
Transverse Transport Effects
Hall Effect
Heat Transport and Crystal Symmetry
Planar Ettingshausen Effect
Turing Patterns
Ultramicroelectrodes
Effusivity

References
Index

483
486
487
492
495

497
498
499
502
506
510
515
525

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Preface

Thermodynamics is a theory which establishes the relationship between the physical
quantities that characterise the macroscopic properties of a system. In this textbook,
thermodynamics is presented as a physical theory which is based upon two fundamental
laws pertaining to energy and entropy, which can be applied to many different systems
in chemistry and physics, including transport phenomena. By asserting that energy and
entropy are state functions, we eliminate the need to master the physical significance of
differentials. Thus, thermodynamics becomes accessible to anyone with an elementary
mathematical background. As the notion of entropy is introduced early on, it is readily
possible to analyse out-of-equilibrium processes taking place in systems composed of
simple blocks.
Students engaging with thermodynamics have the opportunity to discover a broad range
of phenomena. However, they are faced with a challenge. Unlike Newtonian mechanics
where forces are the cause of acceleration, the mathematical formalism of thermodynamics
does not present an explicit link between cause and effect.
Nowadays, it is customary to introduce temperature by referring to molecular agitation
and entropy by invoking Boltzmann’s formula. However, in this book, the intrusion of

notions of statistical physics are deliberately avoided. It is important to start off by
teaching students the meaning of a physical theory and to show them clearly the very
large preliminary conceptual work that establishes the notions and presuppositions of
this theory. Punctual references to notions of statistical physics, which are not formally
presented, give the impression that in science the results from another theoretical body of
knowledge can be borrowed without precaution. By doing so, students might not perceive
thermodynamics as a genuine scientific approach. It is clear that the introduction of
entropy with a mathematical formula is somewhat reassuring. However, it is by performing
calculations of entropy changes in simple thermal processes that students become familiar
with this notion and not by contemplating a formula that is not used in the framework of
thermodynamics.
This book is broken up into four parts. The first part of the book gathers the formal
tools of thermodynamics, such as the thermodynamic potentials and Maxwell relations.
The second part illustrates the thermodynamic approach with a few examples, such as
phase transitions, heat engines and chemical reactions. The third part deals with continuous
media, including a chapter that is devoted to interactions between electromagnetic fields
and matter. A formal development of the thermodynamics of continuous media results in
the description of numerous transport laws, such as the Fourier, Fick or Ohm laws and the
Soret, Dufour or Seebeck effects.
xiii

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xiv

Preface

At the end of each chapter, there are worked solutions that practically demonstrate what
has been presented, and these are followed by several exercises. In the last part of the

book, these exercises are presented with their solutions. Some exercises are inspired by
physics auditorium demonstrations, some by research, for example: the melting point of
nanoparticles, an osmotic power plant, a Kelvin probe, the so-called ZT coefficient of
thermoelectric materials, thermogalvanic cells, ultramicroelectrodes or heat exchangers.
Thanks to the theory of irreversible phenomena which was elaborated in the period
from approximately 1935 to 1965, thermodynamics has become an intelligible theory in
which Newtonian mechanics and transport phenomena are presented in a unified approach.
The book demonstrates that thermodynamics is applicable to many fields of science and
engineering in today’s modern world.

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Acknowledgments

The authors are indebted to their mentor and friend Doctor Franỗois Reuse for the diligence
with which he introduced them to the approach of his master, Professor Stückelberg. The
authors were introduced to a school of thought through numerous discussions with the
students of Professor Stückelberg like Professor Christian Gruber and Professor André
Chatelain and with Professor Jean-Pierre Borel.
The authors gratefully acknowledge the stimulating discussions they had with the specialists whom they invited to contribute to a MOOC on thermodynamics: Chantal Maatouk
and Marwan Brouche of the École Supérieure d’Ingénieurs de Beyrouth, Lebanon; Marthe
Boyomo Onana, Paul-Salomon Ngohe-Ekam, Théophile Mbang and André Talla of the
École Nationale Supérieure Polytechnique at Yaoundé and the Université de Yaoundé I,
Cameroun; Etienne Robert of the École Polytechnique de Montréal, Canada; Miltiadis
Papalexandris of the Université Catholique de Louvain, Belgium; and Michael Grätzel
of the École Polytechnique Fédérale de Lausanne.
Graphic designer Claire-Lise Bandelier took great care in producing the figures according to the authors’ wishes, in particular when making sketches of auditorium experiments.
Professor Christian Gruber proofread the original French manuscript and made critical
suggestions. Editor Evora Dupré secured the English translation and perfected the style of

the text during the many meetings she held with the authors.
Finally, we express our sincere gratitude towards thousands of students and hundreds of
tutors who took part in the course that led to this book. It is in the context of this large and
vigilant audience that this book took shape.

xv

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Part I

FOUNDATIONS

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1

Thermodynamic System and First Law

James Prescott Joule, 1818–1889
J. P. Joule was the owner of a brewery in England and worked as a self-educated scientist making
major contributions to the development of thermodynamics. In 1840, he stated the law that bears his
name on power dissipated by a current passing through a resistance. In 1843, there began a series of
observations on the heat equivalent pertaining to mechanical work.

