jaynes: org/stat-mech.org annotate

annotate org/stat-mech.org @ 5:e7185b523c80 tip

Added Began Gibbs formalism.

author	Dylan Holmes <ocsenave@gmail.com>
date	Mon, 30 Apr 2012 19:10:15 -0500
parents	299a098a30da
children

rev	line source
ocsenave@0	1 #+TITLE: Statistical Mechanics
ocsenave@0	2 #+AUTHOR: E.T. Jaynes; edited by Dylan Holmes
ocsenave@0	3 #+EMAIL: rlm@mit.edu
ocsenave@0	4 #+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
ocsenave@0	5 #+SETUPFILE: ../../aurellem/org/setup.org
ocsenave@0	6 #+INCLUDE: ../../aurellem/org/level-0.org
ocsenave@0	7 #+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
ocsenave@0	8
ocsenave@0	9 # "extensions/eqn-number.js"
ocsenave@0	10
ocsenave@0	11 #+begin_quote
ocsenave@0	12 Note: The following is a typeset version of
ocsenave@0	13 [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]]. I have only made
ocsenave@0	14 minor changes, e.g. to correct typographical errors, add references, or format equations. The
ocsenave@0	15 content itself is intact. --- Dylan
ocsenave@0	16 #+end_quote
ocsenave@0	17
ocsenave@0	18 * Development of Thermodynamics
ocsenave@0	19 Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
ocsenave@0	20 arise from the sensations of warmth and cold associated with our
ocsenave@0	21 sense of touch . Yet science has been able to convert this qualitative
ocsenave@0	22 sensation into an accurately defined quantitative notion,
ocsenave@0	23 which can be applied far beyond the range of our direct experience.
ocsenave@0	24 Today an experimentalist will report confidently that his
ocsenave@0	25 spin system was at a temperature of 2.51 degrees Kelvin; and a
ocsenave@0	26 theoretician will report with almost as much confidence that the
ocsenave@0	27 temperature at the center of the sun is about $2 \times 10^7$ degrees
ocsenave@0	28 Kelvin.
ocsenave@0	29
ocsenave@0	30 The /fact/ that this has proved possible, and the main technical
ocsenave@0	31 ideas involved, are assumed already known to the reader;
ocsenave@0	32 and we are not concerned here with repeating standard material
ocsenave@5	33 already available in a dozen other textbooks. However
ocsenave@0	34 thermodynamics, in spite of its great successes, firmly established
ocsenave@0	35 for over a century, has also produced a great deal of confusion
ocsenave@0	36 and a long list of \ldquo{}paradoxes\rdquo{} centering mostly
ocsenave@0	37 around the second law and the nature of irreversibility.
ocsenave@0	38 For this reason and others noted below, we want to dwell here at
ocsenave@0	39 some length on the /logic/ underlying the development of
ocsenave@0	40 thermodynamics . Our aim is to emphasize certain points which,
ocsenave@0	41 in the writer's opinion, are essential for clearing up the
ocsenave@0	42 confusion and resolving the paradoxes; but which are not
ocsenave@0	43 sufficiently ernphasized---and indeed in many cases are
ocsenave@0	44 totally ignored---in other textbooks.
ocsenave@0	45
ocsenave@0	46 This attention to logic
ocsenave@0	47 would not be particularly needed if we regarded classical
ocsenave@0	48 thermodynamics (or, as it is becoming called increasingly,
ocsenave@0	49 /thermostatics/) as a closed subject, in which the fundamentals
ocsenave@0	50 are already completely established, and there is
ocsenave@0	51 nothing more to be learned about them. A person who believes
ocsenave@0	52 this will probably prefer a pure axiomatic approach, in which
ocsenave@0	53 the basic laws are simply stated as arbitrary axioms, without
ocsenave@0	54 any attempt to present the evidence for them; and one proceeds
ocsenave@0	55 directly to working out their consequences.
ocsenave@0	56 However, we take the attitude here that thermostatics, for
ocsenave@0	57 all its venerable age, is very far from being a closed subject,
ocsenave@0	58 we still have a great deal to learn about such matters as the
ocsenave@0	59 most general definitions of equilibrium and reversibility, the
ocsenave@0	60 exact range of validity of various statements of the second and
ocsenave@0	61 third laws, the necessary and sufficient conditions for
ocsenave@0	62 applicability of thermodynamics to special cases such as
ocsenave@0	63 spin systems, and how thermodynamics can be applied to such
ocsenave@0	64 systems as putty or polyethylene, which deform under force,
ocsenave@0	65 but retain a \ldquo{}memory\rdquo{} of their past deformations.
ocsenave@0	66 Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
ocsenave@0	67 no means rule out the possibility that still more laws of
ocsenave@0	68 thermodynamics exist, as yet undiscovered, which would be
ocsenave@0	69 useful in such applications.
ocsenave@0	70
ocsenave@0	71
ocsenave@0	72 It is only by careful examination of the logic by which
ocsenave@0	73 present thermodynamics was created, asking exactly how much of
ocsenave@0	74 it is mathematical theorems, how much is deducible from the laws
ocsenave@0	75 of mechanics and electrodynamics, and how much rests only on
ocsenave@0	76 empirical evidence, how compelling is present evidence for the
ocsenave@0	77 accuracy and range of validity of its laws; in other words,
ocsenave@0	78 exactly where are the boundaries of present knowledge, that we
ocsenave@0	79 can hope to uncover new things. Clearly, much research is still
ocsenave@0	80 needed in this field, and we shall be able to accomplish only a
ocsenave@0	81 small part of this program in the present review.
ocsenave@0	82
ocsenave@0	83
ocsenave@0	84 It will develop that there is an astonishingly close analogy
ocsenave@0	85 with the logic underlying statistical theory in general, where
ocsenave@0	86 again a qualitative feeling that we all have (for the degrees of
ocsenave@0	87 plausibility of various unproved and undisproved assertions) must
ocsenave@0	88 be convertefi into a precisely defined quantitative concept
ocsenave@0	89 (probability). Our later development of probability theory in
ocsenave@0	90 Chapter 6,7 will be, to a considerable degree, a paraphrase
ocsenave@0	91 of our present review of the logic underlying classical
ocsenave@0	92 thermodynamics.
ocsenave@0	93
ocsenave@3	94 ** The Primitive Thermometer
ocsenave@0	95
ocsenave@0	96 The earliest stages of our
ocsenave@0	97 story are necessarily speculative, since they took place long
ocsenave@0	98 before the beginnings of recorded history. But we can hardly
ocsenave@0	99 doubt that primitive man learned quickly that objects exposed
ocsenave@0	100 to the sun‘s rays or placed near a fire felt different from
ocsenave@0	101 those in the shade away from fires; and the same difference was
ocsenave@0	102 noted between animal bodies and inanimate objects.
ocsenave@0	103
ocsenave@0	104
ocsenave@0	105 As soon as it was noted that changes in this feeling of
ocsenave@0	106 warmth were correlated with other observable changes in the
ocsenave@0	107 behavior of objects, such as the boiling and freezing of water,
ocsenave@0	108 cooking of meat, melting of fat and wax, etc., the notion of
ocsenave@0	109 warmth took its first step away from the purely subjective
ocsenave@0	110 toward an objective, physical notion capable of being studied
ocsenave@0	111 scientifically.
ocsenave@0	112
ocsenave@0	113 One of the most striking manifestations of warmth (but far
ocsenave@0	114 from the earliest discovered) is the almost universal expansion
ocsenave@0	115 of gases, liquids, and solids when heated . This property has
ocsenave@0	116 proved to be a convenient one with which to reduce the notion
ocsenave@0	117 of warmth to something entirely objective. The invention of the
ocsenave@0	118 /thermometer/, in which expansion of a mercury column, or a gas,
ocsenave@0	119 or the bending of a bimetallic strip, etc. is read off on a
ocsenave@0	120 suitable scale, thereby giving us a /number/ with which to work,
ocsenave@0	121 was a necessary prelude to even the crudest study of the physical
ocsenave@0	122 nature of heat. To the best of our knowledge, although the
ocsenave@0	123 necessary technology to do this had been available for at least
ocsenave@0	124 3,000 years, the first person to carry it out in practice was
ocsenave@0	125 Galileo, in 1592.
ocsenave@0	126
ocsenave@0	127 Later on we will give more precise definitions of the term
ocsenave@0	128 \ldquo{}thermometer.\rdquo{} But at the present stage we
ocsenave@0	129 are not in a position to do so (as Galileo was not), because
ocsenave@0	130 the very concepts needed have not yet been developed;
ocsenave@0	131 more precise definitions can be
ocsenave@0	132 given only after our study has revealed the need for them. In
ocsenave@0	133 deed, our final definition can be given only after the full
ocsenave@0	134 mathematical formalism of statistical mechanics is at hand.
ocsenave@0	135
ocsenave@0	136 Once a thermometer has been constructed, and the scale
ocsenave@0	137 marked off in a quite arbitrary way (although we will suppose
ocsenave@0	138 that the scale is at least monotonic: i.e., greater warmth always
ocsenave@0	139 corresponds to a greater number), we are ready to begin scien
ocsenave@0	140 tific experiments in thermodynamics. The number read eff from
ocsenave@0	141 any such instrument is called the /empirical temperature/, and we
ocsenave@0	142 denote it by $t$. Since the exact calibration of the thermometer
ocsenave@0	143 is not specified), any monotonic increasing function
ocsenave@0	144 $t‘ = f(t)$ provides an equally good temperature scale for the
ocsenave@0	145 present.
ocsenave@0	146
ocsenave@0	147
ocsenave@3	148 ** Thermodynamic Systems
ocsenave@0	149
ocsenave@0	150 The \ldquo{}thermodynamic systems\rdquo{} which
ocsenave@0	151 are the objects of our study may be, physically, almost any
ocsenave@0	152 collections of objects. The traditional simplest system with
ocsenave@0	153 which to begin a study of thermodynamics is a volume of gas.
ocsenave@0	154 We shall, however, be concerned from the start also with such
ocsenave@0	155 things as a stretched wire or membrane, an electric cell, a
ocsenave@0	156 polarized dielectric, a paramagnetic body in a magnetic field, etc.
ocsenave@0	157
ocsenave@0	158 The /thermodynamic state/ of such a system is determined by
ocsenave@0	159 specifying (i.e., measuring) certain macrcoscopic physical
ocsenave@0	160 properties. Now, any real physical system has many millions of such
ocsenave@0	161 preperties; in order to have a usable theory we cannot require
ocsenave@0	162 that /all/ of them be specified. We see, therefore, that there
ocsenave@0	163 must be a clear distinction between the notions of
ocsenave@0	164 \ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
ocsenave@0	165 system.\rdquo{}
ocsenave@0	166 A given /physical/ system may correspond to many different
ocsenave@0	167 /thermodynamic systems/, depending
ocsenave@0	168 on which variables we choose to measure or control; and which
ocsenave@0	169 we decide to leave unmeasured and/or uncontrolled.
ocsenave@0	170
ocsenave@0	171
ocsenave@0	172 For example, our physical system might consist of a crystal
ocsenave@0	173 of sodium chloride. For one set of experiments we work with
ocsenave@0	174 temperature, volume, and pressure; and ignore its electrical
ocsenave@0	175 properties. For another set of experiments we work with
ocsenave@0	176 temperature, electric field, and electric polarization; and
ocsenave@0	177 ignore the varying stress and strain. The /physical/ system,
ocsenave@0	178 therefore, corresponds to two entirely different /thermodynamic/
ocsenave@0	179 systems. Exactly how much freedom, then, do we have in choosing
ocsenave@0	180 the variables which shall define the thermodynamic state of our
ocsenave@0	181 system? How many must we choose? What [criteria] determine when
ocsenave@0	182 we have made an adequate choice? These questions cannot be
ocsenave@0	183 answered until we say a little more about what we are trying to
ocsenave@0	184 accomplish by a thermodynamic theory. A mere collection of
ocsenave@0	185 recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and
ocsenave@0	186 Chemistry/]], is a very useful thing, but it hardly constitutes
ocsenave@0	187 a theory. In order to construct anything deserving of such a
ocsenave@0	188 name, the primary requirement is that we can recognize some kind
ocsenave@0	189 of reproducible connection between the different properties con
ocsenave@0	190 sidered, so that information about some of them will enable us
ocsenave@0	191 to predict others. And of course, in order that our theory can
ocsenave@0	192 be called thermodynamics (and not some other area of physics),
ocsenave@0	193 it is necessary that the temperature be one of the quantities
ocsenave@0	194 involved in a nontrivial way.
ocsenave@0	195
ocsenave@0	196 The gist of these remarks is that the notion of
ocsenave@0	197 \ldquo{}thermodynamic system\rdquo{} is in part
ocsenave@0	198 an anthropomorphic one; it is for us to
ocsenave@0	199 say which set of variables shall be used. If two different
ocsenave@0	200 choices both lead to useful reproducible connections, it is quite
ocsenave@0	201 meaningless to say that one choice is any more \ldquo{}correct\rdquo{}
ocsenave@0	202 than the other. Recognition of this fact will prove crucial later in
ocsenave@0	203 avoiding certain ancient paradoxes.
ocsenave@0	204
ocsenave@0	205 At this stage we can determine only empirically which other
ocsenave@0	206 physical properties need to be introduced before reproducible
ocsenave@0	207 connections appear. Once any such connection is established, we
ocsenave@0	208 can analyze it with the hope of being able to (1) reduce it to a
ocsenave@0	209 /logical/ connection rather than an empirical one; and (2) extend
ocsenave@0	210 it to an hypothesis applying beyond the original data, which
ocsenave@0	211 enables us to predict further connections capable of being
ocsenave@0	212 tested by experiment. Examples of this will be given presently.
ocsenave@0	213
ocsenave@0	214
ocsenave@0	215 There will remain, however, a few reproducible relations
ocsenave@0	216 which to the best of present knowledge, are not reducible to
ocsenave@0	217 logical relations within the context of classical thermodynamics
ocsenave@0	218 (and. whose demonstration in the wider context of mechanics,
ocsenave@0	219 electrodynamics, and quantum theory remains one of probability
ocsenave@0	220 rather than logical proof); from the standpoint of thermodynamics
ocsenave@0	221 these remain simply statements of empirical fact which must be
ocsenave@0	222 accepted as such without any deeper basis, but without which the
ocsenave@0	223 development of thermodynamics cannot proceed. Because of this
ocsenave@0	224 special status, these relations have become known as the
ocsenave@0	225 \ldquo{}laws\rdquo{}
ocsenave@0	226 of thermodynamics . The most fundamental one is a qualitative
ocsenave@0	227 rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
ocsenave@0	228
ocsenave@3	229 ** Equilibrium; the Zeroth Law
ocsenave@0	230
ocsenave@0	231 It is a common experience
ocsenave@0	232 that when objects are placed in contact with each other but
ocsenave@0	233 isolated from their surroundings, they may undergo observable
ocsenave@0	234 changes for a time as a result; one body may become warmer,
ocsenave@0	235 another cooler, the pressure of a gas or volume of a liquid may
ocsenave@0	236 change; stress or magnetization in a solid may change, etc. But
ocsenave@0	237 after a sufficient time, the observable macroscopic properties
ocsenave@0	238 settle down to a steady condition, after which no further changes
ocsenave@0	239 are seen unless there is a new intervention from the outside.
ocsenave@0	240 When this steady condition is reached, the experimentalist says
ocsenave@0	241 that the objects have reached a state of /equilibrium/ with each
ocsenave@0	242 other. Once again, more precise definitions of this term will
ocsenave@0	243 be needed eventually, but they require concepts not yet developed.
ocsenave@0	244 In any event, the criterion just stated is almost the only one
ocsenave@0	245 used in actual laboratory practice to decide when equilibrium
ocsenave@0	246 has been reached.
ocsenave@0	247
ocsenave@0	248
ocsenave@0	249 A particular case of equilibrium is encountered when we
ocsenave@0	250 place a thermometer in contact with another body. The reading
ocsenave@0	251 $t$ of the thermometer may vary at first, but eventually it reach es
ocsenave@0	252 a steady value. Now the number $t$ read by a thermometer is always.
ocsenave@0	253 by definition, the empirical temperature /of the thermometer/ (more
ocsenave@0	254 precisely, of the sensitive element of the thermometer). When
ocsenave@0	255 this number is constant in time, we say that the thermometer is
ocsenave@0	256 in /thermal equilibrium/ with its surroundings; and we then extend
ocsenave@0	257 the notion of temperature, calling the steady value $t$ also the
ocsenave@0	258 /temperature of the surroundings/.
ocsenave@0	259
ocsenave@0	260 We have repeated these elementary facts, well known to every
ocsenave@0	261 child, in order to emphasize this point: Thermodynamics can be
ocsenave@0	262 a theory /only/ of states of equilibrium, because the very
ocsenave@0	263 procedure by which the temperature of a system is defined by
ocsenave@0	264 operational means, already presupposes the attainment of
ocsenave@0	265 equilibrium. Strictly speaking, therefore, classical
ocsenave@0	266 thermodynamics does not even contain the concept of a
ocsenave@0	267 \ldquo{}time-varying temperature.\rdquo{}
ocsenave@0	268
ocsenave@0	269 Of course, to recognize this limitation on conventional
ocsenave@0	270 thermodynamics (best emphasized by calling it instead,
ocsenave@0	271 thermostatics) in no way rules out the possibility of
ocsenave@0	272 generalizing the notion of temperature to nonequilibrium states.
ocsenave@0	273 Indeed, it is clear that one could define any number of
ocsenave@0	274 time-dependent quantities all of which reduce, in the special
ocsenave@0	275 case of equilibrium, to the temperature as defined above.
ocsenave@0	276 Historically, attempts to do this even antedated the discovery
ocsenave@0	277 of the laws of thermodynamics, as is demonstrated by
ocsenave@0	278 \ldquo{}Newton's law of cooling.\rdquo{} Therefore, the
ocsenave@0	279 question is not whether generalization is /possible/, but only
ocsenave@0	280 whether it is in any way /useful/; i.e., does the temperature so
ocsenave@0	281 generalized have any connection with other physical properties
ocsenave@0	282 of our system, so that it could help us to predict other things?
ocsenave@0	283 However, to raise such questions takes us far beyond the
ocsenave@0	284 domain of thermostatics; and the general laws of nonequilibrium
ocsenave@0	285 behavior are so much more complicated that it would be virtually
ocsenave@0	286 hopeless to try to unravel them by empirical means alone. For
ocsenave@0	287 example, even if two different kinds of thermometer are calibrated
ocsenave@0	288 so that they agree with each other in equilibrium situations,
ocsenave@0	289 they will not agree in general about the momentary value a
ocsenave@0	290 \ldquo{}time-varying temperature.\rdquo{} To make any real
ocsenave@0	291 progress in this area, we have to supplement empirical observation by the guidance
ocsenave@0	292 of a rather hiqhly-developed theory. The notion of a
ocsenave@0	293 time-dependent temperature is far from simple conceptually, and we
ocsenave@0	294 will find that nothing very helpful can be said about this until
ocsenave@0	295 the full mathematical apparatus of nonequilibrium statistical
ocsenave@0	296 mechanics has been developed.
ocsenave@0	297
ocsenave@0	298 Suppose now that two bodies have the same temperature; i.e.,
ocsenave@0	299 a given thermometer reads the same steady value when in contact
ocsenave@0	300 with either. In order that the statement, \ldquo{}two bodies have the
ocsenave@1	301 same temperature\rdquo{} shall describe a physical property of the bodies,
ocsenave@0	302 and not merely an accidental circumstance due to our having used
ocsenave@0	303 a particular kind of thermometer, it is necessary that /all/
ocsenave@0	304 thermometers agree in assigning equal temperatures to them if
ocsenave@0	305 /any/ thermometer does . Only experiment is competent to determine
ocsenave@0	306 whether this universality property is true. Unfortunately, the
ocsenave@0	307 writer must confess that he is unable to cite any definite
ocsenave@0	308 experiment in which this point was subjected to a careful test.
ocsenave@0	309 That equality of temperatures has this absolute meaning, has
ocsenave@0	310 evidently been taken for granted so much that (like absolute
ocsenave@0	311 sirnultaneity in pre-relativity physics) most of us are not even
ocsenave@0	312 consciously aware that we make such an assumption in
ocsenave@0	313 thermodynamics. However, for the present we can only take it as a familiar
ocsenave@0	314 empirical fact that this condition does hold, not because we can
ocsenave@0	315 cite positive evidence for it, but because of the absence of
ocsenave@0	316 negative evidence against it; i.e., we think that, if an
ocsenave@0	317 exception had ever been found, this would have created a sensation in
ocsenave@0	318 physics, and we should have heard of it.
ocsenave@0	319
ocsenave@0	320 We now ask: when two bodies are at the same temperature,
ocsenave@0	321 are they then in thermal equilibrium with each other? Again,
ocsenave@0	322 only experiment is competent to answer this; the general
ocsenave@0	323 conclusion, again supported more by absence of negative evidence
ocsenave@0	324 than by specific positive evidence, is that the relation of
ocsenave@0	325 equilibrium has this property:
ocsenave@0	326 #+begin_quote
ocsenave@0	327 /Two bodies in thermal equilibrium
ocsenave@0	328 with a third body, are thermal equilibrium with each other./
ocsenave@0	329 #+end_quote
ocsenave@0	330
ocsenave@0	331 This empirical fact is usually called the \ldquo{}zero'th law of
ocsenave@0	332 thermodynamics.\rdquo{} Since nothing prevents us from regarding a
ocsenave@0	333 thermometer as the \ldquo{}third body\rdquo{} in the above statement,
ocsenave@0	334 it appears that we may also state the zero'th law as:
ocsenave@0	335 #+begin_quote
ocsenave@0	336 /Two bodies are in thermal equilibrium with each other when they are
ocsenave@0	337 at the same temperature./
ocsenave@0	338 #+end_quote
ocsenave@0	339 Although from the preceding discussion it might appear that
ocsenave@0	340 these two statements of the zero'th law are entirely equivalent
ocsenave@0	341 (and we certainly have no empirical evidence against either), it
ocsenave@0	342 is interesting to note that there are theoretical reasons, arising
ocsenave@0	343 from General Relativity, indicating that while the first
ocsenave@0	344 statement may be universally valid, the second is not. When we
ocsenave@0	345 consider equilibrium in a gravitational field, the verification
ocsenave@0	346 that two bodies have equal temperatures may require transport
ocsenave@0	347 of the thermometer through a gravitational potential difference;
ocsenave@0	348 and this introduces a new element into the discussion. We will
ocsenave@0	349 consider this in more detail in a later Chapter, and show that
ocsenave@0	350 according to General Relativity, equilibrium in a large system
ocsenave@0	351 requires, not that the temperature be uniform at all points, but
ocsenave@0	352 rather that a particular function of temperature and gravitational
ocsenave@0	353 potential be constant (the function is $T\cdot \exp{(\Phi/c^2})$, where
ocsenave@0	354 $T$ is the Kelvin temperature to be defined later, and $\Phi$ is the
ocsenave@0	355 gravitational potential).
