Thermodynamics and Statistical Mechanics

341 Pages·2016·5.67 MB·English

i
Thermodynamics
and
Statistical Mechanics
Cenalo Vaz
University of Cincinnati
Contents
1 Preliminaries 1
1.1 Equilibrium States and State Variables . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 The Zeroeth Law and Temperature . . . . . . . . . . . . . . . . . . . 5
1.1.3 The Thermodynamic or Absolute Temperature Scale . . . . . . . . . 7
1.2 Thermodynamic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Mechanical Equivalent of Heat . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Energy Transfer as Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1 Conduction: Newton’s Law . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 Conduction: The Heat Equation . . . . . . . . . . . . . . . . . . . . 24
1.5.3 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.4 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 The First Law 33
2.1 The Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Simple Applications to Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Heat Capacities of Gases . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 Adiabatic Equation of State . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Elementary Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . 38
2.3.1 Basic Postulates and Results . . . . . . . . . . . . . . . . . . . . . . 38
2.3.2 Internal Energy and Heat Capacities . . . . . . . . . . . . . . . . . . 40
2.3.3 Law of Equipartition of Energy . . . . . . . . . . . . . . . . . . . . . 41
2.3.4 The Mean Free Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.5 Maxwell-Boltzmann distribution of Molecular Velocities . . . . . . . 44
2.4 Thermodynamic Cycles: Engines and Refrigerators . . . . . . . . . . . . . . 48
2.5 Sample Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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CONTENTS iii
2.5.1 The Carnot Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.2 The Otto Cycle (Gasoline Engine) . . . . . . . . . . . . . . . . . . . 54
2.5.3 The Diesel Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5.4 The Brayton Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5.5 The Stirling Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5.6 The Rankine Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 The Second Law and Entropy 63
3.1 The Kelvin and Clausius statements . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Thermodynamic definition of Temperature . . . . . . . . . . . . . . . . . . . 67
3.3 The Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 The Thermodynamic Phase Space . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Integrability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.1 Internal energy of a Gas . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.2 Phase Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.3 Magnetic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.5.4 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.5.5 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.6 Macroscopic Motion of Systems in Equilibrium . . . . . . . . . . . . . . . . 92
4 Thermodynamic Potentials 94
4.1 Legendre Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.1 Heat Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 The Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.1 The Joule-Thompson (throttling) process . . . . . . . . . . . . . . . 103
4.4 The Helmholz Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4.1 The Van ’t Hoff Equation . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4.2 The Reversible Electrochemical Cell . . . . . . . . . . . . . . . . . . 109
4.4.3 The Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.4 Maxwell’s construction revisited . . . . . . . . . . . . . . . . . . . . 114
4.5 The Gibbs Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5.1 The Clausius-Clapeyron equation . . . . . . . . . . . . . . . . . . . . 116
4.5.2 The Gibbs Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5 Variable Contents 120
5.1 The Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Integrability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Chemical Potential of an ideal gas . . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Applications of the Equilibrium Conditions . . . . . . . . . . . . . . . . . . 125
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5.4.1 The Clausius-Clapeyron Equation . . . . . . . . . . . . . . . . . . . 125
5.4.2 Thermodynamics of Adsorption . . . . . . . . . . . . . . . . . . . . . 126
5.4.3 Equilibrium in Chemical Reactions . . . . . . . . . . . . . . . . . . . 129
5.4.4 Mass Action Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5 New Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . 131
6 The Third Law 133
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 The number of microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2.1 Classical Microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2.2 Quantum microstates . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.3 Nernst’s Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.4.1 Heat Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.4.2 Sublimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4.3 Allotropic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7 Foundations of Equilibrium Statistcal Mechanics 148
7.1 Brief Review of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2 Equilibrium Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Formulation of the Statistical Problem . . . . . . . . . . . . . . . . . . . . . 152
7.3.1 Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3.2 Classical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.4 Statistical Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.5 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.6 Isolated Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.7 Approach to equilibrium: the “H” Theorem . . . . . . . . . . . . . . . . . . 162
