Prinzipien der Thermodynamik und Statistik / Principles of Thermodynamics and Statistics

684 Pages·1959·39.017 MB·German-English

by  E. A. Guggenheim (auth.), S. Flügge (eds.)

ENCYCLOPEDIA OF PHYSICS
EDITED BY
S. FLOCCE
VOLUME 111/2
PRINCIPLES OF THERMODYNAMICS
AND STATISTICS
WITH 25 FIGURES
SPRINGER-VERLAG
BERLIN· GOTTINGEN . HEIDELBERG
1959
HANDBUCH DER PHYSIK
HERAUSGEGEBEN VON
S. FLOGGE
BAND III/2
PRINZIPIEN DER THERMODYNAMIK
UND STATISTIK
MIT 25 FIGUREN
SPRINGER-VERLAG
BERLIN · GOTTINGEN . HEIDELBERG
1959
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So/kover reprint ofthe hardcover 1s t edition 1959
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Inhaltsverzeichnis.
Seite
Thermodynamics, Classical and Statistical. By EDWARD ARMAND GUGGENHEIM, Pro
fessor of Chemistry, University of Reading (Great Britain). (With 4 Figures)
A. Fundamentals of classical thermodynamics
B. Applications of classical thermodynamics 23
C. Fundamentals of statistical mechanics 47
D. Statistical mechanics of molecules 59
E. Statistical thermodynamics. . . 70
F. Degenerate ideal gases and radiation 93
G. External fields 109
H. Historical notes 11 3
Axiomatik der Thermodynamik. Von Dr. GOTTFRIED FALK, Professor der Physik,
Technische Hochschule Karlsruhe (Deutschland) und Dr. HERBERT ]UNG, Technische
Hochschule Aachen (Deutschland). (Mit 5 Figuren) . 119
Einleitung 119
A. Grundbegriffe und formale Beschreibungsmittel 121
B. Die Struktur der Thermodynamik . . . . . 134
I. Entropie und Energie 134
II. Die thermodynamischen Koordinaten 145
Anhang: Zu CARATHEODORYS "Untersuchungen tiber die Grundlagen der Thermo-
Clynamik" .............................. . 161
Prinzipien der statistischen Mechanik. Von Dr. ARNOLD MUNSTER, Professor flir
Physikalische Chemie, U niversitiit Frankfurt/Main (Deutschland). (Mit 12 Figuren) 176
A. Klassische statistische Mechanik. . . . . . . . . . . 178
I. Allgemeine Siitze tiber statistische Gesamtheiten . 178
II. Axiomatische Grundlagen der statistischen Mechanik . 187
III. Die Einstellung des Gleichgewichtes 212
IV. Die mikrokanonische Gesamtheit 239
V. Die kanonische Gesamtheit 262
B. Quantenstatistik 267
1. Allgemeine Siitze tiber quantenstatistische Gesamtheiten 267
II. Axiomatische Grundlagen der Quantenstatistik . . . 278
III. Die Einstellung des Gleichgewichtes . . . . . . . . 288
IV. Die mikrokanonische Gesamtheit der Quantenstatistik 302
V. Die kanonische Gesamtheit der Quantenstatistik. . . 318
VI. Die Berechnung der kanonischen Verteilungsfunktion . 327
VII. Die groJ3e kanonische Gesamtheit 343
VI Inhaltsverzeichnis.