ocsenave@0	356
ocsenave@0	357 Of course, this effect is so small that ordinary terrestrial
ocsenave@0	358 experiments would need to have a precision many orders of
ocsenave@0	359 magnitude beyond that presently possible, before one could hope even
ocsenave@0	360 to detect it; and needless to say, it has played no role in the
ocsenave@0	361 development of thermodynamics. For present purposes, therefore,
ocsenave@0	362 we need not distinguish between the two above statements of the
ocsenave@0	363 zero'th law, and we take it as a basic empirical fact that a
ocsenave@0	364 uniform temperature at all points of a system is an essential
ocsenave@0	365 condition for equilibrium. It is an important part of our
ocsenave@0	366 ivestigation to determine whether there are other essential
ocsenave@0	367 conditions as well. In fact, as we will find, there are many
ocsenave@0	368 different kinds of equilibrium; and failure to distinguish between
ocsenave@0	369 them can be a prolific source of paradoxes.
ocsenave@0	370
ocsenave@0	371 ** Equation of State
ocsenave@0	372 Another important reproducible connection is found when
ocsenave@0	373 we consider a thermodynamic system defined by
ocsenave@0	374 three parameters; in addition to the temperature we choose a
ocsenave@0	375 \ldquo{}displacement\rdquo{} and a conjugate \ldquo{}force.\rdquo{}
ocsenave@0	376 Subject to some qualifications given below, we find experimentally
ocsenave@0	377 that these parameters are not independent, but are subject to a constraint.
ocsenave@0	378 For example, we cannot vary the equilibrium pressure, volume,
ocsenave@0	379 and temperature of a given mass of gas independently; it is found
ocsenave@0	380 that a given pressure and volume can be realized only at one
ocsenave@0	381 particular temperature, that the gas will assume a given tempera~
ocsenave@0	382 ture and volume only at one particular pressure, etc. Similarly,
ocsenave@0	383 a stretched wire can be made to have arbitrarily assigned tension
ocsenave@0	384 and elongation only if its temperature is suitably chosen, a
ocsenave@0	385 dielectric will assume a state of given temperature and
ocsenave@0	386 polarization at only one value of the electric field, etc.
ocsenave@0	387 These simplest nontrivial thermodynamic systems (three
ocsenave@0	388 parameters with one constraint) are said to possess two
ocsenave@0	389 /degrees of freedom/; for the range of possible equilibrium states is defined
ocsenave@0	390 by specifying any two of the variables arbitrarily, whereupon the
ocsenave@0	391 third, and all others we may introduce, are determined.
ocsenave@0	392 Mathematically, this is expressed by the existence of a functional
ocsenave@4	393 relationship of the form[fn:: Edit: The set of solutions to an equation
ocsenave@0	394 like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
ocsenave@0	395 saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
ocsenave@0	396 rule\rdquo{}, so the set of physically allowed combinations of /X/,
ocsenave@0	397 /x/, and /t/ in equilibrium states can be
ocsenave@0	398 expressed as the level set of a function.
ocsenave@0	399
ocsenave@0	400 But not every function expresses a constraint relation; for some
ocsenave@0	401 functions, you can specify two of the variables, and the third will
ocsenave@0	402 still be undetermined. (For example, if f=X^2+x^2+t^2-3,
ocsenave@0	403 the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/
ocsenave@0	404 leaves you with two potential possibilities for /X/ =\pm 1.)
ocsenave@0	405
ocsenave@1	406 A function like /f/ has to possess one more propery in order for its
ocsenave@1	407 level set to express a constraint relationship: it must be monotonic in
ocsenave@0	408 each of its variables /X/, /x/, and /t/.
ocsenave@0	409 #the partial derivatives of /f/ exist for every allowed combination of
ocsenave@0	410 #inputs /x/, /X/, and /t/.
ocsenave@0	411 In other words, the level set has to pass a sort of
ocsenave@0	412 \ldquo{}vertical line test\rdquo{} for each of its variables.]
ocsenave@0	413
ocsenave@0	414 #Edit Here, Jaynes
ocsenave@0	415 #is saying that it is possible to express the collection of allowed combinations $\langle X,x,t \rangle$ of force, quantity, and temperature as a
ocsenave@0	416 #[[http://en.wikipedia.org/wiki/Level_set][level set]] of some function $f$. However, not all level sets represent constraint relations; consider $f(X,x,t)=X^2+x^2+t^2-1$=0.
ocsenave@0	417 #In order to specify
ocsenave@0	418
ocsenave@0	419 \begin{equation}
ocsenave@0	420 f(X,x,t) = O
ocsenave@0	421 \end{equation}
ocsenave@0	422
ocsenave@0	423 where $X$ is a generalized force (pressure, tension, electric or
ocsenave@0	424 magnetic field, etc.), $x$ is the corresponding generalized
ocsenave@0	425 displacement (volume, elongation, electric or magnetic polarization,
ocsenave@1	426 etc.), and $t$ is the empirical temperature. Equation (1-1) is
ocsenave@0	427 called /the equation of state/.
ocsenave@0	428
ocsenave@0	429 At the risk of belaboring it, we emphasize once again that
ocsenave@0	430 all of this applies only for a system in equilibrium; for
ocsenave@0	431 otherwise not only.the temperature, but also some or all of the other
ocsenave@0	432 variables may not be definable. For example, no unique pressure
ocsenave@0	433 can be assigned to a gas which has just suffered a sudden change
ocsenave@0	434 in volume, until the generated sound waves have died out.
ocsenave@0	435
ocsenave@0	436 Independently of its functional form, the mere fact of the
ocsenave@0	437 /existence/ of an equation of state has certain experimental
ocsenave@0	438 consequences. For example, suppose that in experiments on oxygen
ocsenave@0	439 gas, in which we control the temperature and pressure
ocsenave@0	440 independently, we have found that the isothermal compressibility $K$
ocsenave@0	441 varies with temperature, and the thermal expansion coefficient
ocsenave@0	442 \alpha varies with pressure $P$, so that within the accuracy of the data,
ocsenave@0	443
ocsenave@0	444 \begin{equation}
ocsenave@0	445 \frac{\partial K}{\partial t} = - \frac{\partial \alpha}{\partial P}
ocsenave@0	446 \end{equation}
ocsenave@0	447
ocsenave@0	448 Is this a particular property of oxygen; or is there reason to
ocsenave@0	449 believe that it holds also for other substances? Does it depend
ocsenave@0	450 on our particular choice of a temperature scale?
ocsenave@0	451
ocsenave@0	452 In this case, the answer is found at once; for the definitions of $K$,
ocsenave@0	453 \alpha are
ocsenave@0	454
ocsenave@0	455 \begin{equation}
ocsenave@0	456 K = -\frac{1}{V}\frac{\partial V}{\partial P},\qquad
ocsenave@0	457 \alpha=\frac{1}{V}\frac{\partial V}{\partial t}
ocsenave@0	458 \end{equation}
ocsenave@0	459
ocsenave@0	460 which is simply a mathematical expression of the fact that the
ocsenave@0	461 volume $V$ is a definite function of $P$ and $t$; i.e., it depends
ocsenave@0	462 only
ocsenave@0	463 on their present values, and not how those values were attained.
ocsenave@0	464 In particular, $V$ does not depend on the direction in the $(P, t)$
ocsenave@0	465 plane through which the present values were approached; or, as we
ocsenave@0	466 usually say it, $dV$ is an /exact differential/.
ocsenave@0	467
ocsenave@1	468 Therefore, although at first glance the relation (1-2) appears
ocsenave@0	469 nontrivial and far from obvious, a trivial mathematical analysis
ocsenave@0	470 convinces us that it must hold regardless of our particular
ocsenave@0	471 temperature scale, and that it is true not only of oxygen; it must
ocsenave@0	472 hold for any substance, or mixture of substances, which possesses a
ocsenave@0	473 definite, reproducible equation of state $f(P,V,t)=0$.
ocsenave@0	474
ocsenave@0	475 But this understanding also enables us to predict situations in which
ocsenave@1	476 (1-2) will /not/ hold. Equation (1-2), as we have just learned, expresses
ocsenave@0	477 the fact that an equation of state exists involving only the three
ocsenave@0	478 variables $(P,V,t)$. Now suppose we try to apply it to a liquid such
ocsenave@0	479 as nitrobenzene. The nitrobenzene molecule has a large electric dipole
ocsenave@0	480 moment; and so application of an electric field (as in the
ocsenave@0	481 [[http://en.wikipedia.org/wiki/Kerr_effect][electro-optical Kerr cell]]) causes an alignment of molecules which, as
ocsenave@0	482 accurate measurements will verify, changes the pressure at a given
ocsenave@0	483 temperature and volume. Therefore, there can no longer exist any
ocsenave@0	484 unique equation of state involving $(P, V, t)$ only; with
ocsenave@0	485 sufficiently accurate measurements, nitrobenzene must be regarded as a
ocsenave@0	486 thermodynamic system with at least three degrees of freedom, and the
ocsenave@0	487 general equation of state must have at least a complicated a form as
ocsenave@0	488 $f(P,V,t,E) = 0$.
ocsenave@0	489
ocsenave@0	490 But if we introduce a varying electric field $E$ into the discussion,
ocsenave@0	491 the resulting varying electric polarization $M$ also becomes a new
ocsenave@0	492 thermodynamic variable capable of being measured. Experimentally, it
ocsenave@0	493 is easiest to control temperature, pressure, and electric field
ocsenave@0	494 independently, and of course we find that both the volume and
ocsenave@0	495 polarization are then determined; i.e., there must exist functional
ocsenave@0	496 relations of the form $V = V(P,t,E)$, $M = M(P,t,E)$, or in more
ocsenave@0	497 symmetrical form
ocsenave@0	498
ocsenave@0	499 \begin{equation}
ocsenave@0	500 f(V,P,t,E) = 0 \qquad g(M,P,t,E)=0.
ocsenave@0	501 \end{equation}
ocsenave@0	502
ocsenave@0	503 In other words, if we regard nitrobenzene as a thermodynamic system of
ocsenave@0	504 three degrees of freedom (i.e., having specified three parameters
ocsenave@0	505 arbitrarily, all others are then determined), it must possess two
ocsenave@0	506 independent equations of state.
ocsenave@0	507
ocsenave@0	508 Similarly, a thermodynamic system with four degrees of freedom,
ocsenave@0	509 defined by the termperature and three pairs of conjugate forces and
ocsenave@0	510 displacements, will have three independent equations of state, etc.
ocsenave@0	511
ocsenave@0	512 Now, returning to our original question, if nitrobenzene possesses
ocsenave@0	513 this extra electrical degree of freedom, under what circumstances do
ocsenave@0	514 we exprect to find a reproducible equation of state involving
ocsenave@0	515 $(p,V,t)$ only? Evidently, if $E$ is held constant, then the first
ocsenave@0	516 of equations (1-5) becomes such an equation of state, involving $E$ as
ocsenave@0	517 a fixed parameter; we would find many different equations of state of
ocsenave@0	518 the form $f(P,V,t) = 0$ with a different function $f$ for each
ocsenave@0	519 different value of the electric field. Likewise, if $M$ is held
ocsenave@0	520 constant, we can eliminate $E$ between equations (1-5) and find a
ocsenave@0	521 relation $h(P,V,t,M)=0$, which is an equation of state for
ocsenave@0	522 $(P,V,t)$ containing $M$ as a fixed parameter.
ocsenave@0	523
ocsenave@0	524 More generally, if an electrical constraint is imposed on the system
ocsenave@0	525 (for example, by connecting an external charged capacitor to the
ocsenave@0	526 electrodes) so that $M$ is determined by $E$; i.e., there is a
ocsenave@0	527 functional relation of the form
ocsenave@0	528
ocsenave@0	529 \begin{equation}
ocsenave@0	530 g(M,E) = \text{const.}
ocsenave@0	531 \end{equation}
ocsenave@0	532
ocsenave@0	533 then (1-5) and (1-6) constitute three simultaneous equations, from
ocsenave@0	534 which both $E$ and $M$ may be eliminated mathematically, leading
ocsenave@0	535 to a relation of the form $h(P,V,t;q)=0$, which is an equation of
ocsenave@0	536 state for $(P,V,t)$ involving the fixed parameter $q$.
ocsenave@0	537
ocsenave@0	538 We see, then, that as long as a fixed constraint of the form (1-6) is
ocsenave@0	539 imposed on the electrical degree of freedom, we can still observe a
ocsenave@0	540 reproducible equation of state for nitrobenzene, considered as a
ocsenave@0	541 thermodynamic system of only two degrees of freedom. If, however, this
ocsenave@0	542 electrical constraint is removed, so that as we vary $P$ and $t$, the
ocsenave@0	543 values of $E$ and $M$ vary in an uncontrolled way over a
ocsenave@0	544 /two-dimensional/ region of the $(E, M)$ plane, then we will find no
ocsenave@0	545 definite equation of state involving only $(P,V,t)$.
ocsenave@0	546
ocsenave@0	547 This may be stated more colloqually as follows: even though a system
ocsenave@0	548 has three degrees of freedom, we can still consider only the variables
ocsenave@0	549 belonging to two of them, and we will find a definite equation of
ocsenave@0	550 state, /provided/ that in the course of the experiments, the unused
ocsenave@0	551 degree of freedom is not \ldquo{}tampered with\rdquo{} in an
ocsenave@0	552 uncontrolled way.
ocsenave@0	553
ocsenave@0	554 We have already emphasized that any physical system corresponds to
ocsenave@0	555 many different thermodynamic systems, depending on which variables we
ocsenave@0	556 choose to control and measure. In fact, it is easy to see that any
ocsenave@0	557 physical system has, for all practical purposes, an /arbitrarily
ocsenave@0	558 large/ number of degrees of freedom. In the case of nitrobenzene, for
ocsenave@0	559 example, we may impose any variety of nonuniform electric fields on
ocsenave@1	560 our sample. Suppose we place $(n+1)$ different electrodes, labelled
ocsenave@1	561 $\{e_0,e_1, e_2 \ldots e_n\}$ in contact with the liquid in various
ocsenave@1	562 positions. Regarding $e_0$ as the \ldquo{}ground\rdquo{}, maintained
ocsenave@1	563 at zero potential, we can then impose $n$ different potentials
ocsenave@1	564 $\{v_1, \ldots, v_n\}$ on the other electrodes independently, and we
ocsenave@1	565 can also measure the $n$ different conjugate displacements, as the
ocsenave@1	566 charges $\{q_1,\ldots, q_n\}$ accumulated on electrodes
ocsenave@1	567 $\{e_1,\ldots e_n\}$. Together with the pressure (understood as the
ocsenave@1	568 pressure measured at one given position), volume, and temperature, our
ocsenave@1	569 sample of nitrobenzene is now a thermodynamic system of $(n+1)$
ocsenave@1	570 degrees of freedom. This number may be as large as we please, limited
ocsenave@1	571 only by our patience in constructing the apparatus needed to control
ocsenave@1	572 or measure all these quantities.
ocsenave@1	573
ocsenave@5	574 We leave it as an exercise for the reader (Problem 1.1) to find the most
ocsenave@1	575 general condition on the variables \(\{v_1, q_1, v_2, q_2, \ldots
ocsenave@1	576 v_n,q_n\}\) which will ensure that a definite equation of state
ocsenave@1	577 $f(P,V,t)=0$ is observed in spite of all these new degrees of
ocsenave@1	578 freedom. The simplest special case of this relation is, evidently, to
ocsenave@1	579 ground all electrodes, thereby inposing the conditions $v_1 = v_2 =
ocsenave@1	580 \ldots = v_n = 0$. Equally well (if we regard nitrobenzene as having
ocsenave@1	581 negligible electrical conductivity) we may open-circuit all
ocsenave@1	582 electrodes, thereby imposing the conditions $q_i = \text{const.}$ In
ocsenave@1	583 the latter case, in addition to an equation of state of the form
ocsenave@1	584 $f(P,V,t)=0$, which contains these constants as fixed parameters,
ocsenave@1	585 there are $n$ additional equations of state of the form $v_i =
ocsenave@1	586 v_i(P,t)$. But if we choose to ignore these voltages, there will be no
ocsenave@1	587 contradiction in considering our nitrobenzene to be a thermodynamic
ocsenave@1	588 system of two degrees of freedom, involving only the variables
ocsenave@1	589 $P,V,t$.
ocsenave@1	590
ocsenave@1	591 Similarly, if our system of interest is a crystal, we may impose on it
ocsenave@1	592 a wide variety of nonuniform stress fields; each component of the
ocsenave@1	593 stress tensor $T_{ij}$ may bary with position. We might expand each of
ocsenave@1	594 these functions in a complete orthonormal set of functions
ocsenave@1	595 $\phi_k(x,y,z)$:
ocsenave@1	596
ocsenave@1	597 \begin{equation}
ocsenave@1	598 T_{ij}(x,y,z) = \sum_k a_{ijk} \phi_k(x,y,z)
ocsenave@1	599 \end{equation}
ocsenave@1	600
ocsenave@1	601 and with a sufficiently complicated system of levers which in various
ocsenave@1	602 ways squeeze and twist the crystal, we might vary each of the first
ocsenave@1	603 1,000 expansion coefficients $a_{ijk}$ independently, and measure the
ocsenave@1	604 conjugate displacements $q_{ijk}$. Our crystal is then a thermodynamic
ocsenave@1	605 system of over 1,000 degrees of freedom.
ocsenave@1	606
ocsenave@1	607 The notion of \ldquo{}numbers of degrees of freedom\rdquo{} is
ocsenave@1	608 therefore not a /physical property/ of any system; it is entirely
ocsenave@1	609 anthropomorphic, since any physical system may be regarded as a
ocsenave@1	610 thermodynamic system with any number of degrees of freedom we please.
ocsenave@1	611
ocsenave@1	612 If new thermodynamic variables are always introduced in pairs,
ocsenave@1	613 consisting of a \ldquo{}force\rdquo{} and conjugate
ocsenave@1	614 \ldquo{}displacement\rdquo{}, then a thermodynamic system of $n$
ocsenave@1	615 degrees of freedom must possess $(n-1)$ independent equations of
ocsenave@1	616 state, so that specifying $n$ quantities suffices to determine all
ocsenave@1	617 others.
ocsenave@1	618
ocsenave@1	619 This raises an interesting question; whether the scheme of classifying
ocsenave@1	620 thermodynamic variables in conjugate pairs is the most general
ocsenave@1	621 one. Why, for example, is it not natural to introduce three related
ocsenave@1	622 variables at a time? To the best of the writer's knowledge, this is an
ocsenave@1	623 open question; there seems to be no fundamental reason why variables
ocsenave@1	624 /must/ always be introduced in conjugate pairs, but there seems to be
ocsenave@1	625 no known case in which a different scheme suggests itself as more
ocsenave@1	626 appropriate.
ocsenave@1	627
ocsenave@1	628 ** Heat
ocsenave@1	629 We are now in a position to consider the results and interpretation of
ocsenave@1	630 a number of elementary experiments involving
ocsenave@2	631 thermal interaction, which can be carried out as soon as a primitive
ocsenave@2	632 thermometer is at hand. In fact these experiments, which we summarize
ocsenave@2	633 so quickly, required a very long time for their first performance, and
ocsenave@2	634 the essential conclusions of this Section were first arrived at only
ocsenave@2	635 about 1760---more than 160 years after Galileo's invention of the
ocsenave@4	636 thermometer---[[http://web.lemoyne.edu/~giunta/blackheat.html][by Joseph Black]], who was Professor of Chemistry at
ocsenave@2	637 Glasgow University. Black's analysis of calorimetric experiments
ocsenave@2	638 initiated by G. D. Fahrenheit before 1736 led to the first recognition
ocsenave@2	639 of the distinction between temperature and heat, and prepared the way
ocsenave@2	640 for the work of his better-known pupil, James Watt.
ocsenave@1	641
ocsenave@2	642 We first observe that if two bodies at different temperatures are
ocsenave@2	643 separated by walls of various materials, they sometimes maintain their
ocsenave@2	644 temperature difference for a long time, and sometimes reach thermal
ocsenave@2	645 equilibrium very quickly. The differences in behavior observed must be
ocsenave@2	646 ascribed to the different properties of the separating walls, since
ocsenave@2	647 nothing else is changed. Materials such as wood, asbestos, porous
ocsenave@2	648 ceramics (and most of all, modern porous plastics like styrofoam), are
ocsenave@2	649 able to sustain a temperature difference for a long time; a wall of an
ocsenave@2	650 imaginary material with this property idealized to the point where a
ocsenave@2	651 temperature difference is maintained indefinitely is called an
ocsenave@2	652 /adiabatic wall/. A very close approximation to a perfect adiabatic
ocsenave@2	653 wall is realized by the Dewar flask (thermos bottle), of which the
ocsenave@2	654 walls consist of two layers of glass separated by a vacuum, with the
ocsenave@2	655 surfaces silvered like a mirror. In such a container, as we all know,
ocsenave@2	656 liquids may be maintained hot or cold for days.
ocsenave@1	657
ocsenave@2	658 On the other hand, a thin wall of copper or silver is hardly able to
ocsenave@2	659 sustain any temperature difference at all; two bodies separated by
ocsenave@2	660 such a partition come to thermal equilibrium very quickly. Such a wall
ocsenave@2	661 is called /diathermic/. It is found in general that the best
ocsenave@2	662 diathermic materials are the metals and good electrical conductors,
ocsenave@2	663 while electrical insulators make fairly good adiabatic walls. There
ocsenave@2	664 are good theoretical reasons for this rule; a particular case of it is
ocsenave@2	665 given by the [[http://en.wikipedia.org/wiki/Wiedemann_franz_law][Wiedemann-Franz law]] of solid-state theory.
ocsenave@2	666
ocsenave@2	667 Since a body surrounded by an adiabatic wall is able to maintain its
ocsenave@2	668 temperature independently of the temperature of its surroundings, an
ocsenave@2	669 adiabatic wall provides a means of thermally /isolating/ a system from
ocsenave@2	670 the rest of the universe; it is to be expected, therefore, that the
ocsenave@2	671 laws of thermal interaction between two systems will assume the
ocsenave@2	672 simplest form if they are enclosed in a common adiabatic container,
ocsenave@2	673 and that the best way of carrying out experiments on thermal
ocsenave@2	674 peroperties of substances is to so enclose them. Such an apparatus, in
ocsenave@2	675 which systems are made to interact inside an adiabatic container
ocsenave@2	676 supplied with a thermometer, is called a /calorimeter/.
ocsenave@2	677
ocsenave@2	678 Let us imagine that we have a calorimeter in which there is initially
ocsenave@2	679 a volume $V_W$ of water at a temperature $t_1$, and suspended above it
ocsenave@2	680 a volume $V_I$ of some other substance (say, iron) at temperature
ocsenave@2	681 $t_2$. When we drop the iron into the water, they interact thermally
ocsenave@2	682 (and the exact nature of this interaction is one of the things we hope
ocsenave@2	683 to learn now), the temperature of both changing until they are in
ocsenave@2	684 thermal equilibrium at a final temperature $t_0$.