8 The Microcanonical Ensemble 165
8.1 Behavior of the Number of Microstates . . . . . . . . . . . . . . . . . . . . . 166
8.2 The Absolute Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.3 Generalized forces, Work and Energy . . . . . . . . . . . . . . . . . . . . . . 172
8.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.5 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9 The Canonical and Grand Canonical Ensembles 178
9.1 The Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
9.2.1 The Ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
9.2.2 Ideal gas in an external gravitational field . . . . . . . . . . . . . . . 182
CONTENTS v
9.2.3 Spin 1 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . 183
2
9.2.4 Harmonic oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.3 The Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.4 Properties of the Partition Function . . . . . . . . . . . . . . . . . . . . . . 188
9.4.1 Weakly interacting sub-systems. . . . . . . . . . . . . . . . . . . . . 188
9.4.2 Change of zero-point energy. . . . . . . . . . . . . . . . . . . . . . . 190
9.4.3 The Equipartition Theorem . . . . . . . . . . . . . . . . . . . . . . . 190
9.4.4 Galilean Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.5 The Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.6 Thermodynamics in the Grand Canonical Ensemble . . . . . . . . . . . . . 195
9.7 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10 Further Developments 199
10.1 The Gibbs Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.2 Maxwell’s Distribution of Speeds . . . . . . . . . . . . . . . . . . . . . . . . 204
10.3 Osmosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.4 Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
10.5 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10.6 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10.7 Debye’s Theory of Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . 219
10.8 Linear Expansion in Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
10.9 Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
11 Indistinguishability and Quantum Statistics 227
11.1 Quantum Description of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . 227
11.2 Heuristic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
11.3 Combinatoric Derivation of the Distribution Functions . . . . . . . . . . . . 232
11.4 The Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
11.5 Physical consequences of the Distributions . . . . . . . . . . . . . . . . . . . 240
11.5.1 Maxwell-Boltzmann distribution . . . . . . . . . . . . . . . . . . . . 241
11.5.2 Bose-Einstein distribution . . . . . . . . . . . . . . . . . . . . . . . . 242
11.5.3 Fermi-Dirac distribution . . . . . . . . . . . . . . . . . . . . . . . . . 243
12 Quantum Gases 245
12.1 The Photon Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
12.2 Non-relativistic Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
12.3 Bosonic Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.4 Phonons and Rotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
12.5 Bulk motion of Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
12.6 Fermi Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
vi CONTENTS
12.7 Electrons in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
12.7.1 DC Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12.7.2 AC conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
12.7.3 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
12.7.4 Thermionic Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
12.8 Pauli Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
12.9 White Dwarfs and Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . 284
13 The Density Matrix 287
13.1 Closed Quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
13.2 Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
13.3 Additional Properties of the Density Matrix . . . . . . . . . . . . . . . . . . 293
13.4 Density Matrix in the position representation . . . . . . . . . . . . . . . . . 295
13.4.1 One Dimensional Particle . . . . . . . . . . . . . . . . . . . . . . . . 295
13.4.2 One Dimensional Harmonic Oscillator . . . . . . . . . . . . . . . . . 297
13.4.3 N Free Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
13.5 Path integral for the Density Matrix . . . . . . . . . . . . . . . . . . . . . . 303
13.6 Simple Calculations with the Path Integral . . . . . . . . . . . . . . . . . . 307
13.6.1 Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
13.6.2 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 308
14 Order and Disorder 310
14.1 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
14.1.1 Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
14.1.2 The Lattice Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
14.1.3 Binary Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
14.2 The Weiss Mean Field Approximation . . . . . . . . . . . . . . . . . . . . . 315
14.3 Exact Solution: Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
14.3.1 The Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . 322
14.3.2 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 325
14.3.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
14.4 Mean Field Theory: Landau’s Approach . . . . . . . . . . . . . . . . . . . . 329
14.5 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
15 Thermal Field Theory 335
Chapter 1
Preliminaries
The word “Thermodynamics” is derived from the greek words therme, meaning “heat”,
and dynamikos, meaning force or power. It is the area of physics that deals with the
relationship of “heat” to other forms of energy and to mechanical work, and examines how
these quantities are related to the measurable properties of a thermodynamic system.