1.1 Historical Introduction
At the beginning of the nineteenth century, steam engines had been converting heat
into work for about 150 years. Scientific investigations were under way to establish a

quantitative equivalence of heat and work. With time, this concept became the law of
energy conservation, which we will discuss in this chapter.
In 1839, Marc Séguin, nephew of the famous Mongolfier, published his ‘Study on the
influence of railways’ [1]. It was clear for him that the condenser of a heat engine played
a role that was equally important as that of the furnace [2]. Séguin assumed that the steam
that caused the volume of the cylinder to increase performed work that was equal in
3

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4

Figure 1.1

Thermodynamic System and First Law

Joule’s calorimeter: the fall of the weight at nearly constant velocity (because of the fluid viscosity) drives a stirring
mechanism in the calorimeter. The change in potential energy of the weight determines the work performed on
the liquid. The increase in energy of the liquid is deduced from its temperature rise.

magnitude to the heat lost by the steam in the process. He sought to estimate this heat
loss by measuring the heat taken from the furnace and returned to the condenser.
A year later, Julius Robert von Mayer travelled to the tropics as a medical doctor.
He observed that the colour difference between veinous and arterial blood was more
pronounced there than at latitudes where the climate is colder. He attributed this difference
to the heat released by the body. His thoughts on the human body as a heat engine led
him to the idea of an equivalence between heat and work, which he later tested on inert
matter.
In his 1842 treatise [3], he asked the following question: what is the change in

temperature of a stone when it hits the ground after falling from a given height? James
Prescott Joule succeeded in doing this measurement, thanks to his development of highly
sensitive thermometers. Joule also observed that an electrical current dissipates heat,
an effect that bears his name. In 1845, he published his fundamental work on energy
conservation [4]. With a calorimeter (Fig. 1.1), he determined the equivalence between
work (defined by masses going down in the gravitational potential) and heat (corresponding
to the warming of the liquid stirred by the device). Before Joule established this link, it
was customary to measure heat in calories. Joule established the conversion to the unit
that bears his name, the joule [J]: 1 [cal] = 4.1855 [J] .
These are units of energy. Joule and others observed in various circumstances that
the heat provided to a system and the work performed on it are equal and opposite
to each other for every process that brings the system back to its initial state [5]. Von
Mayer, in his treatise of 1842, expressed the idea that every system is characterised by
a quantity, energy, that can only be modified by an external action in the form of work
or heat [6]. Hermann von Helmholtz, in 1847, gives energy conservation the status of a
physical law [7].

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5

1.3 State, Variables and State Functions

1.2 Thermodynamic System
A thermodynamic system consists of matter contained in a region of space delineated by
a closed surface, called an enclosure, that separates the system from its environment or
surroundings. The system is assumed to be large, in the sense that the amount of substance
it contains is typically counted in moles. A mole corresponds to the Avogadro number
6.02 · 1023 that is used to count the number of elementary constituents of matter. This

definition can be extended to a thermodynamic system that includes radiation and any
other kind of physical field. In the first two parts of this book, we will discuss systems
consisting of matter only. We will introduce electromagnetic fields in the third part.
We use the following terms to characterise how a system interacts with its environment.
A system is said to be:
•
•
•
•
•

open, if its enclosure allows convective matter exchange with the environment
closed, if its enclosure does not allow convective matter exchange with the environment
diathermal, if its enclosure allows conductive heat exchange with the environment
adiabatic, if its enclosure does not allow conductive heat exchange with the environment
isolated, if its enclosure does not allow any interaction with the environment

A thermodynamic system can be decomposed in subsystems that can be considered as
thermodynamic systems themselves. The separation between two subsystems is called a
wall. The enclosure between a system and its environment consists of one or several walls.
We use the following terms to characterise a wall. A wall is said to be:
•
•
•
•
•
•

fixed, if it cannot move
movable, if it can move

permeable, if it allows convective matter exchange with the environment
impermeable, if it does not allow convective matter exchange with the environment
diathermal, if it allows conductive heat exchange with the environment
adiabatic, if it does not allow conductive heat exchange with the environment

1.3 State, Variables and State Functions
The state of a system is characterised by physical properties that are described by a set of
state variables. The state is entirely specified by the values of these state variables and it
does not depend on the history of the system. The set of these state variables is written as,
{X1 , X2 , X3 , X4 , X5 , . . . .}
A state function is a physical property that depends only on the state of the system. Thus,
a state function is expressed as,
F (X1 , X2 , X3 , X4 , X5 , . . . .)