ocsenave@2	685
ocsenave@2	686 Now we repeat the experiment with different initial temperatures
ocsenave@2	687 $t_1^\prime$ and $t_2^\prime$, so that a new equilibrium is reached at
ocsenave@2	688 temperature $t_0^\prime$. It is found that, if the temperature
ocsenave@2	689 differences are sufficiently small (and in practice this is not a
ocsenave@2	690 serious limitation if we use a mercury thermometer calibrated with
ocsenave@2	691 uniformly spaced degree marks on a capillary of uniform bore), then
ocsenave@2	692 whatever the values of $t_1^\prime$, $t_2^\prime$, $t_1$, $t_2$, the
ocsenave@2	693 final temperatures $t_0^\prime$, $t_0$ will adjust themselves so that
ocsenave@2	694 the following relation holds:
ocsenave@2	695
ocsenave@2	696 \begin{equation}
ocsenave@2	697 \frac{t_2 - t_0}{t_0 - t_1} = \frac{t_2^\prime -
ocsenave@2	698 t_0^\prime}{t_0^\prime - t_1^\prime}
ocsenave@2	699 \end{equation}
ocsenave@2	700
ocsenave@2	701 in other words, the /ratio/ of the temperature changes of the iron and
ocsenave@2	702 water is independent of the initial temperatures used.
ocsenave@2	703
ocsenave@2	704 We now vary the amounts of iron and water used in the calorimeter. It
ocsenave@2	705 is found that the ratio (1-8), although always independent of the
ocsenave@2	706 starting temperatures, does depend on the relative amounts of iron and
ocsenave@2	707 water. It is, in fact, proportional to the mass $M_W$ of water and
ocsenave@2	708 inversely proportional to the mass $M_I$ of iron, so that
ocsenave@2	709
ocsenave@2	710 \begin{equation}
ocsenave@2	711 \frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_I M_I}
ocsenave@2	712 \end{equation}
ocsenave@2	713
ocsenave@2	714 where $K_I$ is a constant.
ocsenave@2	715
ocsenave@2	716 We next repeat the above experiments using a different material in
ocsenave@2	717 place of the iron (say, copper). We find again a relation
ocsenave@2	718
ocsenave@2	719 \begin{equation}
ocsenave@2	720 \frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_C \cdot M_C}
ocsenave@2	721 \end{equation}
ocsenave@2	722
ocsenave@2	723 where $M_C$ is the mass of copper; but the constant $K_C$ is different
ocsenave@2	724 from the previous $K_I$. In fact, we see that the constant $K_I$ is a
ocsenave@2	725 new physical property of the substance iron, while $K_C$ is a physical
ocsenave@2	726 property of copper. The number $K$ is called the /specific heat/ of a
ocsenave@2	727 substance, and it is seen that according to this definition, the
ocsenave@2	728 specific heat of water is unity.
ocsenave@2	729
ocsenave@2	730 We now have enough experimental facts to begin speculating about their
ocsenave@2	731 interpretation, as was first done in the 18th century. First, note
ocsenave@2	732 that equation (1-9) can be put into a neater form that is symmetrical
ocsenave@2	733 between the two substances. We write $\Delta t_I = t_0 - t_2$, $\Delta
ocsenave@2	734 t_W = t_0 - t_1$ for the temperature changes of iron and water
ocsenave@2	735 respectively, and define $K_W \equiv 1$ for water. Equation (1-9) then
ocsenave@2	736 becomes
ocsenave@2	737
ocsenave@2	738 \begin{equation}
ocsenave@2	739 K_W M_W \Delta t_W + K_I M_I \Delta t_I = 0
ocsenave@2	740 \end{equation}
ocsenave@2	741
ocsenave@2	742 The form of this equation suggests a new experiment; we go back into
ocsenave@2	743 the laboratory, and find $n$ substances for which the specific heats
ocsenave@2	744 $\{K_1,\ldots K_n\}$ have been measured previously. Taking masses
ocsenave@2	745 $\{M_1, \ldots, M_n\}$ of these substances, we heat them to $n$
ocsenave@2	746 different temperatures $\{t_1,\ldots, t_n\}$ and throw them all into
ocsenave@2	747 the calorimeter at once. After they have all come to thermal
ocsenave@2	748 equilibrium at temperature $t_0$, we find the differences $\Delta t_j
ocsenave@2	749 = t_0 - t_j$. Just as we suspected, it turns out that regardless of
ocsenave@2	750 the $K$'s, $M$'s, and $t$'s chosen, the relation
ocsenave@2	751 \begin{equation}
ocsenave@2	752 \sum_{j=0}^n K_j M_j \Delta t_j = 0
ocsenave@2	753 \end{equation}
ocsenave@2	754 is always satisfied. This sort of process is an old story in
ocsenave@2	755 scientific investigations; although the great theoretician Boltzmann
ocsenave@3	756 is said to have remarked: \ldquo{}Elegance is for tailors\rdquo{}, it
ocsenave@2	757 remains true that the attempt to reduce equations to the most
ocsenave@2	758 symmetrical form has often suggested important generalizations of
ocsenave@2	759 physical laws, and is a great aid to memory. Witness Maxwell's
ocsenave@2	760 \ldquo{}displacement current\rdquo{}, which was needed to fill in a
ocsenave@2	761 gap and restore the symmetry of the electromagnetic equations; as soon
ocsenave@2	762 as it was put in, the equations predicted the existence of
ocsenave@2	763 electromagnetic waves. In the present case, the search for a rather
ocsenave@2	764 rudimentary form of \ldquo{}elegance\rdquo{} has also been fruitful,
ocsenave@2	765 for we recognize that (1-12) has the standard form of a /conservation
ocsenave@2	766 law/; it defines a new quantity which is conserved in thermal
ocsenave@2	767 interactions of the type just studied.
ocsenave@2	768
ocsenave@2	769 The similarity of (1-12) to conservation laws in general may be seen
ocsenave@3	770 as follows. Let $A$ be some quantity that is conserved; the $i$th
ocsenave@2	771 system has an amount of it $A_i$. Now when the systems interact such
ocsenave@2	772 that some $A$ is transferred between them, the amount of $A$ in the
ocsenave@3	773 $i$th system is changed by a net amount \(\Delta A_i = (A_i)_{final} -
ocsenave@2	774 (A_i)_{initial}\); and the fact that there is no net change in the
ocsenave@2	775 total amount of $A$ is expressed by the equation \(\sum_i \Delta
ocsenave@3	776 A_i = 0\). Thus, the law of conservation of matter in a chemical
ocsenave@2	777 reaction is expressed by $\sum_i \Delta M_i = 0$, where $M_i$ is the
ocsenave@3	778 mass of the $i$th chemical component.
ocsenave@2	779
ocsenave@3	780 What is this new conserved quantity? Mathematically, it can be defined
ocsenave@3	781 as $Q_i = K_i\cdot M_i \cdot t_i$; whereupon (1-12) becomes
ocsenave@2	782
ocsenave@2	783 \begin{equation}
ocsenave@2	784 \sum_i \Delta Q_i = 0
ocsenave@2	785 \end{equation}
ocsenave@2	786
ocsenave@2	787 and at this point we can correct a slight quantitative inaccuracy. As
ocsenave@2	788 noted, the above relations hold accurately only when the temperature
ocsenave@2	789 differences are sufficiently small; i.e., they are really only
ocsenave@2	790 differential laws. On sufficiently accurate measurements one find that
ocsenave@2	791 the specific heats $K_i$ depend on temperature; if we then adopt the
ocsenave@2	792 integral definition of $\Delta Q_i$,
ocsenave@2	793 \begin{equation}
ocsenave@2	794 \Delta Q_i = \int_{t_{i}}^{t_0} K_i(t) M_i dt
ocsenave@2	795 \end{equation}
ocsenave@2	796
ocsenave@2	797 the conservation law (1-13) will be found to hold in calorimetric
ocsenave@2	798 experiments with liquids and solids, to any accuracy now feasible. And
ocsenave@2	799 of course, from the manner in which the $K_i(t)$ are defined, this
ocsenave@2	800 relation will hold however our thermometers are calibrated.
ocsenave@2	801
ocsenave@2	802 Evidently, the stage is now set for a \ldquo{}new\rdquo{} physical
ocsenave@2	803 theory to account for these facts. In the 17th century, both Francis
ocsenave@2	804 Bacon and Isaac Newton had expressed their opinions that heat was a
ocsenave@2	805 form of motion; but they had no supporting factual evidence. By the
ocsenave@2	806 latter part of the 18th century, one had definite factual evidence
ocsenave@2	807 which seemed to make this view untenable; by the calorimetric
ocsenave@2	808 \ldquo{}mixing\rdquo{} experiments just described, Joseph Black had
ocsenave@2	809 recognized the distinction between temperature $t$ as a measure of
ocsenave@2	810 \ldquo{}hotness\rdquo{}, and heat $Q$ as a measure of /quantity/ of
ocsenave@2	811 something, and introduced the notion of heat capacity. He also
ocsenave@2	812 recognized the latent heats of freezing and vaporization. To account
ocsenave@2	813 for the conservation laws thus discovered, the theory then suggested
ocsenave@2	814 itself, naturally and almost inevitably, that heat was /fluid/,
ocsenave@2	815 indestructable and uncreatable, which had no appreciable weight and
ocsenave@2	816 was attracted differently by different kinds of matter. In 1787,
ocsenave@2	817 Lavoisier invented the name \ldquo{}caloric\rdquo{} for this fluid.
ocsenave@2	818
ocsenave@2	819 Looking down today from our position of superior knowledge (i.e.,
ocsenave@2	820 hindsight) we perhaps need to be reminded that the caloric theory was
ocsenave@2	821 a perfectly respectable scientific theory, fully deserving of serious
ocsenave@2	822 consideration; for it accounted quantitatively for a large body of
ocsenave@2	823 experimental fact, and made new predictions capable of being tested by
ocsenave@2	824 experiment.
ocsenave@2	825
ocsenave@2	826 One of these predictions was the possibility of accounting for the
ocsenave@2	827 thermal expansion of bodies when heated; perhaps the increase in
ocsenave@2	828 volume was just a measure of the volume of caloric fluid
ocsenave@2	829 absorbed. This view met with some disappointment as a result of
ocsenave@2	830 experiments which showed that different materials, on absorbing the
ocsenave@2	831 same quantity of heat, expanded by different amounts. Of course, this
ocsenave@2	832 in itself was not enough to overthrow the caloric theory, because one
ocsenave@2	833 could suppose that the caloric fluid was compressible, and was held
ocsenave@2	834 under different pressure in different media.
ocsenave@2	835
ocsenave@2	836 Another difficulty that seemed increasingly serious by the end of the
ocsenave@2	837 18th century was the failure of all attempts to weigh this fluid. Many
ocsenave@2	838 careful experiments were carried out, by Boyle, Fordyce, Rumford and
ocsenave@2	839 others (and continued by Landolt almost into the 20th century), with
ocsenave@2	840 balances capable of detecting a change of weight of one part in a
ocsenave@2	841 million; and no change could be detected on the melting of ice,
ocsenave@2	842 heating of substances, or carrying out of chemical reactions. But even
ocsenave@2	843 this is not really a conclusive argument against the caloric theory,
ocsenave@2	844 since there is no /a priori/ reason why the fluid should be dense
ocsenave@2	845 enough to weigh with balances (of course, we know today from
ocsenave@2	846 Einstein's $E=mc^2$ that small changes in weight should indeed exist
ocsenave@2	847 in these experiments; but to measure them would require balances about
ocsenave@2	848 10^7 times more sensitive than were available).
ocsenave@2	849
ocsenave@2	850 Since the caloric theory derives entirely from the empirical
ocsenave@2	851 conservation law (1-33), it can be refuted conclusively only by
ocsenave@2	852 exhibiting new experimental facts revealing situations in which (1-13)
ocsenave@2	853 is /not/ valid. The first such case was [[http://www.chemteam.info/Chem-History/Rumford-1798.html][found by Count Rumford (1798)]],
ocsenave@2	854 who was in charge of boring cannon in the Munich arsenal, and noted
ocsenave@2	855 that the cannon and chips became hot as a result of the cutting. He
ocsenave@2	856 found that heat could be produced indefinitely, as long as the boring
ocsenave@2	857 was continued, without any compensating cooling of any other part of
ocsenave@2	858 the system. Here, then, was a clear case in which caloric was /not/
ocsenave@2	859 conserved, as in (1-13); but could be created at will. Rumford wrote
ocsenave@2	860 that he could not conceive of anything that could be produced
ocsenave@2	861 indefinitely by the expenditure of work, \ldquo{}except it be /motion/\rdquo{}.
ocsenave@2	862
ocsenave@2	863 But even this was not enough to cause abandonment of the caloric
ocsenave@2	864 theory; for while Rumford's observations accomplished the negative
ocsenave@2	865 purpose of showing that the conservation law (1-13) is not universally
ocsenave@2	866 valid, they failed to accomplish the positive one of showing what
ocsenave@2	867 specific law should replace it (although he produced a good hint, not
ocsenave@2	868 sufficiently appreciated at the time, in his crude measurements of the
ocsenave@2	869 rate of heat production due to the work of one horse). Within the
ocsenave@2	870 range of the original calorimetric experiments, (1-13) was still
ocsenave@2	871 valid, and a theory successful in a restricted domain is better than
ocsenave@2	872 no theory at all; so Rumford's work had very little impact on the
ocsenave@2	873 actual development of thermodynamics.
ocsenave@2	874
ocsenave@2	875 (This situation is a recurrent one in science, and today physics offers
ocsenave@2	876 another good example. It is recognized by all that our present quantum
ocsenave@2	877 field theory is unsatisfactory on logical, conceptual, and
ocsenave@2	878 mathematical grounds; yet it also contains some important truth, and
ocsenave@2	879 no responsible person has suggested that it be abandoned. Once again,
ocsenave@2	880 a semi-satisfactory theory is better than none at all, and we will
ocsenave@2	881 continue to teach it and to use it until we have something better to
ocsenave@2	882 put in its place.)
ocsenave@2	883
ocsenave@2	884 # what is "the specific heat of a gas at constant pressure/volume"?
ocsenave@2	885 # changed t for temperature below from capital T to lowercase t.
ocsenave@3	886 Another failure of the conservation law (1-13) was [[http://web.lemoyne.edu/~giunta/mayer.html][noted in 1842]] by
ocsenave@2	887 R. Mayer, a German physician, who pointed out that the data already
ocsenave@2	888 available showed that the specific heat of a gas at constant pressure,
ocsenave@2	889 C_p, was greater than at constant volume $C_v$. He surmised that the
ocsenave@2	890 difference was due to the work done in expansion of the gas against
ocsenave@2	891 atmospheric pressure, when measuring $C_p$. Supposing that the
ocsenave@2	892 difference $\Delta Q = (C_p - C_v)\Delta t$ calories, in the heat
ocsenave@2	893 required to raise the temperature by $\Delta t$ was actually a
ocsenave@2	894 measure of amount of energy, he could estimate from the amount
ocsenave@2	895 $P\Delta V$ ergs of work done the amount of mechanical energy (number
ocsenave@2	896 of ergs) corresponding to a calorie of heat; but again his work had
ocsenave@2	897 very little impact on the development of thermodynamics, because he
ocsenave@2	898 merely offered this notion as an interpretation of the data without
ocsenave@2	899 performing or suggesting any new experiments to check his hypothesis
ocsenave@2	900 further.
ocsenave@2	901
ocsenave@2	902 Up to the point, then, one has the experimental fact that a
ocsenave@2	903 conservation law (1-13) exists whenever purely thermal interactions
ocsenave@2	904 were involved; but in processes involving mechanical work, the
ocsenave@2	905 conservation law broke down.
ocsenave@2	906
ocsenave@2	907 ** The First Law
ocsenave@3	908 Corresponding to the partially valid law of \ldquo{}conservation of
ocsenave@3	909 heat\rdquo{}, there had long been known another partially valid
ocsenave@3	910 conservation law in mechanics. The principle of conservation of
ocsenave@3	911 mechanical energy had been given by Leibnitz in 1693 in noting that,
ocsenave@3	912 according to the laws of Newtonian mechanics, one could define
ocsenave@3	913 potential and kinetic energy so that in mechanical processes they were
ocsenave@3	914 interconverted into each other, the total energy remaining
ocsenave@3	915 constant. But this too was not universally valid---the mechanical
ocsenave@3	916 energy was conserved only in the absence of frictional forces. In
ocsenave@3	917 processes involving friction, the mechanical energy seemed to
ocsenave@3	918 disappear.
ocsenave@3	919
ocsenave@3	920 So we had a law of conservation of heat, which broke down whenever
ocsenave@3	921 mechanical work was done; and a law of conservation of mechanical
ocsenave@3	922 energy, which broke down when frictional forces were present. If, as
ocsenave@3	923 Mayer had suggested, heat was itself a form of energy, then one had
ocsenave@3	924 the possibility of accounting for both of these failures in a new law
ocsenave@3	925 of conservation of /total/ (mechanical + heat) energy. On one hand,
ocsenave@3	926 the difference $C_p-C_v$ of heat capacities of gases would be
ocsenave@3	927 accounted for by the mechanical work done in expansion; on the other
ocsenave@3	928 hand, the disappearance of mechanical energy would be accounted for by
ocsenave@3	929 the heat produced by friction.
ocsenave@3	930
ocsenave@3	931 But to establish this requires more than just suggesting the idea and
ocsenave@3	932 illustrating its application in one or two cases --- if this is really
ocsenave@3	933 a new conservation law adequate to replace the two old ones, it must
ocsenave@3	934 be shown to be valid for /all/ substances and /all/ kinds of
ocsenave@3	935 interaction. For example, if one calorie of heat corresponded to $E$
ocsenave@3	936 ergs of mechanical energy in the gas experiments, but to a different
ocsenave@3	937 amoun $E^\prime$ in heat produced by friction, then there would be no
ocsenave@3	938 universal conservation law. This \ldquo{}first law\rdquo{} of
ocsenave@3	939 thermodynamics must therefore take the form:
ocsenave@3	940 #+begin_quote
ocsenave@3	941 There exists a /universal/ mechanical equivalent of heat, so that the
ocsenave@3	942 total (mechanical energy) + (heat energy) remeains constant in all
ocsenave@3	943 physical processes.
ocsenave@3	944 #+end_quote
ocsenave@3	945
ocsenave@3	946 It was James Prescott Joule who provided the [[http://www.chemteam.info/Chem-History/Joule-Heat-1845.html][first experimental data]]
ocsenave@3	947 indicating this universality, and providing the first accurate
ocsenave@3	948 numerical value of this mechanical equivalent. The calorie had been
ocsenave@3	949 defined as the amount of heat required to raise the temperature of one
ocsenave@3	950 gram of water by one degree Centigrade (more precisely, to raise it
ocsenave@3	951 from 14.5 to 15.5$^\circ C$). Joule measured the heating of a number
ocsenave@3	952 of different liquids due to mechanical stirring and electrical
ocsenave@3	953 heating, and established that, within the experimental accuracy (about
ocsenave@3	954 one percent) a /calorie/ of heat always corresponded to the same
ocsenave@3	955 amount of energy. Modern measurements give this numerical value as: 1
ocsenave@3	956 calorie = 4.184 \times 10^7 ergs = 4.184 joules.
ocsenave@3	957 # capitalize Joules? I think the convention is to spell them out in lowercase.
ocsenave@3	958
ocsenave@3	959 The circumstances of this important work are worth noting. Joule was
ocsenave@3	960 in frail health as a child, and was educated by private tutors,
ocsenave@3	961 including the chemist, John Dalton, who had formulated the atomic
ocsenave@3	962 hypothesis in the early nineteenth century. In 1839, when Joule was
ocsenave@3	963 nineteen, his father (a wealthy brewer) built a private laboratory for
ocsenave@3	964 him in Manchester, England; and the good use he made of it is shown by
ocsenave@3	965 the fact that, within a few months of the opening of this laboratory
ocsenave@3	966 (1840), he had completed his first important piece of work, at the
ocsenave@3	967 age of twenty. This was his establishment of the law of \ldquo{}Joule
ocsenave@3	968 heating,\rdquo{} $P=I^2 R$, due to the electric current in a
ocsenave@3	969 resistor. He then used this effect to determine the universality and
ocsenave@3	970 numerical value of the mechanical equivalent of heat, reported
ocsenave@3	971 in 1843. His mechanical stirring experiments reported in 1849 yielded
ocsenave@3	972 the value 1 calorie = 4.154 \times 10^7 ergs, amount 0.7% too low;
ocsenave@3	973 this determination was not improved upon for several decades.
ocsenave@3	974
ocsenave@3	975 The first law of thermodynamics may then be stated mathematically as
ocsenave@3	976 follows:
ocsenave@3	977
ocsenave@3	978 #+begin_quote
ocsenave@3	979 There exists a state function (i.e., a definite function of the
ocsenave@3	980 thermodynamic state) $U$, representing the total energy of any system,
ocsenave@3	981 such that in any process in which we change from one equilibrium to
ocsenave@3	982 another, the net change in $U$ is given by the difference of the heat
ocsenave@3	983 $Q$ supplied to the system, and the mechanical work $W$ done by the
ocsenave@3	984 system.
ocsenave@3	985 #+end_quote
ocsenave@3	986 On an infinitesimal change of state, this becomes
ocsenave@3	987
ocsenave@3	988 \begin{equation}
ocsenave@3	989 dU = dQ - dW.
ocsenave@3	990 \end{equation}
ocsenave@3	991
ocsenave@3	992 For a system of two degrees of freedom, defined by pressure $P$,
ocsenave@3	993 volume $V$, and temperature $t$, we have $dW = PdV$. Then if we regard
ocsenave@3	994 $U$ as a function $U(V,t)$ of volume and temperature, the fact that
ocsenave@3	995 $U$ is a state function means that $dU$ must be an exact differential;
ocsenave@3	996 i.e., the integral
ocsenave@3	997
ocsenave@3	998 \begin{equation}
ocsenave@3	999 \int_1^2 dU = U(V_2,t_2) - U(V_1,t_1)
ocsenave@3	1000 \end{equation}
ocsenave@3	1001 between any two thermodynamic states must be independent of the
ocsenave@3	1002 path. Equivalently, the integral $\oint dU$ over any closed cyclic
ocsenave@3	1003 path (for example, integrate from state 1 to state 2 along path A,
ocsenave@3	1004 then back to state 1 by a different path B) must be zero. From (1-15),
ocsenave@3	1005 this gives for any cyclic integral,
ocsenave@3	1006
ocsenave@3	1007 \begin{equation}
ocsenave@3	1008 \oint dQ = \oint P dV
ocsenave@3	1009 \end{equation}
ocsenave@3	1010
ocsenave@3	1011 another form of the first law, which states that in any process in
ocsenave@3	1012 which the system ends in the same thermodynamic state as the initial
ocsenave@3	1013 one, the total heat absorbed by the system must be equal to the total
ocsenave@3	1014 work done.