Athermodynamicsystemisamacroscopicportionoftheuniverse, bywhichwemeanthat
it consists of a very large number of more elementary constituents, each of which is able to
carry mechanical energy. In contrast, the rest of the universe is called the environment
and the separation between the “system” and the “environment” is generally assumed
to occur via a boundary separating the two. Thermodynamics assumes from the start
that the elementary constituents of the system and its environment are so many that
fluctuations about their average behavior can be completely ignored.
Every thermodynamic system is able to store energy by virtue of the fact that its
elementary constituents possess mechanical energy. A fundamental objective of thermo-
dynamics is to describe the interaction of such a system with the environment or the
interaction between different systems among themselves. Such interactions inevitably
lead to an exchange of energy between them and there are broadly two ways in which this
can occur: (i) either energy enters or leaves the system because an external mechanical
constraint is changed (this occurs, for example, when a gas expands or is compressed and
its volume changes) or (ii) energy spontaneously moves across the boundary by transfer
on the molecular level (for instance, when a hot body is brought in contact with a cold
body). We say that the first kind of energy transfer occurs by useful work, the second
occurs by Heat.
The possibility of transforming heat into useful work and vice-versa had been recog-
nized for a very long time, in fact even before the nature of heat as a transfer of energy
was understood. Heat was originally supposed to be a kind of fluid, called “caloric”,
flowing from one body to another. This understanding was replaced in the mid and late
1
2 CHAPTER 1. PRELIMINARIES
nineteenth century by the more modern mechanical theory of heat, thanks to the work of
Mayer, Joule, Maxwell, Boltzmann and others. Since every macroscopic system is made
up of elementary constituents, exchanges of energy must eventually be mechanical pro-
cesses occurring on a microscopic scale. Understood in this way thermodynmics becomes
a special branch of mechanics: the mechanics, classical or quantum as the need may be,
of very many elementary constituents interacting with each other.
Anyone who has tried to solve the equations of motion for more than two interacting
particles knows already that this is an extremely difficult, if not impossible task. The
general N body problem contains 6N variables. Subtracting the ten first integrals of the
motion (one for energy, three for the motion of the center of mass, three for the total
momentum, three for the total angular momentum) leaves us with 6N − 10 variables,
subject to the same number of initial conditions, to contend with. If the system has
on the order of Avogadro’s number (6.023 × 1023) of elementary constituents then the
problem is computationally unfeasable, even assuming that the initial conditions could
all be obtained experimentally, which itself is possibly still less feasable a task. Yet, one
should be sensitive to the fact that even if such a solution could be obtained, it would
give us information that is far in excess of what we can reasonably measure or control.
In fact, we are able in practice to impose only very coarse constraints on a macroscopic
system and to measure only a very small number of its properties (an example would be
the pressure, volume and temperature of a gas) compared to the number of mechanical
variables involved in a microscopic treatment. The treatment via mechanics, even if it
could be accomplished, would therefore be quite useless from a practical standpoint. A
way out, originally proposed by Boltzmann, is to treat the mechanical system statistically
i.e., asking not detailed questions concerning the motion of the individual constituents
but rather asking questions about the average behavior of the system. This microscopic
approach, combining mechanics and statistics, leads to “Statistical Mechanics”, a part of
which we will examine in the latter half of this text.
Because we measure and control a very small number of properties of the system,
some of which are related only statistically to its underlying microscopic properties, it is
worth asking how far we can go if we simply ignore the underlying mechanics. This is
the approach of Thermodynamics, which concentrates only on relationships between the
measurablepropertiesofthesystem. Ofcourseitrequiresustodefineclearlytheproperties
(variables) we use in our description of the macroscopic system by describing instruments
to precisely measure them. Once this is accomplished, the sole task of thermodynamics
is to codify experiment by postulating a set of principles relating these “thermodynamic
varables”. On the surface at least this sort of description would seem to have little to do
with the theory of dynamical systems. It is an interesting fact then that Thermodynamics
admits a rather elegant formulation as a dynamical system in its own right. We will show
this in a forthcoming chapter.