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6

Thermodynamic System and First Law

1.3.1 Partial Derivatives of a Function
Let f (x, y) be a function of two variables x and y. The partial derivatives with respect to
the variables x and y are defined by,
f (x + Δx, y) − f (x, y)
∂f (x, y)
≡ lim
Δx→0
∂x
Δx

∂f (x, y)
f (x, y + Δy) − f (x, y)
≡ lim
Δy→0
∂y
Δy

(1.1)

Thus, in the calculation of a partial derivative with respect to one variable, the other
variables are kept constant. As an example, let us consider the function f (x, y) = x2 + 3xy.
The partial derivatives of this function are,
∂f (x, y)
= 2x + 3y
∂x

and

∂f (x, y)
= 3x
∂y

1.3.2 Differential of a Function
Let f (x, y) be a function of two variables x and y [8]. The variation of the function f (x, y)
from point (x, y) to point (x + Δx, y + Δy) is written as,
Δf (x, y) = f (x + Δx, y + Δy) − f (x, y)

(1.2)

It can be recast as,

Δf (x, y) = f (x + Δx, y + Δy) − f (x, y + Δy) + f (x, y + Δy) − f (x, y)
=

f (x + Δx, y + Δy) − f (x, y + Δy)
Δx
Δx
f (x, y + Δy) − f (x, y)
+
Δy
Δy

(1.3)

The differential df (x, y) is defined as the infinitesimal limit of the variation Δf (x, y),
df (x, y) ≡ lim

lim Δf (x, y)

Δx→0 Δy→0

(1.4)

Taking into account the limit
lim

Δy→0

f (x + Δx, y + Δy) − f (x, y + Δy)

= f (x + Δx, y) − f (x, y)


(1.5)

in equation (1.3), the differential (1.4) can be written as,
f (x + Δx, y) − f (x, y)
Δx
Δx
f (x, y + Δy) − f (x, y)
+ lim
Δy
Δy→0
Δy

df (x, y) = lim

Δx→0

(1.6)

Using the definition (1.1) of the partial derivatives of a function, the differential (1.6) is
reduced to,
∂f(x, y)
∂f(x, y)
df (x, y) =
dx +
dy
(1.7)
∂x
∂y


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7

1.4 Processes and Change of State

1.3.3 Time Derivative of a Function
Let f (x, y) be a function of two time-dependent variables x and y. The time derivative of
the function is obtained using the chain rule,
∂f (x, y)
∂f (x, y)
x˙ +
y˙
f˙ (x, y) =
∂x
∂y

(1.8)

where
df (x, y)
f˙ (x, y) ≡
dt

x˙ ≡

dx
dt


y˙ ≡

dy
dt

1.4 Processes and Change of State
A thermodynamic system can interact with its environment through processes that change
the state of the system. We distinguish three types of physical processes:
• mechanical processes
• thermal processes
• chemical processes
As we will see, thermodynamics allows us to provide a quantitative characterisation
of the effects that processes have on the state of a system. Mechanical processes
can lead to mechanical or thermal changes of the state. In the experiment of Joule’s
calorimeter (Fig. 1.1), work is performed on the system and its temperature changes. Thus,
a mechanical process can lead to a thermal change of state. Likewise, thermal processes
can lead to thermal or mechanical changes of the system state. In the experiment illustrated
in Fig. 1.2, heat is provided to the system by hand contact. The pressure of the gas rises,
causing a shift of the water levels in the U-shaped tube. Thus, a thermal process can lead
to a mechanical change of state.

Figure 1.2

A vessel contains a fixed amount of gas. The liquid in the tube measures the changes of pressure when the gas is
heated by the hand.

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8


Thermodynamic System and First Law

In the context of describing the types of processes that take place between a system and
its environment, the term “chemical process” refers specifically to matter being exchanged
between a system and its environment. This is not to be confused with chemical reactions
among different substances. Such reactions occurring within the system will be explored
in Chapter 8. The enclosure of the system should be defined so as to encompass the regions
of space where chemical reactions take place.