ocsenave@3	1015
ocsenave@3	1016 Although the equations (1-15)-(1-17) are rather trivial
ocsenave@3	1017 mathematically, it is important to avoid later conclusions that we
ocsenave@3	1018 understand their exact meaning. In the first place, we have to
ocsenave@3	1019 understand that we are now measuring heat energy and mechanical energy
ocsenave@3	1020 in the same units; i.e. if we measured $Q$ in calories and $W$ in
ocsenave@3	1021 ergs, then (1-15) would of course not be correct. It does
ocsenave@3	1022 not matter whether we apply Joule's mechanical equivalent of heat
ocsenave@3	1023 to express $Q$ in ergs, or whether we apply it in the opposite way
ocsenave@3	1024 to express $U$ and $W$ in calories; each procedure will be useful in
ocsenave@3	1025 various problems. We can develop the general equations of
ocsenave@3	1026 thermodynamics
ocsenave@3	1027 without committing ourselves to any particular units,
ocsenave@3	1028 but of course all terms in a given equation must be expressed
ocsenave@3	1029 in the same units.
ocsenave@3	1030
ocsenave@3	1031 Secondly, we have already stressed that the theory being
ocsenave@3	1032 developed must, strictly speaking, be a theory only of
ocsenave@3	1033 equilibrium states, since otherwise we have no operational definition
ocsenave@4	1034 of temperature When we integrate over any \ldquo{}path\rdquo{} in the $(V-t)$
ocsenave@3	1035 plane, therefore, it must be understood that the path of
ocsenave@3	1036 integration is, strictly speaking, just a /locus of equilibrium
ocsenave@3	1037 states/; nonequilibrium states cannot be represented by points
ocsenave@3	1038 in the $(V-t)$ plane.
ocsenave@3	1039
ocsenave@3	1040 But then, what is the relation between path of equilibrium
ocsenave@3	1041 states appearing in our equations, and the sequence of conditions
ocsenave@3	1042 produced experimentally when we change the state of a system in
ocsenave@3	1043 the laboratory? With any change of state (heating, compression,
ocsenave@3	1044 etc.) proceeding at a finite rate we do not have equilibrium in
ocsenave@3	1045 termediate states; and so there is no corresponding \ldquo{}path\rdquo{} in
ocsenave@3	1046 the $(V-t)$ plane ; only the initial and final equilibrium states
ocsenave@3	1047 correspond to definite points. But if we carry out the change
ocsenave@3	1048 of state more and more slowly, the physical states produced are
ocsenave@3	1049 nearer and nearer to equilibrium state. Therefore, we interpret
ocsenave@3	1050 a path of integration in the $(V-t)$ plane, not as representing
ocsenave@3	1051 the intermediate states of any real experiment carried out at
ocsenave@3	1052 a finite rate, but as the /limit/ of this sequence of states, in
ocsenave@3	1053 the limit where the change of state takes place arbitrarily
ocsenave@3	1054 slowly.
ocsenave@3	1055
ocsenave@3	1056 An arbitrarily slow process, so that we remain arbitrarily
ocsenave@3	1057 near to equilibrium at all times, has another important property.
ocsenave@3	1058 If heat is flowing at an arbitrarily small rate, the temperature
ocsenave@3	1059 difference producing it must be arbitrarily small, and therefore
ocsenave@3	1060 an arbitrarily small temperature change would be able to reverse
ocsenave@3	1061 the direction of heat flow. If the Volume is changing very
ocsenave@3	1062 slowly, the pressure difference responsible for it must be very
ocsenave@3	1063 small; so a small change in pressure would be able to reverse
ocsenave@3	1064 the direction of motion. In other words, a process carried out
ocsenave@3	1065 arbitrarily slowly is /reversible/; if a system is arbitrarily
ocsenave@3	1066 close to equilibrium, then an arbitrarily small change in its
ocsenave@3	1067 environment can reverse the direction of the process.
ocsenave@3	1068 Recognizing this, we can then say that the paths of integra
ocsenave@3	1069 tion in our equations are to be interpreted physically as
ocsenave@4	1070 /reversible paths/ In practice, some systems (such as gases)
ocsenave@3	1071 come to equilibrium so rapidly that rather fast changes of
ocsenave@3	1072 state (on the time scale of our own perceptions) may be quite
ocsenave@3	1073 good approximations to reversible changes; thus the change of
ocsenave@3	1074 state of water vapor in a steam engine may be considered
ocsenave@3	1075 reversible to a useful engineering approximation.
ocsenave@3	1076
ocsenave@3	1077
ocsenave@3	1078 ** Intensive and Extensive Parameters
ocsenave@3	1079
ocsenave@3	1080 The literature of thermodynamics has long recognized a distinction between two
ocsenave@3	1081 kinds of quantities that may be used to define the thermodynamic
ocsenave@3	1082 state. If we imagine a given system as composed of smaller
ocsenave@3	1083 subsystems, we usually find that some of the thermodynamic variables
ocsenave@3	1084 have the same values in each subsystem, while others are additive,
ocsenave@3	1085 the total amount being the sum of the values of each subsystem.
ocsenave@3	1086 These are called /intensive/ and /extensive/ variables, respectively.
ocsenave@3	1087 According to this definition, evidently, the mass of a system is
ocsenave@3	1088 always an extensive quantity, and at equilibrium the temperature
ocsenave@3	1089 is an intensive ‘quantity. Likewise, the energy will be extensive
ocsenave@3	1090 provided that the interaction energy between the subsystems can
ocsenave@3	1091 be neglected.
ocsenave@3	1092
ocsenave@3	1093 It is important to note, however, that in general the terms
ocsenave@3	1094 \ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{}
ocsenave@3	1095 so defined cannot be regarded as
ocsenave@3	1096 establishing a real physical distinction between the variables.
ocsenave@3	1097 This distinction is, like the notion of number of degrees of
ocsenave@3	1098 freedom, in part an anthropomorphic one, because it may depend
ocsenave@3	1099 on the particular kind of subdivision we choose to imagine. For
ocsenave@3	1100 example, a volume of air may be imagined to consist of a number
ocsenave@3	1101 of smaller contiguous volume elements. With this subdivision,
ocsenave@3	1102 the pressure is the same in all subsystems, and is therefore in
ocsenave@4	1103 tensive; while the volume is additive and therefore extensive
ocsenave@3	1104 But we may equally well regard the volume of air as composed of
ocsenave@3	1105 its constituent nitrogen and oxygen subsystems (or we could re
ocsenave@3	1106 gard pure hydrogen as composed of two subsystems, in which the
ocsenave@3	1107 molecules have odd and even rotational quantum numbers
ocsenave@4	1108 respectively, etc.) With this kind of subdivision the volume is the
ocsenave@3	1109 same in all subsystems, while the pressure is the sum of the
ocsenave@3	1110 partial pressures of its constituents; and it appears that the
ocsenave@3	1111 roles of \ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{}
ocsenave@3	1112 have been interchanged. Note that this ambiguity cannot be removed by requiring
ocsenave@3	1113 that we consider only spatial subdivisions, such that each sub
ocsenave@4	1114 system has the same local composi tion For, consider a s tressed
ocsenave@3	1115 elastic solid, such as a stretched rubber band. If we imagine
ocsenave@3	1116 the rubber band as divided, conceptually, into small subsystems
ocsenave@3	1117 by passing planes through it normal to its axis, then the tension
ocsenave@3	1118 is the same in all subsystems, while the elongation is additive.
ocsenave@3	1119 But if the dividing planes are parallel to the axis, the elonga
ocsenave@3	1120 tion is the same in all subsystems, while the tension is
ocsenave@3	1121 additive; once again, the roles of \ldquo{}extensive\rdquo{} and
ocsenave@3	1122 \ldquo{}intensive\rdquo{} are
ocsenave@3	1123 interchanged merely by imagining a different kind of subdivision.
ocsenave@3	1124 In spite of the fundamental ambiguity of the usual definitions,
ocsenave@3	1125 the notions of extensive and intensive variables are useful,
ocsenave@3	1126 and in practice we seem to have no difficulty in deciding
ocsenave@3	1127 which quantities should be considered intensive. Perhaps the
ocsenave@3	1128 distinction is better characterized, not by considering
ocsenave@3	1129 subdivisions at all, but by adopting a different definition, in which
ocsenave@3	1130 we recognize that some quantities have the nature of a \ldquo{}force\rdquo{}
ocsenave@3	1131 or \ldquo{}potential\rdquo{}, or some other local physical property, and are
ocsenave@3	1132 therefore called intensive, while others have the nature of a
ocsenave@3	1133 \ldquo{}displacement\rdquo{} or a \ldquo{}quantity\rdquo{} of
ocsenave@3	1134 something (i.e. are proportional to the size of the system),
ocsenave@3	1135 and are therefore called extensive. Admittedly, this definition is somewhat vague, in a
ocsenave@3	1136 way that can also lead to ambiguities ; in any event, let us agree
ocsenave@3	1137 to class pressure, stress tensor, mass density, energy density,
ocsenave@3	1138 particle density, temperature, chemical potential, angular
ocsenave@3	1139 velocity, as intensive, while volume, mass, energy, particle
ocsenave@3	1140 numbers, strain, entropy, angular momentum, will be considered
ocsenave@3	1141 extensive.
ocsenave@3	1142
ocsenave@3	1143 ** The Kelvin Temperature Scale
ocsenave@3	1144 The form of the first law,
ocsenave@3	1145 $dU = dQ - dW$, expresses the net energy increment of a system as
ocsenave@3	1146 the heat energy supplied to it, minus the work done by it. In
ocsenave@3	1147 the simplest systems of two degrees of freedom, defined by
ocsenave@3	1148 pressure and volume as the thermodynamic variables, the work done
ocsenave@3	1149 in an infinitesimal reversible change of state can be separated
ocsenave@3	1150 into a product $dW = PdV$ of an intensive and an extensive quantity.
ocsenave@3	1151 Furthermore, we know that the pressure $P$ is not only the
ocsenave@3	1152 intensive factor of the work; it is also the \ldquo{}potential\rdquo{}
ocsenave@3	1153 which governs mechanical equilibrium (in this case, equilibrium with respect
ocsenave@4	1154 to exchange of volume) between two systems; ie., if they are
ocsenave@3	1155 separated by a flexible but impermeable membrane, the two systems
ocsenave@3	1156 will exchange volume $dV_1 = -dV_2$ in a direction determined by the
ocsenave@3	1157 pressure difference, until the pressures are equalized. The
ocsenave@3	1158 energy exchanged in this way between the systems is a product
ocsenave@3	1159 of the form
ocsenave@3	1160 #+begin_quote
ocsenave@3	1161 (/intensity/ of something) \times (/quantity/ of something exchanged)
ocsenave@3	1162 #+end_quote
ocsenave@3	1163
ocsenave@3	1164 Now if heat is merely a particular form of energy that can
ocsenave@3	1165 also be exchanged between systems, the question arises whether
ocsenave@3	1166 the quantity of heat energy $dQ$ exchanged in an infinitesimal
ocsenave@3	1167 reversible change of state can also be written as a product of one
ocsenave@3	1168 factor which measures the \ldquo{}intensity\rdquo{} of the heat,
ocsenave@3	1169 times another that represents the \ldquo{}quantity\rdquo{}
ocsenave@3	1170 of something exchanged between
ocsenave@3	1171 the systems, such that the intensity factor governs the
ocsenave@3	1172 conditions of thermal equilibrium and the direction of heat exchange,
ocsenave@3	1173 in the same way that pressure does for volume exchange.
ocsenave@3	1174
ocsenave@3	1175
ocsenave@3	1176 But we already know that the /temperature/ is the quantity
ocsenave@3	1177 that governs the heat flow (i.e., heat flows from the hotter to
ocsenave@4	1178 the cooler body until the temperatures are equalized) So the
ocsenave@3	1179 intensive factor in $dQ$ must be essentially the temperature. But
ocsenave@3	1180 our temperature scale is at present still arbitrary, and we can
ocsenave@3	1181 hardly expect that such a factorization will be possible for all
ocsenave@3	1182 calibrations of our thermometers.
ocsenave@3	1183
ocsenave@3	1184 The same thing is evidently true of pressure; if instead of
ocsenave@3	1185 the pressure $P$ as ordinarily defined, we worked with any mono
ocsenave@3	1186 tonic increasing function $P_1 = P_1 (P)$ we would find that $P_1$ is
ocsenave@3	1187 just as good as $P$ for determining the direction of volume
ocsenave@3	1188 exchange and the condition of mechanical equilibrium; but the work
ocsenave@3	1189 done would not be given by $PdV$; in general, it could not even
ocsenave@3	1190 be expressed in the form $P_1 \cdot dF(V)$, where $F(V)$ is some function
ocsenave@3	1191 of V.
ocsenave@3	1192
ocsenave@3	1193
ocsenave@3	1194 Therefore we ask: out of all the monotonic functions $t_1(t)$
ocsenave@3	1195 corresponding to different empirical temperature scales, is
ocsenave@3	1196 there one (which we denote as $T(t)$) which forms a \ldquo{}natural\rdquo{}
ocsenave@3	1197 intensity factor for heat, such that in a reversible change
ocsenave@3	1198 $dQ = TdS$, where $S(U,V)$ is a new function of the thermodynamic
ocsenave@3	1199 state? If so, then the temperature scale $T$ will have a great
ocsenave@3	1200 theoretical advantage, in that the laws of thermodynamics will
ocsenave@3	1201 take an especially simple form in terms of this particular scale,
ocsenave@3	1202 and the new quantity $S$, which we call the /entropy/, will be a
ocsenave@3	1203 kind of \ldquo{}volume\rdquo{} factor for heat.
ocsenave@3	1204
ocsenave@3	1205 We recall that $dQ = dU + PdV$ is not an exact differential;
ocsenave@3	1206 i.e., on a change from one equilibrium state to another the
ocsenave@3	1207 integral
ocsenave@3	1208
ocsenave@3	1209 \[\int_1^2 dQ\]
ocsenave@3	1210
ocsenave@3	1211 cannot be set equal to the difference $Q_2 - Q_1$ of values of any
ocsenave@3	1212 state function $Q(U,V)$, since the integral has different values
ocsenave@3	1213 for different paths connecting the same initial and final states.
ocsenave@3	1214 Thus there is no \ldquo{}heat function\rdquo{} $Q(U,V)$, and the notion of
ocsenave@3	1215 \ldquo{}amount of heat\rdquo{} $Q$ stored in a body has no meaning
ocsenave@3	1216 (nor does the \ldquo{}amount of work\rdquo{} $W$;
ocsenave@3	1217 only the total energy is a well-defined quantity).
ocsenave@3	1218 But we want the entropy $S(U,V)$ to be a definite quantity,
ocsenave@3	1219 like the energy or volume, and so $dS$ must be an exact differential.
ocsenave@3	1220 On an infinitesimal reversible change from one equilibrium state
ocsenave@4	1221 to another, the first law requires that it satisfy[fn:: Edit: The first
ocsenave@3	1222 equality comes from our requirement that $dQ = T\,dS$. The second
ocsenave@3	1223 equality comes from the fact that $dU = dQ - dW$ (the first law) and
ocsenave@3	1224 that $dW = PdV$ in the case where the state has two degrees of
ocsenave@3	1225 freedom, pressure and volume.]
ocsenave@3	1226
ocsenave@3	1227 \begin{equation}
ocsenave@3	1228 dS(U,V) = \frac{dQ}{T} = \frac{dU}{T} + \frac{P}{T}dV
ocsenave@3	1229 \end{equation}
ocsenave@3	1230
ocsenave@3	1231 Thus $(1/T)$ must be an /integrating factor/ which converts $dQ$ into
ocsenave@4	1232 an exact differential [[fn::Edit: A differential $M(x,y)dx +
ocsenave@3	1233 N(x,y)dy$ is called /exact/ if there is a scalar function
ocsenave@3	1234 $\Phi(x,y)$ such that $M = \frac{\partial \Phi}{\partial x}$ and
ocsenave@3	1235 $N=\frac{\partial \Phi}{\partial y}$. If there is, \Phi is called the
ocsenave@3	1236 /potential function/ of the differential, Conceptually, this means
ocsenave@3	1237 that M(x,y)dx + N(x,y) dy is the derivative of a scalar potential and
ocsenave@3	1238 so consequently corresponds to a conservative field.
ocsenave@3	1239
ocsenave@3	1240 Even if there is no such potential function
ocsenave@3	1241 \Phi for the given differential, it is possible to coerce an
ocsenave@3	1242 inexact differential into an exact one by multiplying by an unknown
ocsenave@3	1243 function $\mu(x,y)$ (called an /integrating factor/) and requiring the
ocsenave@3	1244 resulting differential $\mu M\, dx + \mu N\, dy$ to be exact.
ocsenave@3	1245
ocsenave@3	1246 To complete the analogy, here we have the differential $dQ =
ocsenave@3	1247 dU + PdV$ (by the first law) which is not exact---conceptually, there
ocsenave@3	1248 is no scalar potential nor conserved quantity corresponding to
ocsenave@3	1249 $dQ$. We have introduced a new differential $dS = \frac{1}{T}dQ$, and we
ocsenave@3	1250 are searching for the temperature scale $T(U,V)$ which makes $dS$
ocsenave@3	1251 exact (i.e. which makes $S$ correspond to a conserved quantity). This means
ocsenave@3	1252 that $\frac{1}{T}$ is playing the role of the integrating factor
ocsenave@3	1253 \ldquo{}\mu\rdquo{} for the differential $dQ$.]]
ocsenave@3	1254
ocsenave@3	1255 Now the question of the existence and properties of
ocsenave@3	1256 integrating factors is a purely mathematical one, which can be
ocsenave@3	1257 investigated independently of the properties of any particular
ocsenave@3	1258 substance. Let us denote this integrating factor for the moment
ocsenave@3	1259 by $w(U,V) = T^{-1}$; then the first law becomes
ocsenave@3	1260
ocsenave@3	1261 \begin{equation}
ocsenave@3	1262 dS(U,V) = w dU + w P dV
ocsenave@3	1263 \end{equation}
ocsenave@3	1264
ocsenave@3	1265 from which the derivatives are
ocsenave@3	1266
ocsenave@3	1267 \begin{equation}
ocsenave@3	1268 \left(\frac{\partial S}{\partial U}\right)_V = w, \qquad
ocsenave@3	1269 \left(\frac{\partial S}{\partial V}\right)_U = wP.
ocsenave@3	1270 \end{equation}
ocsenave@3	1271
ocsenave@3	1272 The condition that $dS$ be exact is that the cross-derivatives be
ocsenave@3	1273 equal, as in (1-4):
ocsenave@3	1274
ocsenave@3	1275 \begin{equation}
ocsenave@3	1276 \frac{\partial^2 S}{\partial U \partial V} = \frac{\partial^2
ocsenave@3	1277 S}{\partial V \partial U},
ocsenave@3	1278 \end{equation}
ocsenave@3	1279
ocsenave@3	1280 or
ocsenave@3	1281
ocsenave@3	1282 \begin{equation}
ocsenave@3	1283 \left(\frac{\partial w}{\partial V}\right)_U = \left(\frac{\partial
ocsenave@3	1284 P}{\partial U}\right)_V + P\cdot \left(\frac{\partial w}{\partial U}\right)_V.
ocsenave@3	1285 \end{equation}
ocsenave@3	1286
ocsenave@3	1287 Any function $w(U,V)$ satisfying this differential equation is an
ocsenave@3	1288 integrating factor for $dQ$.
ocsenave@3	1289
ocsenave@3	1290 But if $w(U,V)$ is one such integrating factor, which leads
ocsenave@3	1291 to the new state function $S(U,V)$, it is evident that
ocsenave@3	1292 $w_1(U,V) \equiv w \cdot f(S)$ is an equally good integrating factor, where
ocsenave@3	1293 $f(S)$ is an arbitrary function. Use of $w_1$ will lead to a
ocsenave@3	1294 different state function
ocsenave@3	1295
ocsenave@3	1296 #what's with the variable collision?
ocsenave@3	1297 \begin{equation}
ocsenave@3	1298 S_1(U,V) = \int^S f(S) dS
ocsenave@3	1299 \end{equation}
ocsenave@3	1300
ocsenave@3	1301 The mere conversion of into an exact differential is, therefore,
ocsenave@3	1302 not enough to determine any unique entropy function $S(U,V)$.
ocsenave@3	1303 However, the derivative
ocsenave@3	1304
ocsenave@3	1305 \begin{equation}
ocsenave@3	1306 \left(\frac{dU}{dV}\right)_S = -P
ocsenave@3	1307 \end{equation}
ocsenave@3	1308
ocsenave@3	1309 is evidently uniquely determined; so also, therefore, is the
ocsenave@3	1310 family of lines of constant entropy, called /adiabats/, in the
ocsenave@3	1311 $(U-V)$ plane. But, as (1-24) shows, the numerical value of $S$ on
ocsenave@3	1312 each adiabat is still completely undetermined.
ocsenave@3	1313
ocsenave@3	1314 In order to fix the relative values of $S$ on different
ocsenave@3	1315 adiabats we need to add the condition, not yet put into the equations,
ocsenave@3	1316 that the integrating factor $w(U,V) = T^{-1}$ is to define a new
ocsenave@4	1317 temperature scale In other words, we now ask: out of the
ocsenave@3	1318 infinite number of different integrating factors allowed by
ocsenave@3	1319 the differential equation (1-23), is it possible to find one
ocsenave@3	1320 which is a function only of the empirical temperature $t$? If
ocsenave@3	1321 $w=w(t)$, we can write
ocsenave@3	1322
ocsenave@3	1323 \begin{equation}
ocsenave@3	1324 \left(\frac{\partial w}{\partial V}\right)_U = \frac{dw}{dt}\left(\frac{\partial
ocsenave@3	1325 t}{\partial V}\right)_U
ocsenave@3	1326 \end{equation}
ocsenave@3	1327 \begin{equation}
ocsenave@3	1328 \left(\frac{\partial w}{\partial U}\right)_V = \frac{dw}{dt}\left(\frac{\partial
ocsenave@3	1329 t}{\partial U}\right)_V
ocsenave@3	1330 \end{equation}
ocsenave@3	1331
ocsenave@3	1332
ocsenave@3	1333 and (1-23) becomes
ocsenave@3	1334 \begin{equation}
ocsenave@3	1335 \frac{d}{dt}\log{w} = \frac{\left(\frac{\partial P}{\partial
ocsenave@3	1336 U}\right)_V}{\left(\frac{\partial t}{\partial V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V}
ocsenave@3	1337 \end{equation}
ocsenave@3	1338
ocsenave@3	1339
ocsenave@3	1340 which shows that $w$ will be determined to within a multiplicative
ocsenave@3	1341 factor.