Both approaches have their advantages and disadvantages. In the microscopic ap-
1.1. EQUILIBRIUM STATES AND STATE VARIABLES 3
proach (Statistical Mechanics) the system must be modeled mechanically in a fairly de-
tailedfashionandourunderstandingofitisproportionallymoreprofound, butthemathe-
maticaldifficultiesinvolvedintheactualtransitionfromtheoreticalmodeltoexperimental
predictions and vice versa are also greater. In the macroscopic approach (Thermodynam-
ics, as conceived by Mayer, Carnot, Clausius, Kelvin and many others) our understanding
of the actual processes between the elementary constituents is sacrificed for mathematical
simplicity. It is remarkable indeed that many deep results can yet be obtained without any
referencewhatsoevertotheunderlyingmechanics. Thismustbeunderstoodasevidenceof
the power of statistical averaging: when very large numbers are involved statistical fluctu-
ations become vanishingly small and Thermodynamics becomes an excellent effective but
nevertheless complete and internally consistent theory, within its realm of applicability.
1.1 Equilibrium States and State Variables
A thermodynamic state is the macroscopic condition of a thermodynamic system as de-
scribed by the values of a small number of variables, such as the temperature, pressure,
density, volume, composition and others that will be defined in the following chapters.
These are called the state variables and together the state variables span the thermo-
dynamic phase space of the system. They are always few in number when compared to
the number of mechanical variables that would be required, and generally may be strictly
defined only when the system is in “equilibrium” (equilibrium refers to a situation in
which there is no change of the state variables in time.) They define a space of possible
equilibrium states of the system. An essential task of thermodynamics is to discover a
reasonably “complete” set of state variables for the given system.
Thermodynamic variables are often divided into two categories: the intensive vari-
ables, which are independent of the system size or the quantity of matter contained in the
system and the extensive variables whose values, by contrast, do depend on the system
size or the amount of matter in the system. For example, temperature is an intensive
property of a thermodynamic system. The same applies to the density of a homogeneous
system since if the size (volume) of such a system is scaled by some factor, then its mass
is scaled by the same factor and therefore the density stays constant. On the other hand,
the total energy of a system of weakly interacting particles, or of particles with very short
range interactions, is extensive.
Now volume is a geometric quantity for which we have a well developed intuition and
the mass density, ρ((cid:126)r), defined according to
(cid:90)
M = d3(cid:126)rρ((cid:126)r), (1.1.1)
V
is likewise well understood. Let us precisely define two other basic of state variables, viz.,
the pressure, and the temperature. Others will be defined as the subject unfolds.
4 CHAPTER 1. PRELIMINARIES
Figure 1.1: A gas is confined in a chamber with a piston at one end
1.1.1 Pressure
Imagine that the gas is confined in a container with a piston at one end. The piston gives
us some measure of control over the volume of the gas (see figure 1.1). The gas exerts
a force on the piston, which can be measured by placing weights on the piston until the
upward force of the gas on the piston balances its weight. In this state of mechanical
equilibrium we may crudely define the pressure of the gas as the force, F, exerted by it
per unit area, A, of the piston
F
p = (1.1.2)
A
but this expression is really only the definition of the average pressure. Note that the
direction of the force is perpendicular (normal) to the surface bounding the gas, equiva-
lently that of the piston. A more general definition of pressure may be given by expressing
the above relationship in terms of infinitesimal quantities. Consider any macroscopic,
continuous system and a surface within the system or bounding it. Let dS(cid:126) be an infinites-
imal, directed element of this surface (the direction is taken to be the outward normal,
the word “outward” signifying that it points into the environment). Let the force on this
infinitesimal element of surface and normal to it be dF(cid:126) , then define the pressure as the
n
proportionality in the relationship
dF(cid:126) = pdS(cid:126) (1.1.3)
n
between F(cid:126) and dS(cid:126). Thus the force on a finite area of any surface would be
n
(cid:90)
F(cid:126) = pdS(cid:126) (1.1.4)
n
S
where the integral is taken over the portion of the surface under consideration. If the
pressure is constant we obviously recover the simple definition in (1.1.3). We say that