Seite
C. Allgemeine Begriindung der Thermodynamik. Schwankungen. Phasenumwand-
lungen . . . . . . . . . . . . . . . . . . . . . . . . . 350
I. Begriindung der Thermodynamik. Theorie der Schwankungen 350
II. Phasenumwandlungen. Stabilitatsbedingungen 371
Verzeichnis der Formelsymbole 410
Literatur . 412
Thermodynamik der irreversiblen Prozesse. Von Professor Dr. JOSEF MEIXNER,
Direktor des Instituts flir Theoretische Physik der Technischen Hochschulc Aachen
und Dozent Dr. HELMUT GOTTLIEB REIK, Technische Hochschule Aachen (Deutsch-
land). (Mit 4 Figuren) 413
A. Einleitung . . . . . . . . . . . . 413
B. Die Thermodynamik der irreversiblen Prozesse in kontinuierlichen Medien mit
inneren Umwandlungen . . . . . . . . . . . . . . . . . . . . . 417
C. Anwendung der Thermodynamik irreversibler Prozesse in fluiden Medien 442
D. Eigenschaften von Medien mit inneren Variablen . . . . 470
E. Relativistische Thermodynamik der irreversiblen Prozesse 494
F. Zur statistischen Theorie der irreversiblen Prozesse 505
Probability and Stochastic Processes. By ALLADI RAMAKRISHNAN, Professor of
Physics, University of Madras (India) . 524
Prefatory note 524
Introduction 524
A. Probability . 525
I. Probability and measure theory 525
II. Probability frequency functions 530
III. Functions associated with the frequency functions 538
a) Discrete random variables 539
b) Continuous random variables . . . . 540
IV. Standard probability frequency functions 542
a) Discrete distributions . . . . . . . 543
b) Continuous distributions . . . . . . 546
V. Product densities, Janossy densities and the characteristic functional. 549
VI. Combinatorial analysis 553
B. Stochastic processes . . . . 558
I. A physical approach to stochastic processes 558
II. Stochastic process: Measure theoretical approach 572
III. Calculus of random functions . . . . . 573
IV. Physical examples of stochastic processes 579
Class A: x discrete, t discrete . . 579
Class B: x discrete, t continuous .. 580
Class C: x continuous, t discrete .. 587
Class D: x continuous, t continuous . " 591
V. Cascade processes involving continuous parameters. 598
VI. Stochastic problems in astrophysics . . . . . . 603
VII. Statistical theory of the structure of simple fluids 611
VIII. Differential equations involving random functions of time. 615
IX. Stationary processes . . . . . . . . . . . . . . . . . 622
Inhaltsverzeichnis. VII
Seite
X. On the computation of infinitesimal transition probabilities 632
XI. Applications to quantum mechanics . . . . . . . 634
XII. Random functions of a many-dimensional parameter 640
XIII. Equations involving random parameters . . . . 642
XIV. Inverse probability. . . . . . . . . . . . . . 643
XV. Stochastic processes involving "back-scattering" 646
XVI. Some concluding remarks . 648
Acknowledgment. 649
References 649
Sachverzeichnis (Deutsch/Englisch) 652
Subject Index (English/German) .. 665
Thermodynamics, Classical and Statistical.
By
E. A. GUGGENHEIM.
"\Vith 4 Figures.
A. Fundamentals of classical thermodynamics.
1. Scope of thermodynamics. The most important conception in thermo
dynamics is temperature. The essential properties of temperature will be described
below. Anticipating this we may define thermodynamics as that part of physics
concerned with the dependence on temperature of any equilibrium property.
This definition may be illustrated by a simple example. Consider the distribution
of two immiscible liquids such as mercury and water in a gravitational field.
The equilibrium distribution is that in which the heavier liquid, mercury, occupies
the part of accessible space where the gravitational potential is lowest and the
lighter liquid, water, occupies the part of the remaining accessible space where
the gravitational potential is lowest. This equilibrium distribution is, apart
from the effect of thermal expansion which we neglect, independent of temper
ature. Consequently the problem does not involve thermodynamics, but only
hydrostatics. Now consider by contrast the distribution in a gravitational field
of two completely miscible fluids such as bromine and carbon disulphide. The
relative proportions of the two substances will vary from place to place, the
proportion of the heavier liquid, bromine, being greatest at the lowest gravitational
potential and conversely. The precise relation between the composition and the
gravitational potential depends on the temperature, assumed uniform, of the
mixture. Clearly this is a problem in thermodynamics, not merely hydrostatics.
We shall now mention a few other typical examples to show that thermo
dynamics has a bearing on most branches of physics, including elasticity, hydro
dynamic~, elEctrostatics and electrodynamics. In the relation, known as Hooke's
law, of proportionality between tension and extension the coefficient of pro
portionality will in general be temperature dependent. In so far as its variation
with temperature is relevant thermodynamics is involved. To study the temper
ature dependence of the compressibility of a fluid, that of the dielectric coefficient
of a dielectric, that of the permeability of a paramagnetic material, that of the
electromotive force of a cell and in fact the temperature dependence of any
equilibrium property thermodynamics is needed.