ocsenave@3	1342
ocsenave@3	1343 Is the temperature scale thus defined independent of the
ocsenave@3	1344 empirical scale from which we started? To answer this, let
ocsenave@3	1345 $t_1 = t_1(t)$ be any monotonic function which defines a different
ocsenave@3	1346 empirical temperature scale. In place of (1-28), we then have
ocsenave@3	1347
ocsenave@3	1348 \begin{equation}
ocsenave@3	1349 \frac{d}{dt_1}\log{w} \quad=\quad \frac{\left(\frac{\partial P}{\partial
ocsenave@3	1350 U}\right)_V}{\left(\frac{\partial t_1}{\partial V}\right)_U-P\left(\frac{\partial t_1}{\partial U}\right)_V}
ocsenave@3	1351 \quad = \quad
ocsenave@3	1352 \frac{\left(\frac{\partial P}{\partial
ocsenave@3	1353 U}\right)_V}{\frac{dt_1}{dt}\left[ \left(\frac{\partial t}{\partial
ocsenave@3	1354 V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V\right]},
ocsenave@3	1355 \end{equation}
ocsenave@3	1356 or
ocsenave@3	1357 \begin{equation}
ocsenave@3	1358 \frac{d}{dt_1}\log{w_1} = \frac{dt}{dt_1}\frac{d}{dt}\log{w}
ocsenave@3	1359 \end{equation}
ocsenave@3	1360
ocsenave@3	1361 which reduces to $d \log{w_1} = d \log{w}$, or
ocsenave@3	1362 \begin{equation}
ocsenave@3	1363 w_1 = C\cdot w
ocsenave@3	1364 \end{equation}
ocsenave@3	1365
ocsenave@3	1366 Therefore, integrating factors derived from whatever empirical
ocsenave@3	1367 temperature scale can differ among themselves only by a
ocsenave@3	1368 multiplicative factor. For any given substance, therefore, except
ocsenave@3	1369 for this factor (which corresponds just to our freedom to choose
ocsenave@3	1370 the size of the units in which we measure temperature), there is
ocsenave@3	1371 only /one/ temperature scale $T(t) = 1/w$ with the property that
ocsenave@3	1372 $dS = dQ/T$ is an exact differential.
ocsenave@3	1373
ocsenave@3	1374 To find a feasible way of realizing this temperature scale
ocsenave@3	1375 experimentally, multiply numerator and denominator of the right
ocsenave@3	1376 hand side of (1-28) by the heat capacity at constant volume,
ocsenave@3	1377 $C_V^\prime = (\partial U/\partial t) V$, the prime denoting that
ocsenave@3	1378 it is in terms of the empirical temperature scale $t$.
ocsenave@3	1379 Integrating between any two states denoted 1 and 2, we have
ocsenave@3	1380
ocsenave@3	1381 \begin{equation}
ocsenave@3	1382 \frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}
ocsenave@3	1383 \frac{\left(\frac{\partial P}{\partial t}\right)_V dt}{P - C_V^\prime
ocsenave@3	1384 \left(\frac{\partial t}{\partial V}\right)_U} \right\}
ocsenave@3	1385 \end{equation}
ocsenave@3	1386
ocsenave@3	1387 If the quantities on the right-hand side have been determined
ocsenave@3	1388 experimentally, then a numerical integration yields the ratio
ocsenave@3	1389 of Kelvin temperatures of the two states.
ocsenave@3	1390
ocsenave@3	1391 This process is particularly simple if we choose for our
ocsenave@3	1392 system a volume of gas with the property found in Joule's famous
ocsenave@3	1393 expansion experiment; when the gas expands freely into a vacuum
ocsenave@3	1394 (i.e., without doing work, or $U = \text{const.}$), there is no change in
ocsenave@3	1395 temperature. Real gases when sufficiently far from their condensation
ocsenave@3	1396 points are found to obey this rule very accurately.
ocsenave@3	1397 But then
ocsenave@3	1398
ocsenave@3	1399 \begin{equation}
ocsenave@3	1400 \left(\frac{dt}{dV}\right)_U = 0
ocsenave@3	1401 \end{equation}
ocsenave@3	1402
ocsenave@3	1403 and on a change of state in which we heat this gas at constant
ocsenave@3	1404 volume, (1-31) collapses to
ocsenave@3	1405
ocsenave@3	1406 \begin{equation}
ocsenave@3	1407 \frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}
ocsenave@3	1408 \frac{1}{P}\left(\frac{\partial P}{\partial t}\right)_V dt\right\} = \frac{P_2}{P_1}.
ocsenave@3	1409 \end{equation}
ocsenave@3	1410
ocsenave@3	1411 Therefore, with a constant-volume ideal gas thermometer, (or more
ocsenave@3	1412 generally, a thermometer using any substance obeying (1-32) and
ocsenave@3	1413 held at constant volume), the measured pressure is directly
ocsenave@3	1414 proportional to the Kelvin temperature.
ocsenave@3	1415
ocsenave@3	1416 For an imperfect gas, if we have measured $(\partial t /\partial
ocsenave@3	1417 V)_U$ and $C_V^\prime$, Eq. (1-31) determines the necessary
ocsenave@3	1418 corrections to (1-33). However, an alternative form of (1-31), in
ocsenave@3	1419 which the roles of pressure and volume are interchanged, proves to be
ocsenave@3	1420 more convenient for experimental determinations. To derive it, introduce the
ocsenave@3	1421 enthalpy function
ocsenave@3	1422
ocsenave@3	1423 \begin{equation}H = U + PV\end{equation}
ocsenave@3	1424
ocsenave@3	1425 with the property
ocsenave@3	1426
ocsenave@3	1427 \begin{equation}
ocsenave@3	1428 dH = dQ + VdP
ocsenave@3	1429 \end{equation}
ocsenave@3	1430
ocsenave@3	1431 Equation (1-19) then becomes
ocsenave@3	1432
ocsenave@3	1433 \begin{equation}
ocsenave@3	1434 dS = \frac{dH}{T} - \frac{V}{T}dP.
ocsenave@3	1435 \end{equation}
ocsenave@3	1436
ocsenave@3	1437 Repeating the steps (1-20) to (1-31) of the above derivation
ocsenave@3	1438 starting from (1-36) instead of from (1-19), we arrive at
ocsenave@3	1439
ocsenave@3	1440 \begin{equation}
ocsenave@3	1441 \frac{T_2}{T_1} = \exp\left\{\int_{t_1}^{t_2}
ocsenave@3	1442 \frac{\left(\frac{dV}{dt}\right)_P dt}{V + C_P^\prime
ocsenave@3	1443 \left(\frac{\partial t}{\partial P}\right)_H}\right\}
ocsenave@3	1444 \end{equation}
ocsenave@3	1445
ocsenave@3	1446 or
ocsenave@3	1447
ocsenave@3	1448 \begin{equation}
ocsenave@3	1449 \frac{T_2}{T_1} = \exp\left\{\frac{\alpha^\prime
ocsenave@3	1450 dt}{1+\left(C_P^\prime \cdot \mu^\prime / V\right)}\right\}
ocsenave@3	1451 \end{equation}
ocsenave@3	1452
ocsenave@3	1453 where
ocsenave@3	1454 \begin{equation}
ocsenave@3	1455 \alpha^\prime \equiv \frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P
ocsenave@3	1456 \end{equation}
ocsenave@3	1457 is the thermal expansion coefficient,
ocsenave@3	1458 \begin{equation}
ocsenave@3	1459 C_P^\prime \equiv \left(\frac{\partial H}{\partial t}\right)_P
ocsenave@3	1460 \end{equation}
ocsenave@3	1461 is the heat capacity at constant pressure, and
ocsenave@3	1462 \begin{equation}
ocsenave@3	1463 \mu^\prime \equiv \left(\frac{dt}{dP}\right)_H
ocsenave@3	1464 \end{equation}
ocsenave@3	1465
ocsenave@3	1466 is the coefficient measured in the Joule-Thompson porous plug
ocsenave@3	1467 experiment, the primes denoting again that all are to be measured
ocsenave@3	1468 in terms of the empirical temperature scale $t$.
ocsenave@3	1469 Since $\alpha^\prime$, $C_P^\prime$, $\mu^\prime$ are all
ocsenave@3	1470 easily measured in the laboratory, Eq. (1-38) provides a
ocsenave@3	1471 feasible way of realizing the Kelvin temperature scale experimentally,
ocsenave@3	1472 taking account of the imperfections of real gases.
ocsenave@3	1473 For an account of the work of Roebuck and others based on this
ocsenave@3	1474 relation, see [[http://books.google.com/books?id=KKJKAAAAMAAJ][Zemansky (1943)]]; pp. 252-255.
ocsenave@3	1475
ocsenave@3	1476 Note that if $\mu^\prime = O$ and we heat the gas at constant
ocsenave@3	1477 pressure, (1-38) reduces to
ocsenave@3	1478
ocsenave@3	1479 \begin{equation}
ocsenave@3	1480 \frac{T_2}{T_1} = \exp\left\{ \int_{t_1}^{t_2}
ocsenave@3	1481 \frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P dt \right\} = \frac{V_2}{V_1}
ocsenave@3	1482 \end{equation}
ocsenave@3	1483
ocsenave@3	1484 so that, with a constant-pressure gas thermometer using a gas for
ocsenave@3	1485 which the Joule-Thomson coefficient is zero, the Kelvin temperature is
ocsenave@3	1486 proportional to the measured volume.
ocsenave@3	1487
ocsenave@3	1488 Now consider another empirical fact, [[http://en.wikipedia.org/wiki/Boyle%27s_law][Boyle's law]]. For gases
ocsenave@3	1489 sufficiently far from their condensation points---which is also
ocsenave@3	1490 the condition under which (1-32) is satisfied---Boyle found that
ocsenave@3	1491 the product $PV$ is a constant at any fixed temperature. This
ocsenave@3	1492 product is, of course proportional to the number of moles $n$
ocsenave@3	1493 present, and so Boyle's equation of state takes the form
ocsenave@3	1494
ocsenave@3	1495 \begin{equation}PV = n \cdot f(t)\end{equation}
ocsenave@3	1496
ocsenave@3	1497 where f(t) is a function that depends on the particular empirical
ocsenave@3	1498 temperature scale used. But from (1-33) we must then have
ocsenave@3	1499 $f(t) = RT$, where $R$ is a constant, the universal gas constant whose
ocsenave@4	1500 numerical value (1.986 calories per mole per degree K), depends
ocsenave@3	1501 on the size of the units in which we choose to measure the Kelvin
ocsenave@3	1502 temperature $T$. In terms of the Kelvin temperature, the ideal gas
ocsenave@3	1503 equation of state is therefore simply
ocsenave@3	1504
ocsenave@3	1505 \begin{equation}
ocsenave@3	1506 PV = nRT
ocsenave@3	1507 \end{equation}
ocsenave@3	1508
ocsenave@3	1509
ocsenave@3	1510 The relations (1-32) and (1-44) were found empirically, but
ocsenave@3	1511 with the development of thermodynamics one could show that they
ocsenave@3	1512 are not logically independent. In fact, all the material needed
ocsenave@3	1513 for this demonstration is now at hand, and we leave it as an
ocsenave@3	1514 exercise for the reader to prove that Joule‘s relation (1-32) is
ocsenave@3	1515 a logical consequence of Boyle's equation of state (1-44) and the
ocsenave@3	1516 first law.
ocsenave@3	1517
ocsenave@3	1518
ocsenave@3	1519 Historically, the advantages of the gas thermometer were
ocsenave@3	1520 discovered empirically before the Kelvin temperature scale was
ocsenave@3	1521 defined; and the temperature scale \theta defined by
ocsenave@3	1522
ocsenave@3	1523 \begin{equation}
ocsenave@3	1524 \theta = \lim_{P\rightarrow 0}\left(\frac{PV}{nR}\right)
ocsenave@3	1525 \end{equation}
ocsenave@3	1526
ocsenave@3	1527 was found to be convenient, easily reproducible, and independent
ocsenave@3	1528 of the properties of any particular gas. It was called the
ocsenave@3	1529 /absolute/ temperature scale; and from the foregoing it is clear
ocsenave@3	1530 that with the same choice of the numerical constant $R$, the
ocsenave@3	1531 absolute and Kelvin scales are identical.
ocsenave@3	1532
ocsenave@3	1533
ocsenave@3	1534 For many years the unit of our temperature scale was the
ocsenave@3	1535 Centigrade degree, so defined that the difference $T_b - T_f$ of
ocsenave@3	1536 boiling and freezing points of water was exactly 100 degrees.
ocsenave@3	1537 However, improvements in experimental techniques have made another
ocsenave@3	1538 method more reproducible; and the degree was redefined by the
ocsenave@3	1539 Tenth General Conference of Weights and Measures in 1954, by
ocsenave@3	1540 the condition that the triple point of water is at 273.l6^\circ K,
ocsenave@3	1541 this number being exact by definition. The freezing point, 0^\circ C,
ocsenave@3	1542 is then 273.15^\circ K. This new degree is called the Celsius degree.
ocsenave@3	1543 For further details, see the U.S. National Bureau of Standards
ocsenave@3	1544 Technical News Bulletin, October l963.
ocsenave@3	1545
ocsenave@3	1546
ocsenave@3	1547 The appearance of such a strange and arbitrary-looking
ocsenave@3	1548 number as 273.16 in the /definition/ of a unit is the result of
ocsenave@3	1549 the historical development, and is the means by which much
ocsenave@3	1550 greater confusion is avoided. Whenever improved techniques make
ocsenave@3	1551 possible a new and more precise (i.e., more reproducible)
ocsenave@3	1552 definition of a physical unit, its numerical value is of course chosen
ocsenave@3	1553 so as to be well inside the limits of error with which the old
ocsenave@3	1554 unit could be defined. Thus the old Centigrade and new Celsius
ocsenave@3	1555 scales are the same, within the accuracy with which the
ocsenave@3	1556 Centigrade scale could be realized; so the same notation, ^\circ C, is used
ocsenave@4	1557 for both Only in this way can old measurements retain their
ocsenave@3	1558 value and accuracy, without need of corrections every time a
ocsenave@3	1559 unit is redefined.
ocsenave@3	1560
ocsenave@3	1561 #capitalize Joules?
ocsenave@3	1562 Exactly the same thing has happened in the definition of
ocsenave@3	1563 the calorie; for a century, beginning with the work of Joule,
ocsenave@3	1564 more and more precise experiments were performed to determine
ocsenave@4	1565 the mechanical equivalent of heat more and more accurately But
ocsenave@3	1566 eventually mechanical and electrical measurements of energy be
ocsenave@3	1567 came far more reproducible than calorimetric measurements; so
ocsenave@3	1568 recently the calorie was redefined to be 4.1840 Joules, this
ocsenave@3	1569 number now being exact by definition. Further details are given
ocsenave@3	1570 in the aforementioned Bureau of Standards Bulletin.
ocsenave@3	1571
ocsenave@3	1572
ocsenave@3	1573 The derivations of this section have shown that, for any
ocsenave@3	1574 particular substance, there is (except for choice of units) only
ocsenave@3	1575 one temperature scale $T$ with the property that $dQ = TdS$ where
ocsenave@3	1576 $dS$ is the exact differential of some state function $S$. But this
ocsenave@3	1577 in itself provides no reason to suppose that the /same/ Kelvin
ocsenave@3	1578 scale will result for all substances; i.e., if we determine a
ocsenave@3	1579 \ldquo{}helium Kelvin temperature\rdquo{} and a
ocsenave@3	1580 \ldquo{}carbon dioxide Kelvin temperature\rdquo{} by the measurements
ocsenave@3	1581 indicated in (1-38), and choose the units so that they agree numerically at one point, will they then
ocsenave@3	1582 agree at other points? Thus far we have given no reason to
ocsenave@3	1583 expect that the Kelvin scale is /universal/, other than the empirical
ocsenave@3	1584 fact that the limit (1-45) is found to be the same for all gases.
ocsenave@3	1585 In section 2.0 we will see that this universality is a conse
ocsenave@3	1586 quence of the second law of thermodynamics (i.e., if we ever
ocsenave@3	1587 find two substances for which the Kelvin scale as defined above
ocsenave@3	1588 is different, then we can take advantage of this to make a
ocsenave@3	1589 perpetual motion machine of the second kind).
ocsenave@3	1590
ocsenave@3	1591
ocsenave@3	1592 Usually, the second law is introduced before discussing
ocsenave@3	1593 entropy or the Kelvin temperature scale. We have chosen this
ocsenave@3	1594 unusual order so as to demonstrate that the concepts of entropy
ocsenave@3	1595 and Kelvin temperature are logically independent of the second
ocsenave@3	1596 law; they can be defined theoretically, and the experimental
ocsenave@3	1597 procedures for their measurement can be developed, without any
ocsenave@3	1598 appeal to the second law. From the standpoint of logic, there
ocsenave@3	1599 fore, the second law serves /only/ to establish that the Kelvin
ocsenave@3	1600 temperature scale is the same for all substances.
ocsenave@3	1601
ocsenave@3	1602
ocsenave@4	1603 ** Entropy of an Ideal Boltzmann Gas
ocsenave@3	1604
ocsenave@3	1605 At the present stage we are far from understanding the physical
ocsenave@4	1606 meaning of the function $S$ defined by (1-19); but we can investigate
ocsenave@4	1607 its mathematical form and numerical values. Let us do this for a
ocsenave@4	1608 system
ocsenave@4	1609 consisting of $n$ moles of a substance which obeys the ideal gas
ocsenave@3	1610 equation of state
ocsenave@4	1611
ocsenave@4	1612 \begin{equation}PV = nRT\end{equation}
ocsenave@4	1613
ocsenave@4	1614 and for which the heat capacity at constant volume
ocsenave@4	1615 $C_V$ is a constant. The difference in entropy between any two states (1)
ocsenave@3	1616 and (2) is from (1-19),
ocsenave@4	1617
ocsenave@4	1618 \begin{equation}
ocsenave@4	1619 S_2 - S_1 = \int_1^2 \frac{dQ}{T} = \int_1^2
ocsenave@4	1620 \left[\left(\frac{\partial S}{\partial V}\right)+\left(\frac{\partial S}{\partial T}\right)_V dT\right]
ocsenave@4	1621 \end{equation}
ocsenave@3	1622
ocsenave@3	1623 where we integrate over any reversible path connecting the two
ocsenave@4	1624 states. From the manner in which $S$ was defined, this integral
ocsenave@3	1625 must be the same whatever path we choose. Consider, then, a
ocsenave@4	1626 path consisting of a reversible expansion at constant
ocsenave@4	1627 temperature to a state 3 which has the initial temperature $T_1$, and the
ocsenave@4	1628 the final volume $V_2$; followed by heating at constant volume to the
ocsenave@4	1629 final temperature $T_2$.
ocsenave@4	1630 Then (1-47) becomes
ocsenave@4	1631
ocsenave@4	1632 \begin{equation}
ocsenave@4	1633 S_2 - S_1 = \int_1^3 \left(\frac{\partial S}{\partial V}\right)_T dV +
ocsenave@4	1634 \int_3^2 \left(\frac{\partial S}{\partial T}\right)_V dT
ocsenave@4	1635 \end{equation}
ocsenave@4	1636
ocsenave@4	1637 To evaluate the integral over $(1\rightarrow 3)$, note that since $dU
ocsenave@4	1638 = TdS - PdV$, the Helmholtz free energy function $F \equiv U -TS$ has
ocsenave@4	1639 the property $dF = -SdT - PdV$; and of course $dF$ is an exact
ocsenave@4	1640 differential since $F$ is a definite state function. The condition
ocsenave@4	1641 that $dF$ be exact is, analogous to (1-22),
ocsenave@4	1642
ocsenave@4	1643 \begin{equation}
ocsenave@4	1644 \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial
ocsenave@4	1645 P}{\partial T}\right)_V
ocsenave@4	1646 \end{equation}
ocsenave@4	1647
ocsenave@3	1648 which is one of the Maxwell relations, discussed further in
ocsenave@4	1649 Chapter 2. But [the value of this expression] is determined by the equation of state
ocsenave@4	1650 (1-46):
ocsenave@4	1651
ocsenave@4	1652 \begin{equation}
ocsenave@4	1653 \left(\frac{\partial S}{\partial V}\right)_T = \frac{nR}{V}
ocsenave@4	1654 \end{equation}
ocsenave@4	1655
ocsenave@4	1656 Likewise, along the path $(3\rightarrow 2)$, we have
ocsenave@4	1657
ocsenave@4	1658 \begin{equation}
ocsenave@4	1659 \left(\frac{\partial S}{\partial T}\right)_V = \frac{n C_V}{T}
ocsenave@4	1660 \end{equation}
ocsenave@4	1661
ocsenave@4	1662 where $C_V$ is the molar heat capacity at constant volume.
ocsenave@4	1663 Collecting these results, we have
ocsenave@4	1664
ocsenave@4	1665 \begin{equation}
ocsenave@4	1666 S_2 - S_1 = \int_1^3 \frac{nR}{V} dV + \int_2^3 \frac{n C_V}{T} dT =
ocsenave@4	1667 nR\log{(V_2/V_1)} + nC_V \log{(T_2/T_1)}
ocsenave@4	1668 \end{equation}
ocsenave@4	1669
ocsenave@4	1670 since $C_V$ was assumed independent of $T$. Thus the entropy function
ocsenave@3	1671 must have the form
ocsenave@4	1672
ocsenave@4	1673 \begin{equation}
ocsenave@4	1674 S(n,V,T) = nR \log{V} + n C_V \log{T} + (\text{const.})
ocsenave@4	1675 \end{equation}
ocsenave@3	1676
ocsenave@3	1677 From the derivation, the additive constant must be independent
ocsenave@3	1678 of V and T; but it can still depend on n. We indicate this by
ocsenave@3	1679 writing
ocsenave@4	1680
ocsenave@4	1681 \begin{equation}
ocsenave@4	1682 S(n,V,T) = n\left[R \log{V} + C_V \log{T}\right] + f(n)
ocsenave@4	1683 \end{equation}
ocsenave@4	1684
ocsenave@4	1685 where $f(n)$ is a function not determined by the definition (1-47).