1.5 Extensive and Intensive Quantities
A quantity is called extensive when it has the following property: its value for the whole
system is equal to the sum of its values for every subsystem. The following are extensive
quantities:
•
•
•
•
•

mass
momentum
angular momentum
energy
volume

Sometimes we refer to extensive quantities divided by the volume, the mass or the
number of moles of the system. They are called a volume density, a mass density or a
molar density. In that case we speak of a specific quantity or a reduced extensive quantity.
The following are densities:

•
•
•
•

mass density
momentum density
angular momentum density
energy density

A quantity is called intensive when it is conjugated to an extensive quantity, which
means that it is defined as the partial derivative of the energy with respect to this extensive
quantity. The following are intensive quantities:
• velocity
• pressure
• temperature
To determine if a quantity is extensive or intensive, it is useful to imagine what happens
to this quantity when the size of the system doubles. If the quantity is extensive, its value
doubles, but if it is intensive, its value does not change. It is also useful to clarify that
certain quantities are neither extensive nor intensive. This is the case of the entropy
production rate, which we will introduce in Chapter 2.
A system is called homogeneous when all the intensive scalar functions conjugated to
the extensive scalar state variables do not depend on position. This means that they have the
same value for every subsystem. A system is called uniform when the intensive vectorial

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9


1.6 First Law of Thermodynamics

functions conjugated to the extensive vectorial state variables do not depend on position.
This means that they have the same norm and orientation in every subsystem.

1.6 First Law of Thermodynamics
Thermodynamics is based on two fundamental laws. Their justification is based on the
empirical validity of their implications. In this chapter, we discuss the first law. We will
present the second law in Chapter 2.
The first law of thermodynamics states that:
For every system, there is a scalar extensive state function called energy (E). When
the system is isolated, the energy is conserved.
The energy conservation law is mathematically written as,
E˙ = 0

(isolated system)

(1.9)

where E˙ ≡ dE/dt. This conservation is related to time homogeneity [9]. It implies that
energy is defined up to a constant.
When the system interacts with its environment, the energy evolution results from
the power of the processes exerted on the system. We distinguish four types of external
processes and write [10]:
E˙ = P ext + PW + PQ + PC

(open system)

(1.10)


• P ext represents the power associated with the external forces and torques that modify
the translational kinetic energy of the centre of mass and the rotational kinetic energy
around the centre of mass. These forces and torques do not modify the shape of the
system.
• PW represents the mechanical power associated with the work performed by the
environment on the system that results in a deformation of the system without any change
in its state of motion, in particular its kinetic energy.
• PQ represents the thermal power associated with heat exchange with the environment
through conduction.
• PC represents the chemical power associated with matter exchange with the environment
through convection.
Any physical process performing work is called a mechanical action. Any physical
process in which heat is exchanged is called a heat transfer. A physical process in which
matter is exchanged is called a matter transfer or mass transfer. When a heat transfer
takes place through a matter transfer, it is called a heat transfer by convection. When a
heat transfer occurs without matter transfer, it is called a heat transfer by conduction. In
general, a matter transfer leads simultaneously to a mechanical action and to a heat transfer.

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10

Thermodynamic System and First Law

We have characterised systems according to ways in which they interact with their
environment. We can now specify such characteristics in terms of the powers of the various
processes considered here. Hence, a system is called:
•
•

•
•

rigid if no work by deformation is possible, i.e. PW = 0.
closed if there is no matter transfer, i.e. PC = 0, and open otherwise.
adiabatic if there is no heat transfer, i.e. PQ = 0 and diathermal in the opposite case.
isolated if it is rigid, adiabatic and closed in the absence of external forces and torques,
i.e. P ext = PW = PQ = PC = 0.
When a system is closed, the energy evolution equation (1.10) reduces to,
E˙ = P ext + PW + PQ

(closed system)

(1.11)

The first law can be expanded on to include two other conservation laws that impose
additional constraints on the possible states of the system. The first is related to the
translational state of motion and the second to the rotational state of motion.
Concerning translations, we have the following conservation law:
For every system, there is a vectorial extensive state function called momentum (P).
When the system is isolated, the momentum is conserved.
The momentum conservation law is mathematically written as,
P˙ = 0

(isolated system)

(1.12)

where P˙ ≡ dP/dt. This conservation law is related to the homogeneity of space [9]. It
implies that the momentum is defined up to a constant.

In the case of a system interacting with the environment, the evolution of the momentum
with respect to an inertial frame of reference is given by the centre-of-mass theorem [11],
P˙ = F ext

(1.13)

where F ext is the net external force exerted on the system. If the system undergoes a
uniform translational motion, the momentum is constant. A net external force causes a
departure from the state of uniform translation.
Concerning rotations, we have the following conservation law:
For every system, there is a vectorial extensive state function called angular
momentum (L). When the system is isolated, the angular momentum is conserved.
The angular momentum conservation law is mathematically written as,
L˙ = 0

(isolated system)

(1.14)

where L˙ ≡ dL/dt. This conservation law is related to the isotropy of space. It implies that
the angular momentum is defined up to a constant.
In the case of a system interacting with the environment, the evolution of the angular
momentum with respect to an inertial frame of reference is given by the angular momentum
theorem [11],
(1.15)
L˙ = M ext

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