2. Energy and heat. First law. Leaving temperature for the moment, we
must now say something about energy. The conception of energy arose first
in mechanics and was extended to electrostatics and electrodynamics. \\Then
these branches of physics are idealized so as to exclude friction, ~viscosity, hyste
resis, temperature gradients, temperature dependence of properties and related
phenomena the fundamental property of energy can be described in two alter
native ways.
1. When several systems interact in any way with one another, the whole
set of systems being isolated from the rest of the universe, the sum of the energies
of the several systems remains constant.
Handbuch der Physik, Bel. Il 1/2.
2 E. A. GUGGENHEIM: Thermodynamics, Classical and Statistical. Sect. 2.
II. When a single system interacts with the rest of the universe (its surround
ings) the increase of the energy of this system is equal to the work done on the
system by the rest of the universe (its surroundings).
Under the idealized conditions mentioned above these two descriptions are
equivalent, but when these restrictions are removed the two descriptions are no
longer equivalent and we have to make a choice between them. Of the alternatives
we choose I and with this choice the extended conception of energy is termed
total energy and is denoted by U. The formulation I is then a statement of
the conservation of total energy. It is based indirectly on the classical experi
ments of JOULE.
Let us now consider in greater detail the interaction between a pair of systems,
supposed isolated from the rest of the universe. Using subscripts A, B to relate
to the two systems we have
dD:4+ dUB=0 (2.1)
or
dUA=-dUB (2.2)
but in general this is not equal to the work WBA done by B on A. In other words
there can be exchange of energy between A and B of a kind other than work.
Such an exchange of energy is that determined by a temperature difference and
is called heat. If then we denote the heat flow from B to A by qBA, we have the
following relations
+
dUA = WBA qBA , (2·3)
dUB=WAB+qAB, (2.4)
(2.5)
(2.6)
This set of relations together constitutes the first law of thermodynamics.
The sign of q is determined by the temperature difference between A and B,
and the universal convention is to define the sign of a temperature difference in
such a way that heat always flows from the higher to the lower temperature.
The above analysis of the most general interactiOl:!. between two systems can
immediately be extended to the most general interaction between a given system
and the rest of the universe. If we denote by U the total energy of the system,
by q the heat flow from the surroundings to the system and by W the work done
on the system we have
dU=q+w. (2.7)
When two systems are separated by a boundary or wall which allows heat
to flow freely from the one to the other, the two systems are said to be in thermal
contact. When on the contrary the boundary or wall is of such a nature as to
prevent any flow of heat the wall is said to be insulating. When a system is
completely surrounded by an insulating boundary, the system is said to be therm
ally insulated and any process taking place in the system is called adiabatic.
We thus have
q = 0, dU = w, adiabatic process. (2.8)
We have already mentioned that if two systems have different temperatures,
then over and above any work done by the one system on the other there can be
a further exchange of energy, called a flow of heat, but only in one direction
Sects. 3-6. Natural and reversible processes. 3
namely from the higher to the lower temperature. It is a corollary of this that
if the two systems have the same temperature, there will be no flow of heat be
tween them. They are then said to be in thermal equilibrium.
3. Thermostats and thermometers. Consider now two systems in thermal
contact, one very much smaller than the other, for example a short thin metallic
wire immersed in a large quantity of water. If the quantity of water is large
enough (or the wire small enough), then in the process of attaining thermal equi
librium the change in the physical state of the water will be entirely negligible
compared with that of the wire. This situation is described differently according
as we are primarily interested in the small system or in the large one.
If we are primarily interested in the small system, the wire, then we regard
the water as a means of controlling the temperature of the wire and we refer
to the water as a temperature bath or thermostat. .
If on the other hartd we are primarily interested in the large system, th~. water,
we regard the wire as an instrument for recording the temperature of the ,vater
and we refer to the wire as a thermometer. This recording of temperature can be
rendered quantitative by measuring some property of the thermometer, such as
. its electrical resistance, which varies with temperature.
4. Absolute temperature. The choice of thermometers is very wide especially
as there is a choice both of the substance constituting the thermometer and of
the property measured. Consequently there is a wide, effectively infinite, choice
of temperature scales. There is however one particular scale which has outst;md
ingly simple characteristics which can be described in a manner independent
of the properties of any particular substance or class of substances. This temper
ature is called absolute temperature. It was first defined by KELVIN and is denoted
by T. It is the only scale that we shall use. It will be defined by its properties,
especially its relation to entropy. The question how T can best be measured
must necessarily be postponed to the next chapter.