ocsenave@4	1686 The form of $f(n)$ is, however, restricted by the condition that
ocsenave@4	1687 the entropy be an extensive quantity; i.e., two identical systems
ocsenave@4	1688 placed together should have twice the entropy of a single system; or
ocsenave@4	1689 more generally,
ocsenave@4	1690
ocsenave@4	1691 \begin{equation}
ocsenave@4	1692 S(qn, qV, T) = q\cdot S(n,v,T),\qquad 0<q<\infty
ocsenave@4	1693 \end{equation}
ocsenave@4	1694
ocsenave@4	1695 Substituting (1-54) into (1-55), we find that $f(n)$ must satisfy
ocsenave@4	1696 the functional equation
ocsenave@4	1697
ocsenave@4	1698 \begin{equation}
ocsenave@4	1699 f(q\cdot n) = q\cdot f(n) - R\cdot n\cdot q\log{q}\end{equation}
ocsenave@4	1700
ocsenave@4	1701
ocsenave@4	1702 To solve this, one can differentiate with respect to $q$ and set
ocsenave@4	1703 $q = 1$; we then obtain the differential equation
ocsenave@4	1704
ocsenave@4	1705 \begin{equation}
ocsenave@4	1706 n\cdot f^\prime(n) - f(n) + R\cdot n = 0
ocsenave@4	1707 \end{equation}
ocsenave@4	1708 # xy' - y + rx = 0
ocsenave@4	1709 which is readily solved; alternatively, just set $n = 1$ in (1-56)
ocsenave@4	1710 and replace $q$ by $n$ By either procedure we find
ocsenave@4	1711
ocsenave@4	1712 \begin{equation}
ocsenave@4	1713 f(n) = n\cdot f(1) - R\cdot n \log{n} (1-58)
ocsenave@4	1714 \end{equation}
ocsenave@4	1715
ocsenave@4	1716 As a check, it is easily verified that this is the solution of (1-56)
ocsenave@4	1717 and (1-57). We then have finally,
ocsenave@4	1718
ocsenave@4	1719 \begin{equation}
ocsenave@4	1720 S(n,V,t) = n\left[C_v\cdot\log{t} + R\cdot \log{\left(\frac{V}{n}\right)} +
ocsenave@4	1721 A\right]
ocsenave@4	1722 \end{equation}
ocsenave@4	1723
ocsenave@4	1724 where $A\equiv f(1)$ is still an arbitrary constant, not determined
ocsenave@4	1725 by the definition (1-19), or by the condition (1-55) that $S$ be
ocsenave@4	1726 extensive. However, $A$ is not without physical meaning; we will
ocsenave@4	1727 see in the next Section that the vapor pressure of this
ocsenave@4	1728 substance (and more generally, its chemical potential) depends on
ocsenave@4	1729 $A$. Later, it will appear that the numerical value of $A$ involves
ocsenave@3	1730 Planck's constant, and its theoretical determination therefore
ocsenave@4	1731 requires quantum statistics.
ocsenave@3	1732
ocsenave@4	1733 #edit: "is constant"
ocsenave@4	1734 We conclude from this that, in any region where experimentally
ocsenave@4	1735 $C_V$ is constant, and the ideal gas equation of state is
ocsenave@4	1736 obeyed, the entropy must have the form (1-59) The fact that
ocsenave@3	1737 classical statistical mechanics does not lead to this result,
ocsenave@4	1738 the term $n\cdot R \cdot \log{(1/n)}$ being missing (Gibbs paradox),
ocsenave@4	1739 was historically one of the earliest clues indicating the need for the
ocsenave@3	1740 quantum theory.
ocsenave@4	1741
ocsenave@4	1742 In the case of a liquid, the volume does not change
ocsenave@4	1743 appreciably on heating, and so $dS = n\cdot C_V\cdot dT/T$, and if
ocsenave@4	1744 $C_V$ is independent of temperature, we would have in place of (1-59),
ocsenave@4	1745
ocsenave@4	1746 \begin{equation}
ocsenave@4	1747 S = n\left[C_V\ln{T}+A_\ell\right]
ocsenave@4	1748 \end{equation}
ocsenave@4	1749
ocsenave@4	1750 where $A_\ell$ is an integration constant, which also has physical
ocsenave@3	1751 meaning in connection with conditions of equilibrium between
ocsenave@3	1752 two different phases.
ocsenave@4	1753
ocsenave@4	1754 ** The Second Law: Definition
ocsenave@4	1755
ocsenave@4	1756 Probably no proposition in physics has been the subject of more deep
ocsenave@4	1757 and sustained confusion
ocsenave@4	1758 than the second law of thermodynamics It is not in the province
ocsenave@3	1759 of macroscopic thermodynamics to explain the underlying reason
ocsenave@3	1760 for the second law; but at this stage we should at least be able
ocsenave@4	1761 to /state/ this law in clear and experimentally meaningful terms.
ocsenave@3	1762 However, examination of some current textbooks reveals that,
ocsenave@3	1763 after more than a century, different authors still disagree as
ocsenave@3	1764 to the proper statement of the second law, its physical meaning,
ocsenave@3	1765 and its exact range of validity.
ocsenave@4	1766
ocsenave@3	1767 Later on in this book it will be one of our major objectives
ocsenave@4	1768 to show, from several different viewpoints, how much clearer and
ocsenave@3	1769 simpler these problems now appear in the light of recent develop
ocsenave@4	1770 ments in statistical mechanics For the present, however, our
ocsenave@3	1771 aim is only to prepare the way for this by pointing out exactly
ocsenave@3	1772 what it is that is to be proved later. As a start on this at
ocsenave@3	1773 tempt, we note that the second law conveys a certain piece of
ocsenave@4	1774 informations about the /direction/ in which processes take place.
ocsenave@3	1775 In application it enables us to predict such things as the final
ocsenave@3	1776 equilibrium state of a system, in situations where the first law
ocsenave@3	1777 alone is insufficient to do this.
ocsenave@4	1778
ocsenave@4	1779
ocsenave@3	1780 A concrete example will be helpful. We have a vessel
ocsenave@4	1781 equipped with a piston, containing $N$ moles of carbon dioxide.
ocsenave@3	1782
ocsenave@4	1783 #changed V_f to V_1
ocsenave@4	1784 The system is initially at thermal equilibrium at temperature $T_0$,
ocsenave@4	1785 volume $V_0$ and pressure $P_O$; and under these conditions it contains
ocsenave@4	1786 $n$ moles of CO_2 in the vapor phase and $N-n$ moles in the liquid
ocsenave@4	1787 phase The system is now thermally insulated from its
ocsenave@4	1788 surroundings, and the piston is moved rapidly (i.e., so that $n$ does not
ocsenave@3	1789 change appreciably during the motion) so that the system has a
ocsenave@4	1790 new volume $V_1$; and immediately after the motion, a new pressure
ocsenave@4	1791 $P_1$ The piston is now held fixed in its new position, and the
ocsenave@3	1792 system allowed to come once more to equilibrium. During this
ocsenave@4	1793 process, will the CO_2 tend to evaporate further, or condense further?
ocsenave@4	1794 What will be the final equilibrium temperature $T_{eq}$,
ocsenave@4	1795 the final pressure $P_eq$, and final value of $n_{eq}$?
ocsenave@4	1796
ocsenave@4	1797 It is clear that the first law alone is incapable of answering
ocsenave@3	1798 these questions; for if the only requirement is conservation of
ocsenave@4	1799 energy, then the CO_2 might condense, giving up its heat of
ocsenave@4	1800 vaporization and raising the temperature of the system; or it might
ocsenave@3	1801 evaporate further, lowering the temperature. Indeed, all values
ocsenave@4	1802 of $n_{eq}$ in $O \leq n_{eq} \leq N$ would be possible without any
ocsenave@4	1803 violation of the first law. In practice, however, this process will be found
ocsenave@4	1804 to go in only one direction and the system will reach a definite
ocsenave@3	1805 final equilibrium state with a temperature, pressure, and vapor
ocsenave@3	1806 density predictable from the second law.
ocsenave@4	1807
ocsenave@4	1808
ocsenave@3	1809 Now there are dozens of possible verbal statements of the
ocsenave@3	1810 second law; and from one standpoint, any statement which conveys
ocsenave@4	1811 the same information has equal right to be called \ldquo{}the second
ocsenave@4	1812 law.\rdquo{} However, not all of them are equally direct statements of
ocsenave@3	1813 experimental fact, or equally convenient for applications, or
ocsenave@3	1814 equally general; and it is on these grounds that we ought to
ocsenave@4	1815 choose among them.
ocsenave@4	1816
ocsenave@4	1817 Some of the mos t popular statements of the second law
ocsenave@4	1818 belong to the class of the well-known \ldquo{}impossibility\rdquo{}
ocsenave@4	1819 assertions; i.e., it is impossible to transfer heat from a lower to a higher
ocsenave@3	1820 temperature without leaving compensating changes in the rest of
ocsenave@4	1821 the universe, it is impossible to convert heat into useful work
ocsenave@3	1822 without leaving compensating changes, it is impossible to make
ocsenave@3	1823 a perpetual motion machine of the second kind, etc.
ocsenave@3	1824
ocsenave@3	1825 Suoh formulations have one clear logical merit; they are
ocsenave@3	1826 stated in such a way that, if the assertion should be false, a
ocsenave@4	1827 single experiment would suffice to demonstrate that fact
ocsenave@4	1828 conclusively. It is good to have our principles stated in such a
ocsenave@3	1829 clear, unequivocal way.
ocsenave@4	1830
ocsenave@4	1831 However, impossibility statements also have some
ocsenave@4	1832 disadvantages In the first place, /they are not, and by their very
ocsenave@4	1833 nature cannot be, statements of eiperimental fact/. Indeed, we
ocsenave@3	1834 can put it more strongly; we have no record of anyone having
ocsenave@3	1835 seriously tried to do any of the various things which have been
ocsenave@3	1836 asserted to be impossible, except for one case which actually
ocsenave@4	1837 succeeded. In the experimental realization of negative spin
ocsenave@4	1838 temperatures, one can transfer heat from a lower to a higher
ocsenave@3	1839 temperature without external changes; and so one of the common
ocsenave@3	1840 impossibility statements is now known to be false [for a clear
ocsenave@4	1841 discussion of this, see the [[../sources/Ramsey.pdf][article of N. F. Ramsey (1956)]];
ocsenave@3	1842 experimental details of calorimetry with negative temperature
ocsenave@4	1843 spin systems are given by Abragam and Proctor (1958)]
ocsenave@4	1844
ocsenave@4	1845
ocsenave@3	1846 Finally, impossibility statements are of very little use in
ocsenave@4	1847 /applications/ of thermodynamics; the assertion that a certain kind
ocsenave@4	1848 of machine cannot be built, or that a certain laboratory feat
ocsenave@3	1849 cannot be performed, does not tell me very directly whether my
ocsenave@3	1850 carbon dioxide will condense or evaporate. For applications,
ocsenave@3	1851 such assertions must first be converted into a more explicit
ocsenave@3	1852 mathematical form.
ocsenave@4	1853
ocsenave@4	1854
ocsenave@3	1855 For these reasons, it appears that a different kind of
ocsenave@3	1856 statement of the second law will be, not necessarily more
ocsenave@4	1857 \ldquo{}correct\rdquo{}, but more useful in practice. Now both Clausius (1875)
ocsenave@3	1858 and Planck (1897) have laid great stress on their conclusion
ocsenave@3	1859 that the most general statement, and also the most immediately
ocsenave@3	1860 useful in applications, is simply the existence of a state
ocsenave@3	1861 function, called the entropy, which tends to increase. More
ocsenave@3	1862 precisely: in an adiabatic change of state, the entropy of
ocsenave@3	1863 a system may increase or may remain constant, but does not
ocsenave@3	1864 decrease. In a process involving heat flow to or from the
ocsenave@3	1865 system, the total entropy of all bodies involved may increase
ocsenave@4	1866 or may remain constant; but does not decrease; let us call this
ocsenave@4	1867 the \ldquo{}weak form\rdquo{} of the second law.
ocsenave@3	1868
ocsenave@3	1869 The weak form of the second law is capable of answering the
ocsenave@4	1870 first question posed above; thus the carbon dioxide will
ocsenave@4	1871 evaporate further if, and only if, this leads to an increase in the
ocsenave@4	1872 total entropy of the system This alone, however, is not enough
ocsenave@4	1873 to answer the second question; to predict the exact final
ocsenave@4	1874 equilibrium state, we need one more fact.
ocsenave@4	1875
ocsenave@3	1876 The strong form of the second law is obtained by adding the
ocsenave@4	1877 further assertion that the entropy not only \ldquo{}tends\rdquo{} to increase;
ocsenave@4	1878 in fact it /will/ increase, /to the maximum value permitted by the
ocsenave@4	1879 constraints imposed[fn::Note, however, that the second law has
ocsenave@4	1880 nothing to say about how rapidly this approach to equilibrium takes place.]/. In the case of the carbon dioxide, these
ocsenave@4	1881 constraints are: fixed total energy (first law), fixed total
ocsenave@4	1882 amount of carbon dioxide, and fixed position of the piston. The
ocsenave@3	1883 final equilibrium state is the one which has the maximum entropy
ocsenave@4	1884 compatible with these constraints, and it can be predicted
ocsenave@4	1885 quantitatively from the strong form of the second law if we know,
ocsenave@3	1886 from experiment or theory, the thermodynamic properties of carbon
ocsenave@4	1887 dioxide (ie, heat capacity, equation of state, heat of vapor
ocsenave@4	1888 ization)
ocsenave@3	1889
ocsenave@4	1890 To illustrate this, we set up the problem in a crude
ocsenave@4	1891 approximation which supposes that (l) in the range of conditions
ocsenave@4	1892 of interest, the molar heat capacity $C_v$ of the vapor, and $C_\ell$ of
ocsenave@4	1893 the liquid, and the molar heat of vaporization $L$, are all con
ocsenave@4	1894 stants, and the heat capacities of cylinder and piston are
ocsenave@4	1895 negligible; (2) the liquid volume is always a small fraction of the
ocsenave@4	1896 total $V$, so that changes in vapor volume may be neglected; (3) the
ocsenave@4	1897 vapor obeys the ideal gas equation of state $PV = nRT$. The
ocsenave@4	1898 internal energy functions of liquid and vapor then have the form
ocsenave@4	1899
ocsenave@4	1900 \begin{equation}
ocsenave@4	1901 U_\ell = (N-n)\left[C_\ell\cdot T + A\right]
ocsenave@4	1902 \end{equation}
ocsenave@4	1903 \begin{equation}
ocsenave@4	1904 U_v = n\left[C_v\cdot T + A + L\right]
ocsenave@4	1905 \end{equation}
ocsenave@4	1906
ocsenave@4	1907 where $A$ is a constant which plays no role in the problem. The
ocsenave@4	1908 appearance of $L$ in (1-62) recognizes that the zero from which we
ocsenave@3	1909 measure energy of the vapor is higher than that of the liquid by
ocsenave@4	1910 the energy $L$ necessary to form the vapor. On evaporation of $dn$
ocsenave@4	1911 moles of liquid, the total energy increment is $dU = dU_\ell + dU_v =
ocsenave@4	1912 0$; or
ocsenave@4	1913
ocsenave@4	1914 \begin{equation}
ocsenave@4	1915 \left[n\cdot C_v + (N-n)C_\ell\right] dT + \left[(C_v-C_\ell)T + L\right]dn = 0
ocsenave@4	1916 \end{equation}
ocsenave@4	1917
ocsenave@3	1918 which is the constraint imposed by the first law. As we found
ocsenave@4	1919 previously (1-59), (1-60) the entropies of vapor and liquid are
ocsenave@3	1920 given by
ocsenave@4	1921
ocsenave@4	1922 \begin{equation}
ocsenave@4	1923 S_v = n\left[C_v\cdot\ln{T} + R\cdot \ln{\left(V/n\right)} + A_v\right]
ocsenave@4	1924 \end{equation}
ocsenave@4	1925 \begin{equation}
ocsenave@4	1926 S_\ell = (N-n)\left[C_\ell\cdot \ln{T}+A_\ell\right]
ocsenave@4	1927 \end{equation}
ocsenave@4	1928
ocsenave@4	1929 where $A_v$, $A_\ell$ are the constants of integration discussed in the
ocsenave@3	1930 last Section.
ocsenave@4	1931
ocsenave@4	1932
ocsenave@3	1933 We leave it as an exercise for the reader to complete the
ocsenave@4	1934 derivation from this point, and show that the total entropy
ocsenave@4	1935 $S = S_\ell + S_v$ is maximized subject to the constraint (1-63), when
ocsenave@4	1936 the values $n_{eq}$, $T_{eq}$ are related by
ocsenave@4	1937
ocsenave@4	1938 \begin{equation}
ocsenave@4	1939 \frac{n_{eq}}{V}= B\cdot T_{eq}^a\cdot \exp{\left(-\frac{L}{RT_{eq}}\right)}
ocsenave@4	1940 \end{equation}
ocsenave@4	1941
ocsenave@4	1942 where $B\equiv \exp{(-1-a-\frac{A_\ell-A_v}{R})}$ and $a\equiv
ocsenave@4	1943 (C_v-C_\ell)/R$ are constants.
ocsenave@4	1944
ocsenave@4	1945
ocsenave@3	1946 Equation (1-66) is recognized as an approximate form of the Vapor
ocsenave@4	1947 pressure formula
ocsenave@5	1948 We note that $A_\ell$, $A_v$, which appeared first as integration
ocsenave@5	1949 constants for the entropy with no particular physical meaning,
ocsenave@3	1950 now play a role in determining the vapor pressure.
ocsenave@4	1951
ocsenave@4	1952 ** The Second Law: Discussion
ocsenave@4	1953
ocsenave@4	1954 We have emphasized the distinction between the weak and strong forms
ocsenave@4	1955 of the second law
ocsenave@4	1956 because (with the exception of Boltzmann's original unsuccessful
ocsenave@4	1957 argument based on the H-theorem), most attempts to deduce the
ocsenave@4	1958 second law from statistical mechanics have considered only the
ocsenave@3	1959 weak form; whereas it is evidently the strong form that leads
ocsenave@3	1960 to definite quantitative predictions, and is therefore needed
ocsenave@4	1961 for most applications. As we will see later, a demonstration of
ocsenave@4	1962 the weak form is today almost trivial---given the Hamiltonian form
ocsenave@4	1963 of the equations of motion, the weak form is a necessary
ocsenave@4	1964 condition for any experiment to be reproducible. But demonstration
ocsenave@4	1965 of the strong form is decidedly nontrivial; and we recognize from
ocsenave@4	1966 the start that the job of statistical mechanics is not complete
ocsenave@4	1967 until that demonstration is accomplished.
ocsenave@2	1968
ocsenave@2	1969
ocsenave@4	1970 As we have noted, there are many different forms of the
ocsenave@4	1971 seoond law, that have been favored by various authors. With
ocsenave@4	1972 regard to the entropy statement of the second law, we note the
ocsenave@4	1973 following. In the first place, it is a direct statement of
ocsenave@4	1974 experimental fact, verified in many thousands of quantitative
ocsenave@4	1975 measurements, /which have actually been performed/. This is worth a
ocsenave@4	1976 great deal in an age when theoretical physics tends to draw
ocsenave@4	1977 sweeping conclusions from the assumed outcomes of
ocsenave@4	1978 \ldquo{}thought-experiments.\rdqquo{} Secondly, it has stood the test
ocsenave@4	1979 of time; it is the entropy statement which remained valid in the case
ocsenave@4	1980 of negative spin temperatures, where some others failed. Thirdly, it
ocsenave@4	1981 is very easy to apply in practice, the weak form leading
ocsenave@4	1982 immediately to useful predictions as to which processes will go and
ocsenave@4	1983 which will not; the strong form giving quantitative predictions
ocsenave@4	1984 of the equilibrium state. At the present time, therefore, we
ocsenave@4	1985 cannot understand what motivates the unceasing attempts of many
ocsenave@4	1986 textbook authors to state the second law in new and more
ocsenave@4	1987 complicated ways.
ocsenave@4	1988
ocsenave@4	1989 One of the most persistent of these attempts involves the
ocsenave@4	1990 use of [[http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Principle_of_Carath.C3.A9odory][Carath\eacute{}odory's principle]]. This states that, in the
ocsenave@4	1991 neighborhood of any thermodynamic state there are other states which
ocsenave@4	1992 cannot be reached by an adiabatic process. After some mathematical
ocsenave@4	1993 analysis
ocsenave@4	1994 [Margenau and Murphy (1943), pp. 26-31; or Wannier (1966),
ocsenave@4	1995 pp. 126-132]
ocsenave@4	1996 one infers the existence of a state function (entropy) which tends
ocsenave@4	1997 to increase; or at least, cannot decrease. From a /mathematical/
ocsenave@4	1998 standpoint there can be no objection at all to this; the analysis
ocsenave@4	1999 is quite rigorous. But from a /physical/ standpoint it is subject
ocsenave@4	2000 to the same objection that its premise is an impossibility
ocsenave@4	2001 statement, and therefore not an experimental fact.
ocsenave@4	2002 Indeed, the conclusion of Carath\eacute{}odory's
ocsenave@4	2003 argument is a far more direct statement of observed fact than its
ocsenave@4	2004 premise; and so it would seem more logical to use the argument
ocsenave@4	2005 backwards. Thus, from the experimental fact that the entropy
ocsenave@4	2006 tends to increase, we would infer that there must exist
ocsenave@4	2007 neighboring states inaccessible in an adiabatic process; but the
ocsenave@4	2008 result is then trivial. In a similar way, other impossibility
ocsenave@4	2009 statements follow trivially from the entropy statement of the
ocsenave@4	2010 second law.
ocsenave@4	2011
ocsenave@4	2012
ocsenave@4	2013 Finally, we note that all statements of the second law are
ocsenave@4	2014 subject to a very important qualification, not always sufficiently
ocsenave@4	2015 emphasized. As we stress repeatedly, conventional thermodynamics
ocsenave@4	2016 is a theory only of states of thermal equilibrium; such concepts
ocsenave@4	2017 as temperature and entropy are not even defined for others.
ocsenave@4	2018 Therefore, all the above statements of the second law must be under
ocsenave@4	2019 stood as describing only the /net result/ of processes /which begin
ocsenave@4	2020 and end in states of complete thermal equilibrium/. Classical
ocsenave@4	2021 thermodynamics has nothing to say about processes that do not
ocsenave@4	2022 meet this condition, or about intermediate states of processes
ocsenave@4	2023 that do. Again, it is nuclear magnetic resonance (NMR)
ocsenave@4	2024 experiments which provide the most striking evidence showing how
ocsenave@4	2025 essential this qualification is; the spin-echo experiment
ocsenave@4	2026 (Hahn, 1950) is, as we will see in detail later, a gross violation of
ocsenave@4	2027 any statement of the second law that fails to include it.
ocsenave@4	2028
ocsenave@4	2029
ocsenave@4	2030 This situation has some interesting consequences, in that
ocsenave@4	2031 impossibility statements may be misleading if we try to read too
ocsenave@4	2032 much into them. From classical thermodynamics alone, we cannot
ocsenave@4	2033 logically infer the impossibility of a \ldquo{}perpetual motion machine\rdquo{}
ocsenave@4	2034 of the second kind (i.e., a machine which converts heat energy
ocsenave@4	2035 into useful work without requiring any low temperature heat sink,
ocsenave@4	2036 as does the Carnot engine); we can infer only that such a machine
ocsenave@4	2037 cannot operate between equilibrium states. More specifically, if
ocsenave@4	2038 the machine operates by carrying out some cyclic process, then
ocsenave@4	2039 the states of (machine + environment) at the beginning and end
ocsenave@4	2040 of a cycle cannot be states of complete thermal equilibrium, as
ocsenave@4	2041 in the reversible Carnot engine. But no real machine operates
ocsenave@4	2042 between equilibrium states anyway. Without some further analysis
ocsenave@4	2043 involving statistical mechanics, we cannot be at all certain that
ocsenave@4	2044 a sufficiently clever inventor could not find a way to convert
ocsenave@4	2045 heat energy into useful work on a commercially profitable scale;
ocsenave@4	2046 the energy is there, and the only question is whether we could
ocsenave@4	2047 persuade it to \ldquo{}organize\rdquo{} itself enough to perform useful work
ocsenave@4	2048 against pistons, magnets, gravitational or electric fields,
ocsenave@4	2049 chemical activation energy hills, etc.
ocsenave@4	2050
ocsenave@4	2051
ocsenave@5	2052 It was Maxwell himself who first ([[../sources/Maxwell-Heat.pdf][1871]])[fn::Edit: See also, the [[http://openlibrary.org/books/OL7243600M/Theory_of_heat][Open Library
ocsenave@4	2053 page]], where you can read and download Maxwell's book in a variety of formats.] suggested such
ocsenave@4	2054 possibilities, in his invention of the \ldquo{}Maxwell Demon\rdquo{},
ocsenave@4	2055 an imaginary being (or mechanism) which can regulate valves so as to allow
ocsenave@4	2056 fast molecules to pass through a partition in one direction only,
ocsenave@4	2057 thus heating up one side at the expense of the other. We could
ocsenave@4	2058 then allow the heat to flow back from the hot side to the cold
ocsenave@4	2059 through a conventional Carnot engine, generating useful work; and
ocsenave@4	2060 the whole arrangement would constitute a perpetual motion machine
ocsenave@4	2061 of the second kind.
ocsenave@4	2062
ocsenave@4	2063 #http://naca.larc.nasa.gov/search.jsp?R=19760010893&qs=Ns%3DLoaded-Date\|1%26N%3D4294709597
ocsenave@4	2064
ocsenave@4	2065 Maxwell did not regard such a device as impossible in principle;
ocsenave@4	2066 only very difficult technically. Later authors ([[../sources/Szilard.pdf][Szilard, 1929]];
ocsenave@4	2067 Brillouin, 1951, 1956)
ocsenave@4	2068 have argued, on the basis of quantum
ocsenave@4	2069 theory or connections between entropy and information, that it
ocsenave@4	2070 fundamentally impossible. However, all these arguments seem
ocsenave@4	2071 to contain just enough in the way of questionable assumptions or
ocsenave@4	2072 loopholes in the logic, as to leave the critical reader not quite
ocsenave@4	2073 convinced. This is particularly so when we recall the lessons
ocsenave@4	2074 of history; clever experimenters have, over and over again, made
ocsenave@4	2075 fools of theorists who were too quick to assert that something
ocsenave@4	2076 cannot be done.
ocsenave@4	2077
ocsenave@4	2078 A recent example worth recalling concerns the Overhauser
ocsenave@4	2079 effect in magnetic resonance (enhancement of the polarization
ocsenave@4	2080 of one set of spins by irradiation of another set coupled to them).
ocsenave@4	2081 When this effect was first proposed, several well-known
ocsenave@4	2082 authorities on thermodynamics and statistical mechanics ridiculed the
ocsenave@4	2083 suggestion and asserted that the effect could not possibly exist,
ocsenave@4	2084 because it violated the second law of thermodynamics. This
ocsenave@4	2085 incident is a valuable reminder of how little we really understand
ocsenave@4	2086 the second law, or how to apply it in new situations.
ocsenave@4	2087
ocsenave@4	2088 In this connection, there is a fascinating little gadget
ocsenave@5	2089 known as the [[http://en.wikipedia.org/wiki/Vortex_tube][Hilsch tube]] or Vortex tube, in which a jet of
ocsenave@4	2090 compressed air is injected into a pipe at right angles to its
ocsenave@4	2091 axis, but off center so that it sets up a rapid rotational
ocsenave@4	2092 motion of the gas. In some manner, this causes a separation of
ocsenave@4	2093 the fast and slow molecules, cold air collecting along the axis
ocsenave@4	2094 of the tube, and hot air at the walls. On one side of the jet,
ocsenave@4	2095 a diaphragm with a small hole at the center allows only the cold
ocsenave@4	2096 air to escape, the other side is left open so that the hot air
ocsenave@4	2097 can escape. The result is that when compressed air at room
ocsenave@4	2098 temperature is injected, one can obtain air from the hot side
ocsenave@5	2099 at $+100^\circ$ F from the cold side at $-70^\circ$ F, in sufficient quantities
ocsenave@4	2100 to be used for quick-freezing small objects, or for cooling
ocsenave@5	2101 photomultiplier tubes [for construction drawings and
ocsenave@5	2102 experimental data, see [[http://books.google.com/books?id=yOUWAAAAIAAJ][Stong (1960)]]; for a partial thermodynamic
ocsenave@5	2103 analysis, see Hilsch (1947)[fn::Edit: Hilsch's paper is entitled /The use of the expansion of gases in
ocsenave@5	2104 a centrifugal field as a cooling process./]].
ocsenave@4	2105
ocsenave@4	2106 Of course, the air could also be cooled by adiabatic expansion
ocsenave@4	2107 (i.e., by doing work against a piston); and it appears that
ocsenave@4	2108 the amount of cooling achieved in vortex tubes is comparable to,
ocsenave@4	2109 but somewhat less than, what could be obtained this way for the
ocsenave@4	2110 same pressure drop. However, the operation of the vortex tube
ocsenave@4	2111 is manifestly not simple adiabatic since no work is
ocsenave@4	2112 done; rather, part of the gas is heated up, at the cost of cooling
ocsenave@4	2113 the rest; i.e., fast and slow molecules are separated spatially.
ocsenave@4	2114 There is, apparently, no violation of the laws of thermodynamics,
ocsenave@4	2115 since work must be supplied to compress the air; nevertheless,
ocsenave@4	2116 the device resembles the Maxwell Demon so much as to make one
ocsenave@4	2117 uncomfortable.. This is so particularly because of our
ocsenave@4	2118 embarrassing inability to explain in detail (i.e., in molecular terms)
ocsenave@4	2119 how such asimple device works. If we did understand it, would
ocsenave@4	2120 we be able to see still more exciting possibilities? No one
ocsenave@4	2121 knows.
ocsenave@4	2122
ocsenave@4	2123
ocsenave@4	2124 It is interesting to note in passing that such considerations
ocsenave@4	2125 were very much in Planck's mind also; in his [[http://books.google.com/books?id=kOjy3FQqXPQC&printsec=frontcover][/Treatise on Thermodynamics/]] (Planck, 1897; 116), he begins his discussion
ocsenave@4	2126 of the second law in these words (translation of A. Ogg):
ocsenave@4	2127 #+begin_quote
ocsenave@4	2128 \ldquo{}We
ocsenave@4	2129 $\ldots$ put forward the following proposition $\ldots$ :
ocsenave@4	2130 /it is impossible to construct an engine which will work a complete cycle,
ocsenave@4	2131 and produce no effect except the raising of a weight and the cooling of a heat-reservoir./ Such an engine could be used simultaneously
ocsenave@4	2132 as a motor and a refrigerator without any waste of energy or
ocsenave@4	2133 material, and would in any case be the most profitable engine
ocsenave@4	2134 ever made. It would, it is true, not be equivalent to perpetual
ocsenave@4	2135 motion, for it does not produce work from nothing, but from the
ocsenave@4	2136 heat which it draws from the reservoir. It would not, therefore,
ocsenave@4	2137 like perpetual motion, contradict the principle of energy, but
ocsenave@4	2138 would nevertheless possess for man the essential advantage of
ocsenave@4	2139 perpetual motion, the supply of work without cost; for the in
ocsenave@4	2140 exhaustible supply of heat in the earth, in the atmosphere, and
ocsenave@4	2141 in the sea, would, like the oxygen of the atmosphere, be at
ocsenave@4	2142 everybody ‘s immediate disposal. For this reason we take the
ocsenave@4	2143 above proposition as our starting point. Since we are to deduce
ocsenave@4	2144 the second law from it, we expect, at the same time, to make a
ocsenave@4	2145 most serviceable application of any natural phenomenon which may
ocsenave@4	2146 be discovered to deviate from the second law.\rdquo{}
ocsenave@4	2147 #+end_quote
ocsenave@4	2148 The ammonia maser ([[../sources/Townes-Maser.pdf][Townes, 1954]]) is another example of an
ocsenave@4	2149 experimental device which, at first glance, violates the second
ocsenave@4	2150 law by providing \ldquo{}useful work\rdquo{} in the form of coherent microwave
ocsenave@4	2151 radiation at the expense of thermal energy. The ammonia molecule
ocsenave@4	2152 has two energy levels separated by 24.8 GHz, with a large electric
ocsenave@4	2153 dipole moment matrix element connecting them. We cannot obtain
ocsenave@4	2154 radiation from ordinary ammonia gas because the lower state
ocsenave@4	2155 populations are slightly greater than the upper, as given by
ocsenave@4	2156 the usual Boltzmann factors. However, if we release ammonia gas
ocsenave@4	2157 slowly from a tank into a vacuum so that a well-collimated jet
ocsenave@4	2158 of gas is produced, we can separate the upper state molecules
ocsenave@4	2159 from the lower. In an electric field, there is a quadratic
ocsenave@4	2160 Stark effect, the levels \ldquo{}repelling\rdquo{} each other according to
ocsenave@4	2161 the well-known rule of second-order perturbation theory. Thus,
ocsenave@4	2162 the thermally excited upper-state molecules have their energy
ocsenave@4	2163 raised further by a strong field; and vice versa for the lower
ocsenave@4	2164 state molecules. If the field is inhomogeneous, the result is
ocsenave@4	2165 that upper-state molecules experience a force drawing them into
ocsenave@4	2166 regions of weak field; and lower-state molecules are deflected
ocsenave@4	2167 toward strong field regions. The effect is so large that, in a
ocsenave@4	2168 path length of about 15 cm, one can achieve an almost complete
ocsenave@4	2169 spatial separation. The upper-state molecules then pass through
ocsenave@4	2170 a small hole into a microwave cavity, where they give up their
ocsenave@4	2171 energy in the form of coherent radiation.
ocsenave@4	2172
ocsenave@4	2173
ocsenave@4	2174 Again, we have something very similar to a Maxwell Demon;
ocsenave@4	2175 for without performing any work (since no current flows to the
ocsenave@4	2176 electrodes producing the deflecting field) we have separated
ocsenave@4	2177 the high-energy molecules from the low-energy ones, and obtained
ocsenave@4	2178 useful work from the former. This, too, was held to be
ocsenave@4	2179 impossible by some theorists before the experiment succeeded!
ocsenave@4	2180
ocsenave@4	2181 Later in this course, when we have learned how to formulate
ocsenave@4	2182 a general theory of irreversible processes, we will see that the
ocsenave@4	2183 second law can be extended to a new principle that tells us which
ocsenave@4	2184 nonequilibrium states can be reached, reproducibly, from others;
ocsenave@4	2185 and this will of course have a direct bearing on the question of
ocsenave@4	2186 perpetual motion machines of the second kind. However, the full
ocsenave@4	2187 implications of this generalized second law have not yet been
ocsenave@4	2188 worked out; our understanding has advanced just to the point
ocsenave@4	2189 where confident, dogmatic statements on either side now seem
ocsenave@4	2190 imprudent. For the present, therefore, we leave it as an open
ocsenave@4	2191 question whether such machines can or cannot be made.
ocsenave@4	2192
ocsenave@2	2193
ocsenave@5	2194
ocsenave@5	2195
ocsenave@5	2196
ocsenave@5	2197
ocsenave@5	2198
ocsenave@5	2199
ocsenave@5	2200
ocsenave@5	2201
ocsenave@5	2202
ocsenave@5	2203
ocsenave@5	2204
ocsenave@5	2205
ocsenave@5	2206
ocsenave@5	2207
ocsenave@5	2208
ocsenave@5	2209
ocsenave@5	2210
ocsenave@5	2211
ocsenave@5	2212
ocsenave@5	2213
ocsenave@5	2214
ocsenave@5	2215
ocsenave@5	2216
ocsenave@5	2217
ocsenave@5	2218
ocsenave@5	2219
ocsenave@5	2220 * COMMENT Use of Jacobians in Thermodynamics
ocsenave@5	2221
ocsenave@5	2222 Many students find that thermodynamics, although mathematically almost
ocsenave@5	2223 trivial, is nevertheless one of the most difficult subjects in their program.
ocsenave@5	2224 A large part of the blame for this lies in the extremely cumbersome partial
ocsenave@5	2225 derivative notation. In this chapter we develop a different mathematical
ocsenave@5	2226 scheme, with which thermodynamic derivations can be carried out more easily,
ocsenave@5	2227 and which gives a better physical insight into the meaning of thermodynamic
ocsenave@5	2228 relations.
ocsenave@5	2229
ocsenave@5	2230 *** COMMENT Editor's addendum
ocsenave@5	2231 #+begin_quote
ocsenave@5	2232 In order to help readers with the Jacobian material that follows, I
ocsenave@5	2233 have included this section of supplementary material. --- Dylan
ocsenave@5	2234 #+end_quote}
ocsenave@5	2235
ocsenave@5	2236 Suppose your experimental parameters consist of three variables
ocsenave@5	2237 $X,Y,Z$---say, volume, pressure, and temperature. Then the
ocsenave@5	2238 physically allowed combinations $\langle x,y,z\rangle$ of $X,Y,Z$
ocsenave@5	2239 comprise the /(equilibrium) state space/
ocsenave@5	2240 of your thermodynamic system; the set of these combinations forms a
ocsenave@5	2241 subset $\Omega$ of $\mathbb{R}^3$. (If there were four experimental
ocsenave@5	2242 parameters, the state space would be a subset of $\mathbb{R}^4$, and
ocsenave@5	2243 so on).
ocsenave@5	2244
ocsenave@5	2245 You can represent the flux of some physical quantities (such as
ocsenave@5	2246 heat, entropy, or number of moles) as a vector field spread throughout
ocsenave@5	2247 $\Omega$, i.e., a function $F:\Omega\rightarrow \mathbb{R}^n$ sending
ocsenave@5	2248 each state to the value of the vector at that state.
ocsenave@5	2249 When you trace out different paths through the state space
ocsenave@5	2250 $\gamma:[a,b]\rightarrow \Omega$, you can measure the net quantity
ocsenave@5	2251 exchanged by
ocsenave@5	2252
ocsenave@5	2253 \begin{equation}
ocsenave@5	2254 \text{net exchange} = \int_a^b (F\circ \gamma)\cdot \gamma^\prime.
ocsenave@5	2255 \end{equation}
ocsenave@5	2256
ocsenave@5	2257 Some quantities are conservative.
ocsenave@5	2258
ocsenave@5	2259 - If the vector field $F$ (representing the flux of a physical
ocsenave@5	2260 quantity) is in fact the gradient of some function
ocsenave@5	2261 $\varphi:\Omega\rightarrow \mathbb{R}$, then $F$ is conservative and
ocsenave@5	2262 $\varphi$ represents the value of the conserved quantity at each state.
ocsenave@5	2263 - In this case, the value of $\varphi$ is completely determined by
ocsenave@5	2264 specifying the values of the experimental parameters $X, Y, Z$. In
ocsenave@5	2265 particular, it doesn't matter by which path the state was reached.
ocsenave@5	2266
ocsenave@5	2267
ocsenave@5	2268 Some physical quantities (such as entropy or number of moles) are
ocsenave@5	2269 completely determined by your experimental parameters $X, Y, Z$. Others (such as
ocsenave@5	2270 heat) are not. For those quantities that are,
ocsenave@5	2271 you have functions $\phi:\Omega\rightarrow \mathbb{R}$ sending each state
ocsenave@5	2272 to the value of the quantity at that state.
ocsenave@5	2273
ocsenave@5	2274
ocsenave@5	2275
ocsenave@5	2276 and measure the change in physical
ocsenave@5	2277 quantities (like entropy or number of moles)
ocsenave@5	2278
ocsenave@5	2279
ocsenave@5	2280 Given your experimental parameters $X,Y,Z$, there may be other
ocsenave@5	2281 physical quantities (such as entropy or number of moles) which are uniquely
ocsenave@5	2282 defined by each combination of $\langle x,y,z\rangle$. Stated
ocsenave@5	2283 mathematically, there is a function $f:\Omega\rightarrow \mathbb{R}$
ocsenave@5	2284 sending each state to the value of the quantity at that state.
ocsenave@5	2285
ocsenave@5	2286
ocsenave@5	2287
ocsenave@5	2288 Now, sometimes you would like to use a different coordinate system to
ocsenave@5	2289 describe the same physical situation.
ocsenave@5	2290 A /change of variables/ is an
ocsenave@5	2291 invertible differentiable transformation $g:\mathbb{R}^n\rightarrow
ocsenave@5	2292 \mathbb{R}^n$---a function with $n$ input components (the $n$ old
ocsenave@5	2293 variables) and $n$ output components (the $n$ new variables), where
ocsenave@5	2294 each output component can depend on any number of the input components. For
ocsenave@5	2295 example, in two dimensions you can freely switch between Cartesian
ocsenave@5	2296 coordinates and polar coordinates; the familiar transformation is
ocsenave@5	2297
ocsenave@5	2298 $g\langle x, y\rangle \mapsto \langle \sqrt{x^2+y^2}, \arctan{(y/x)}\rangle$
ocsenave@5	2299
ocsenave@5	2300
ocsenave@5	2301
ocsenave@5	2302
ocsenave@5	2303
ocsenave@5	2304 ** Statement of the Problem
ocsenave@5	2305 In fields other than thermodynamics , one usually starts out by stating
ocsenave@5	2306 explicitly what variables shall be considered the independent ones, and then
ocsenave@5	2307 uses partial derivatives without subscripts, the understanding being that all
ocsenave@5	2308 independent variables other than the ones explicitly present are held constant
ocsenave@5	2309 in the differentiation. This convention is used in most of mathematics and
ocsenave@5	2310 physics without serious misunderstandings. But in thermodynamics, one never
ocsenave@5	2311 seems to be able to maintain a fixed set of independent variables throughout
ocsenave@5	2312 a derivation, and it becomes necessary to add one or more subscripts to every
ocsenave@5	2313 derivative to indicate what is being held constant. The often-needed
ocsenave@5	2314 transformation from one constant quantity to another involves the
ocsenave@5	2315 relation
ocsenave@5	2316
ocsenave@5	2317 \begin{equation}
ocsenave@5	2318 \left(\frac{\partial A}{\partial B}\right)_C = \left(\frac{\partial
ocsenave@5	2319 A}{\partial B}\right)_D + \left(\frac{\partial A}{\partial D}\right)_B \left(\frac{\partial D}{\partial B}\right)_C
ocsenave@5	2320 \end{equation}
ocsenave@5	2321
ocsenave@5	2322 which, although it expresses a fact that is mathematically trivial, assumes
ocsenave@5	2323 such a complicated form in the usual notation that few people can remember it
ocsenave@5	2324 long enough to write it down after the book is closed.
ocsenave@5	2325
ocsenave@5	2326 As a further comment on notation, we note that in thermodynamics as well
ocsenave@5	2327 as in mechanics and electrodynamics, our equations are made cumbersome if we
ocsenave@5	2328 are forced to refer at all times to some particular coordinate system (i.e.,
ocsenave@5	2329 set of independent variables). In the latter subjects this needless
ocsenave@5	2330 complication has long since been removed by the use of vector
ocsenave@5	2331 notation,
ocsenave@5	2332 which enables us to describe physical relationships without reference to any particular
ocsenave@5	2333 coordinate system. A similar house-cleaning can be effected for thermodynamics
ocsenave@5	2334 by use of jacobians, which enable us to express physical relationships without
ocsenave@5	2335 committing ourselves to any particular set of independent variables.
ocsenave@5	2336 We have here an interesting example of retrograde progress in science:
ocsenave@5	2337 for the historical fact is that use of jacobians was the original mathematical
ocsenave@5	2338 method of thermodynamics. They were used extensively by the founder of modern
ocsenave@5	2339 thermodynamics, Rudolph Clausius, in his work dating from about 1850. He used
ocsenave@5	2340 the notation
ocsenave@5	2341
ocsenave@5	2342 \begin{equation}
ocsenave@5	2343 D_{xy} \equiv \frac{\partial^2 Q}{\partial x\partial y} -
ocsenave@5	2344 \frac{\partial^2 Q}{\partial y \partial x}
ocsenave@5	2345 \end{equation}
ocsenave@5	2346
ocsenave@5	2347
ocsenave@5	2348 where $Q$ stands, as always, for heat, and $x$, $y$ are any
ocsenave@5	2349 two thermodynamic quantities. Since $dQ$ is not an exact differential,
ocsenave@5	2350 $D_{xy}$ is not identically zero. It is understandable that this notation, used in his published works, involved
ocsenave@5	2351 Clausius in many controversies, which in retrospect appear highly amusing. An
ocsenave@5	2352 account of some of them may be found in his book (Clausius, 1875). On the
ocsenave@5	2353 other hand, it is unfortunate that this occurred, because it is probably for
ocsenave@5	2354 this reason that the quantities $D_{xy}$ went out of general use for many years,
ocsenave@5	2355 with only few exceptions (See comments at the end of this chapter).
ocsenave@5	2356 In a footnote in Chapter II of Planck's famous treatise (Planck, 1897), he explains
ocsenave@5	2357 that he avoids using $dQ$ to represent an infinitesimal quantity of heat, because
ocsenave@5	2358 that would imply that it is the differential of some quantity $Q$. This in turn
ocsenave@5	2359 leads to the possibility of many fallacious arguments, all of which amount to
ocsenave@5	2360 setting $D_{xy}=0$. However, a reading of Clausius‘ works makes it clear that
ocsenave@5	2361 the quantities $D_{xy}$, when properly used, form the natural medium for discussion
ocsenave@5	2362 of thermodynamics. They enabled him to carry out certain derivations with a
ocsenave@5	2363 facility and directness which is conspicuously missing in most recent
ocsenave@5	2364 expositions. We leave it as an exercise for the reader to prove that $D_{xy}$ is a
ocsenave@5	2365 jacobian (Problem 2.1).
ocsenave@5	2366
ocsenave@5	2367 We now develop a condensed notation in which the algebra of jacobians
ocsenave@5	2368 may be surveyed as a whole, in a form easy to remember since the abstract
ocsenave@5	2369 relations are just the ones with which we are familiar in connection with the
ocsenave@5	2370 properties of commutators in quantum mechanics.