5. Phase. Extensive and intensive properties. At this stage it is convenient
to define phase, extensive property and intensive property. A phase is a system
or part of a system which is completely homogeneous. An extensive property
is any property, such as mass, whose value for the whole system is equal to
the sum of its values for the separate pha~es. Its value for a phase of given
nature is proportional to the amount of the phase. Important examples of
extensive properties are volume and total energy. An intensive property is
any property, such as density, whose value is constant throughout a phase. Its
value in a phase of given nature is independent of the amount of the phase.
Important examples of intensive properties are pressure and temperature.
6. Natural and reversible processes. We must now consider a classification
of processes due to PLANCK. All the independent infinitesimal processes that
might conceivably take place may be divided into three types: natural processes,
unnatural processes and reversible processes.
Natural processes are all. such as actually do occur in nature; they proceed
in a direction towards equilibrium.
An unnatural process is one in a direction away from equilibrium; such a
process never occurs in nature.
As a limiting case between natural and unnatural processes we have reversible
processes, which consist of the passage in either direction through a continuous
series of equilibrium states. Reversible processes do not actually occur in nature,
but in whichever direction we contemplate a reversible process we can by a small
1*
4 E. A. GUGGENHEIM: Thermodynamics, Classical and Statistical. Sect. 7.
change in the conditions produce a natural process differing as little as we choose
from the reversible process contemplated.
We shall illustrate the three types by examples. Consider a system consisting
of a liquid together with its vapour at a pressure P. Let the equilibrium vapour
pressure of the liquid be p. Consider now the process of the evaporation of a
small quantity of the liquid. If P< p, this is a natural process and will in fact
take place. If on the other hand p>p, the process contemplated is unnatural
and cannot take place; in fact the contrary process of condensation will take
place. If P = P then the process contemplated and its converse are reversible,
for by slightly decreasing or increasing P we can make either occur. The last
case may be described in an alternative manner as follows. If P = P- ~, where
~> 0, then the process of evaporation is a natural one. Now suppose ~ gradually
decreased. In the limit ~ -';> 0, the process becomes reversible.
We have defined a reversible process as a hypothetical passage through equi
librium states. If we have a system interacting with its surroundings either
through the performance of work or through the flow of heat, we shall use the
term reversible process only if there is throughout the process equilibrium be
tween the system and its surroundings. If we wish to refer to the hypothetical
passage of the system through a sequence of internal equilibrium states, without
necessarily being in equilibrium with its surroundings we shall refer to a reversible
change. We shall illustrate this distinction by examples.
Consider a system consisting of a liquid and its vapour in mutual equilibrium
in a cylinder closed by a piston opposed by a pressure equal to the equilibrium
vapour pressure corresponding to the temperature of the system.) Suppose now
that there is a flow of heat through the walls of the cylinder, with a consequent
evaporation of liquid and work done on the piston at constant temperature
and pressure. The change in the system is a reversible change, but the whole
process is a reversible process only if the medium surrounding the cylinder is
at the same temperature as the liquid and vapour; otherwise the flow of heat
through the walls of the cylinder is not reversible and so the process as a whole
is not reversible, though the change in the system within the cylinder is reversible.
As a second example consider a flow of heat from one system in complete
internal equilibrium to another system in complete internal equilibrium. Pro
vided both systems remain in internal equilibrium then the change which each
system undergoes is a reversible change, but the whole process of heat flow is
not a reversible process unless the two systems are at the same temperature.
If a system is in complete equilibrium, any conceivable infinitesimal change
in it must be reversible. For a natural process is an approach towards equili
brium, and as the system is already in equilibrium the change cannot be a natural
one. Nor can it be an unnatural one, for in that case the opposite infinitesimal
change would be a natural one, and this would contradict the supposition that
the system is already in equilibrium. The only remaining possibility is that,
if the system is in complete equilibrium, any conceivable infinitesimal change
must be reversible.
7. Entropy. Second law. vVe are now ready to introduce the quantity,
invented by CLAUSIUS, called entropy, the enumeration of whose properties
constitutes the second law of thermodynamics. The entropy 5 is an extensive
property of a system which can change in two distinct ways namely by inter
action with the surroundings and by changes taking place inside the system.
Symbolically we may write this as
(7.1)