ocsenave@5	2371
ocsenave@5	2372 ** Formal Properties of Jacobians[fn::For any function $F:\mathbb{R}^n\rightarrow \mathbb{R}^n$, $F:\langle x_1,\ldots, x_n\rangle \mapsto \langle F_1(x), F_2(x),\ldots F_n(x)\rangle$ we can define the Jacobian matrix of $F$ to be $JF = \begin{bmatrix}\partial_1{F_1}&\ldots& \partial_n{F_n}\\\vdots&\ddots&\vdots\\\partial_1 F_n & \ldots & \partial_n F_n\\\end{bmatrix}$, and the Jacobian (determinant) of $f$ to be the determinant of this matrix (provided all partial derivatives exist). ]
ocsenave@5	2373 Consider first a system with only two degrees of freedom. We define
ocsenave@5	2374
ocsenave@5	2375 \begin{equation}
ocsenave@5	2376 [A,B] \equiv \frac{\partial(A,B)}{\partial(x,y)} =
ocsenave@5	2377 \left\|\begin{matrix}\frac{\partial A}{\partial x}& \frac{\partial
ocsenave@5	2378 A}{\partial y} \\
ocsenave@5	2379 \frac{\partial B}{\partial x} & \frac{\partial B}{\partial y} \end{matrix}\right\|
ocsenave@5	2380 \end{equation}
ocsenave@5	2381 where $x$, $y$ are any variables adequate to determine the state of the system.
ocsenave@5	2382
ocsenave@5	2383 Since for any change of variables, $x,y \mapsto x^\prime, y^\prime$ we
ocsenave@5	2384 have
ocsenave@5	2385
ocsenave@5	2386 \begin{equation}
ocsenave@5	2387 \frac{\partial(A,B)}{\partial(x^\prime,y^\prime)} = \frac{\partial(A,B)}{\partial(x,y)}\frac{\partial(x,y)}{\partial(x^\prime,y^\prime)}
ocsenave@5	2388 \end{equation}
ocsenave@5	2389
ocsenave@5	2390 or, in an easily understandable condensed notation,
ocsenave@5	2391
ocsenave@5	2392 \begin{equation}
ocsenave@5	2393 [A,B]^\prime = [A,B][x,y]^\prime
ocsenave@5	2394 \end{equation}
ocsenave@5	2395
ocsenave@5	2396 It follows that any equations that are homogeneous in the jacobians are in
ocsenave@5	2397 variant in form under "coordinate transformations“, so that we can suppress
ocsenave@5	2398 the independent variables x, y and carry out derivations without committing
ocsenave@5	2399 ourselves to any particular set.
ocsenave@5	2400 The algebra of these symbols is characterized by the following identities
ocsenave@5	2401 (the comma may be omitted if A, B are single letters). The properties of
ocsenave@5	2402 antisymmetry, linearity, and composition have the familiar form
ocsenave@5	2403 In addition we have three cyclic identities, easily proved:
ocsenave@5	2404 These relations are not all independent; for example, (2—ll) follows from
ocsenave@5	2405 (2-9) and (2-13).
ocsenave@5	2406 Putting dC = O in (2-9) , we obtain the rule
ocsenave@5	2407 by means of which equations are translated from one language to the other.
ocsenave@5	2408
ocsenave@5	2409
ocsenave@5	2410 From it one sees that the transformation law (2-l) now appears as a special
ocsenave@5	2411 case of the identity (2-11) . Writing for the enthalpy, free energy, and Gibbs
ocsenave@5	2412 function respectively ,
ocsenave@5	2413 where U is the internal energy with the property dU = t :35 — P (N, we have as
ocsenave@5	2414 consequences of (2-13) the relations
ocsenave@5	2415 The advantages of this notation is shown particularly when we consider the
ocsenave@5	2416 four Maxwe ll equati ons
ocsenave@5	2417 Applying (2-14) , we see that each reduces to the single identity
ocsenave@5	2418
ocsenave@5	2419
ocsenave@5	2420 Thus, all of the Maxwell equations are expressions in different "coordinate
ocsenave@5	2421 systems" of the same basic fact (2-18) , which will receive a physical inter
ocsenave@5	2422 pretation in Sec. 2.4. In a derivation, such as that of Eq. (1-49) , every
ocsenave@5	2423 thing that can be gained by using any of the equations (2-17) is already
ocsenave@5	2424 accomplished by application of the single relation (2-18).
ocsenave@5	2425 Jacobians which involve the entropy in combinations other than are
ocsenave@5	2426 related to various specific heats. The heat capacity at constant X is
ocsenave@5	2427 and, using (2-14) we obtain the identity
ocsenave@5	2428 C
ocsenave@5	2429 In the simplest derivations, application of (2-18) or (2—20) is the essential
ocsenave@5	2430 step.
ocsenave@5	2431 In his well-known textbook, Zemansky (1943) shows that many of the ele
ocsenave@5	2432 mentary derivations in thermodynamics may be reduced to application of the
ocsenave@5	2433 In the above notation these equations are far from obvious and not easy to
ocsenave@5	2434 remember. Note, however, that the T :38 equations are special cases of the
ocsenave@5	2435 cyclic identity (2-9) for the sets of variables {TVS}, respectively,
ocsenave@5	2436 while the energy equation is a consequence of (2-13) and the Maxwell relation:
ocsenave@5	2437
ocsenave@5	2438
ocsenave@5	2439 From (2~l4) we see that this is the energy equation in jacobian notation.
ocsenave@5	2440 2 .3 Elementary Examples
ocsenave@5	2441 In a large class of problems, the objective is to express some quantity
ocsenave@5	2442 of interest, or some condition of interest, in terms of experimentally mea
ocsenave@5	2443 surable quantities. Therefore, the “sense of direction“ in derivations is
ocsenave@5	2444 provided by the principle that we want to get rid of any explicit appearance
ocsenave@5	2445 of the entropy and the various energies U, H, F, G. Thus, if the entropy
ocsenave@5	2446 appears in the combination [TS], we use the Maxwell relation to replace it
ocsenave@5	2447 with . If it appears in some other combination , we can use the
ocsenave@5	2448 identity (2-20) .
ocsenave@5	2449 Similarly, if combinations such as or [UX] appear, we can use (2-16)
ocsenave@5	2450 and replace them with
ocsenave@5	2451 it cannot be eliminated in this way. However, since in phenomenological
ocsenave@5	2452 thermodynamics the absolute value of the entropy has no meaning, this situa
ocsenave@5	2453 tion cannot arise in any expression representing a definite physical quantity.
ocsenave@5	2454 For problems of this simplest type, the jacobian formalism works like a
ocsenave@5	2455 well-oiled machine, as the following examples show. We denote the isothermal
ocsenave@5	2456 compressibility, thermal expansion coefficient, and ratio of specific heats
ocsenave@5	2457 bY K1 5: Y, réspectively:
ocsenave@5	2458
ocsenave@5	2459
ocsenave@5	2460 and note that from (2-27) and (2-28) we have
ocsenave@5	2461 (2-30)
ocsenave@5	2462 Several derivatives, chosen at random, are now evaluated in terms of these
ocsenave@5	2463 quantities:
ocsenave@5	2464 A more difficult type of problem is the following: We have given a num
ocsenave@5	2465 ber of quantities and wish to find the general relation, if any, connecting
ocsenave@5	2466 them. In one sense, the question whether relations exist can be answered
ocsenave@5	2467
ocsenave@5	2468
ocsenave@5	2469 immediately; for any two quantities A, B a necessary and sufficient condition
ocsenave@5	2470 for the existence of a functional relation A f(B) in a region R is:
ocsenave@5	2471 = O in R}. In a system of two degrees of freedom it is clear that between
ocsenave@5	2472 any three quantities A, B, C there is necessarily at least one functional
ocsenave@5	2473 relation f(A,B,C) = O, as is implied by the identity (2-9) [Problem 2.2] . An
ocsenave@5	2474 example is the equation of state f(PVT) = O. This , however, is not the type
ocsenave@5	2475 of relation one usually has in mind. For each choice of A, B, C and each
ocsenave@5	2476 particular system of two degrees of freedom, some functional relationship
ocsenave@5	2477 must exist, but in general it will depend on the physical nature of the system
ocsenave@5	2478 and can be obtained only when one has sufficient information, obtained from
ocsenave@5	2479 measurement or theory, about the system.
ocsenave@5	2480 The problem is rather to find those relations between various quantities
ocsenave@5	2481 which hold generally, regardless of the nature of the particular system.
ocsenave@5	2482 Mathematically, all such relations are trivial in the sense that they must be
ocsenave@5	2483 special cases of the basic identities already given. Their physical meaning
ocsenave@5	2484 may, however, be far from trivial and they may be difficult to find. Note,
ocsenave@5	2485 for example, that the derivative computed in (2-35) is just the Joule—Thomson
ocsenave@5	2486 coefficient 11. Suppose the problem had been stated as: "Given the five
ocsenave@5	2487 quantities V, Cp, 8, determine whether there is a general relation
ocsenave@5	2488 between them and if so find it." Now, although a repetition of the argument
ocsenave@5	2489 of (2-35) would be successful in this case, this success must be viewed as a
ocsenave@5	2490 lucky accident from ‘the standpoint of the problem just formulated. It is not
ocsenave@5	2491 a general rule for attacking this type of problem because there is no way of
ocsenave@5	2492 ensuring that the answer will come out in terms of the desired quantities.
ocsenave@5	2493 To illustrate a general rule of procedure, consider the problem of find
ocsenave@5	2494 ing a relationship, if any, between iCp, CV, V, T, B, K}. First we write
ocsenave@5	2495 these quantities in terms of jacobians.
ocsenave@5	2496
ocsenave@5	2497
ocsenave@5	2498 At this point we make a definite choice of some coordinate system. Since
ocsenave@5	2499 [TP] occurs more often than any other jacobian, we adopt x = T, y = P as the
ocsenave@5	2500 The variables in jacobians are P, V, T, S, for which (2-11) gives
ocsenave@5	2501 [PV][TS] + [VT] [PS] + = 0 (2-40)
ocsenave@5	2502 or, in this case
ocsenave@5	2503 Substituting the expressions (2-39) into this we obtain
ocsenave@5	2504 or, rearranging, we have the well—known law
ocsenave@5	2505 which is now seen as a special case of (2-11).
ocsenave@5	2506 There are several points to notice in this derivation: (1) no use has
ocsenave@5	2507 been made of the fact that the quantities T, V were given explicitly; the
ocsenave@5	2508
ocsenave@5	2509
ocsenave@5	2510
ocsenave@5	2511
ocsenave@5	2512
ocsenave@5	2513
ocsenave@5	2514
ocsenave@5	2515
ocsenave@5	2516
ocsenave@5	2517
ocsenave@5	2518
ocsenave@5	2519
ocsenave@5	2520
ocsenave@5	2521
ocsenave@5	2522
ocsenave@5	2523
ocsenave@5	2524
ocsenave@5	2525
ocsenave@5	2526
ocsenave@5	2527
ocsenave@5	2528
ocsenave@5	2529
ocsenave@5	2530
ocsenave@5	2531
ocsenave@5	2532
ocsenave@5	2533
ocsenave@5	2534
ocsenave@5	2535
ocsenave@5	2536
ocsenave@5	2537
ocsenave@5	2538
ocsenave@5	2539
ocsenave@5	2540
ocsenave@5	2541
ocsenave@5	2542
ocsenave@5	2543
ocsenave@5	2544
ocsenave@5	2545
ocsenave@5	2546
ocsenave@5	2547
ocsenave@5	2548
ocsenave@5	2549 * Gibbs Formalism \mdash{} Physical Derivation
ocsenave@5	2550
ocsenave@5	2551
ocsenave@5	2552 In this Chapter we present physical arguments by which the Gibbs
ocsenave@5	2553 formalism can be derived and justified, deliberately avoiding all use
ocsenave@5	2554 of probability theory. This will serve to convince us of the /validity/ of Gibbs’ formalism
ocsenave@5	2555 for the particular applications given by Gibbs, and will give us an intuitive
ocsenave@5	2556 physical understanding of the second law, as well as the physical meaning of
ocsenave@5	2557 the Kelvin temperature.
ocsenave@5	2558
ocsenave@5	2559 Later on (Chapter 9) we will present an entirely different derivation in
ocsenave@5	2560 terms of a general problem of statistical estimation, deliberately avoiding
ocsenave@5	2561 all use of physical ideas, and show that the identical mathematical formalism
ocsenave@5	2562 emerges. This will serve to convince us of the /generality/ of the
ocsenave@5	2563 Gibbs methods, and show that their applicability is in no way restricted to equilibrium
ocsenave@5	2564 problems; or indeed, to physics.
ocsenave@5	2565
ocsenave@5	2566
ocsenave@5	2567 It is interesting to note that most of Gibbs‘ important results were
ocsenave@5	2568 found independently and almost simultaneously by Einstein (1902); but it is
ocsenave@5	2569 to Gibbs that we owe the elegant mathematical formulation of the theory. In
ocsenave@5	2570 the following we show how, from mechanical considerations involving
ocsenave@5	2571 the microscopic state of a system, the Gibbs rules emerge as a
ocsenave@5	2572 description of equilibrium macroscopic properties. Having this, we can then reason
ocsenave@5	2573 backwards, and draw inferences about microscopic conditions from macroscopic experimental
ocsenave@5	2574 data. We will consider only classical mechanics here; however, none of this
ocsenave@5	2575 classical theory will have to be unlearned later, because the Gibbs formalism
ocsenave@5	2576 lost none of its validity through the development of quantum theory. Indeed,
ocsenave@5	2577 the full power of Gibbs‘ methods has been realized only through their
ocsenave@5	2578 successful application to quantum theory.
ocsenave@5	2579
ocsenave@5	2580 ** COMMENT Review of Classical Mechanics (SICM)
ocsenave@5	2581 In [[http://mitpress.mit.edu/sicm/][classical mechanics]] a complete description of the state of a system is
ocsenave@5	2582 given by specifying $n$ coordinates $q_1\ldots q_n$, and the corresponding velocities
ocsenave@5	2583 $D{q}_1\ldots Dq_n$. The equations of motion are then determined by a Lagrangian function
ocsenave@5	2584 which in simple mechanical problems is
ocsenave@5	2585
ocsenave@5	2586 \begin{equation}
ocsenave@5	2587 L(t,q(t),Dq(t)) = T - V
ocsenave@5	2588 \end{equation}
ocsenave@5	2589
ocsenave@5	2590
ocsenave@5	2591 where $T$ and $V$ are the kinetic and potential energies. In problems involving
ocsenave@5	2592 coupling of particles to an electromagnetic field, the Lagrangian function
ocsenave@5	2593 takes a more general form, as we will see later. In either case, the
ocsenave@5	2594 equations of motion are
ocsenave@5	2595
ocsenave@5	2596 \begin{equation}
ocsenave@5	2597 D(\partial_2 L \circ \Gamma[q]) - \partial_1 L \circ \Gamma[q] = 0
ocsenave@5	2598 \end{equation}
ocsenave@5	2599
ocsenave@5	2600 where $\Gamma[q]$ is the function $t\mapsto \langle
ocsenave@5	2601 t,q(t),Dq(t)\rangle$, and $\partial_i$ denotes the derivative with
ocsenave@5	2602 respect to the $i$th argument ($i=0,1,2,\ldots$).
ocsenave@5	2603
ocsenave@5	2604 The advantage of the Lagrangian form (5-2) over the original Newtonian form
ocsenave@5	2605 (to which it is completely equivalent in simple mechanical problems)
ocsenave@5	2606
ocsenave@5	2607 \begin{equation}
ocsenave@5	2608 D^2 (m\cdot x(t)) = -\partial_1 V \circ \Gamma[x]
ocsenave@5	2609 \end{equation}
ocsenave@5	2610
ocsenave@5	2611 is that (5-2) holds for arbitrary choices of the coordinates $q_i$;
ocsenave@5	2612 they can include angles, or any other parameters which serve to locate a particle in
ocsenave@5	2613 space. The Newtonian equations (5-3), on the other hand, hold only when the
ocsenave@5	2614 $x_i$ are rectangular (cartesian) coordinates of a particle.
ocsenave@5	2615 Still more convenient for our purposes is the Hamiltonian form of the
ocsenave@5	2616 equations of motion. Define the momentum \ldquo{}canonically
ocsenave@5	2617 conjugate\rdquo{} to the
ocsenave@5	2618 coordinate $q$ by
ocsenave@5	2619
ocsenave@5	2620 \begin{equation}
ocsenave@5	2621 p(t) \equiv \partial_1 L \circ \Gamma[q]
ocsenave@5	2622 \end{equation}
ocsenave@5	2623
ocsenave@5	2624 let $\mathscr{V}(t,q,p) = Dq$, and define a Hamiltonian function $H$ by
ocsenave@5	2625
ocsenave@5	2626 \begin{equation}
ocsenave@5	2627 H(t,q,p) = p \cdot V(t,q,p) - L(t,q, V(t,q,p)
ocsenave@5	2628 \end{equation}
ocsenave@5	2629
ocsenave@5	2630 the notation indicating that after forming the right-hand side of (5-5) the
ocsenave@5	2631 velocities $\dot{q}_i$ are eliminated mathematically, so that the
ocsenave@5	2632 Hamiltonian is
ocsenave@5	2633 expressed as a function of the coordinates and momenta only.
ocsenave@5	2634
ocsenave@5	2635 #+begin_quote
ocsenave@5	2636 ------
ocsenave@5	2637 Problem (5.1). A particle of mass $m$ is located by specifying
ocsenave@5	2638 $(q_1,q_2,q_3)=(r,\theta,z)$ respectively, where $r$, $\theta$, $z$
ocsenave@5	2639 are a cylindrical coordinate system
ocsenave@5	2640 related to the cartesian $x, y, z$ by $x + iy = re^{i\theta}$, $z=z$. The
ocsenave@5	2641 particle moves in a potential $V(q_1,q_2,q_3)$. Show that the
ocsenave@5	2642 Hamiltonian in this coordinate system is
ocsenave@5	2643
ocsenave@5	2644 \begin{equation}
ocsenave@5	2645 H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m}+\frac{p_3^2}{2m} + V(q_1,q_2,q_3)
ocsenave@5	2646 \end{equation}
ocsenave@5	2647
ocsenave@5	2648 and discuss the physical meaning of $p_1$, $p_2$, $p_3$.
ocsenave@5	2649 ------
ocsenave@5	2650
ocsenave@5	2651
ocsenave@5	2652
ocsenave@5	2653 Problem (5.2). Find the Hamiltonian for the same particle, in the spherical
ocsenave@5	2654 coordinate system $(q_1,q_2,q_3) = (r,\theta,\phi)$ related to the
ocsenave@5	2655 Cartesian by
ocsenave@5	2656 $x + iy = r\,\sin{\theta}\,e^{i\phi}$, $z=r\,\cos{\theta}$., and again
ocsenave@5	2657 discuss the physical meaning of $p_1$, $p_2$, $p_3$ .
ocsenave@5	2658 ------
ocsenave@5	2659 #+end_quote
ocsenave@5	2660
ocsenave@5	2661
ocsenave@5	2662
ocsenave@5	2663
ocsenave@5	2664
ocsenave@5	2665
ocsenave@5	2666
ocsenave@5	2667
ocsenave@5	2668
ocsenave@5	2669
ocsenave@5	2670 ** Review of Classical Mechanics
ocsenave@5	2671 In [[http://mitpress.mit.edu/sicm/][classical mechanics]] a complete description of the state of a system is
ocsenave@5	2672 given by specifying $n$ coordinates $q_1\ldots q_n$, and the corresponding velocities
ocsenave@5	2673 $\dot{q}_1\ldots \dot{q}_n$. The equations of motion are then determined by a Lagrangian function
ocsenave@5	2674 which in simple mechanical problems is
ocsenave@5	2675
ocsenave@5	2676 \begin{equation}
ocsenave@5	2677 L(q_i,\dot{q}_i) = T - V
ocsenave@5	2678 \end{equation}
ocsenave@5	2679
ocsenave@5	2680
ocsenave@5	2681 where $T$ and $V$ are the kinetic and potential energies. In problems involving
ocsenave@5	2682 coupling of particles to an electromagnetic field, the Lagrangian function
ocsenave@5	2683 takes a more general form, as we will see later. In either case, the
ocsenave@5	2684 equations of motion are
ocsenave@5	2685
ocsenave@5	2686 \begin{equation}
ocsenave@5	2687 \frac{\partial L}{\partial q_i} - \frac{d}{dt}\frac{\partial
ocsenave@5	2688 L}{\partial \dot{q}_i} = 0.
ocsenave@5	2689 \end{equation}
ocsenave@5	2690
ocsenave@5	2691 The advantage of the Lagrangian form (5-2) over the original Newtonian form
ocsenave@5	2692 (to which it is completely equivalent in simple mechanical problems)
ocsenave@5	2693
ocsenave@5	2694 \begin{equation}
ocsenave@5	2695 m\ddot{x}_i = -\frac{\partial V}{\partial x_i}
ocsenave@5	2696 \end{equation}
ocsenave@5	2697
ocsenave@5	2698 is that (5-2) holds for arbitrary choices of the coordinates $q_i$;
ocsenave@5	2699 they can include angles, or any other parameters which serve to locate a particle in
ocsenave@5	2700 space. The Newtonian equations (5-3), on the other hand, hold only when the
ocsenave@5	2701 $x_i$ are rectangular (cartesian) coordinates of a particle.
ocsenave@5	2702 Still more convenient for our purposes is the Hamiltonian form of the
ocsenave@5	2703 equations of motion. Define the momentum \ldquo{}canonically
ocsenave@5	2704 conjugate\rdquo{} to the
ocsenave@5	2705 coordinate $q_i$ by
ocsenave@5	2706
ocsenave@5	2707 \begin{equation}
ocsenave@5	2708 p_i \equiv \frac{\partial L}{\partial q_i}
ocsenave@5	2709 \end{equation}
ocsenave@5	2710
ocsenave@5	2711 and a Hamiltonian function $H$ by
ocsenave@5	2712
ocsenave@5	2713 \begin{equation}
ocsenave@5	2714 H(q_1,p_1;\cdots ; q_n,p_n) \equiv \sum_{i=1}^n p\cdot \dot{q}_i -
ocsenave@5	2715 L(q_1,\ldots, q_n).
ocsenave@5	2716 \end{equation}
ocsenave@5	2717
ocsenave@5	2718 the notation indicating that after forming the right-hand side of (5-5) the
ocsenave@5	2719 velocities $\dot{q}_i$ are eliminated mathematically, so that the Hamiltonian is ex
ocsenave@5	2720 pressed as a function of the coordinates and momenta only.
ocsenave@5	2721
ocsenave@5	2722 #+begin_quote
ocsenave@5	2723 ------
ocsenave@5	2724 Problem (5.1). A particle of mass $m$ is located by specifying
ocsenave@5	2725 $(q_1,q_2,q_3)=(r,\theta,z)$ respectively, where $r$, $\theta$, $z$
ocsenave@5	2726 are a cylindrical coordinate system
ocsenave@5	2727 related to the cartesian $x, y, z$ by $x + iy = re^{i\theta}$, $z=z$. The
ocsenave@5	2728 particle moves in a potential $V(q_1,q_2,q_3)$. Show that the
ocsenave@5	2729 Hamiltonian in this coordinate system is
ocsenave@5	2730
ocsenave@5	2731 \begin{equation}
ocsenave@5	2732 H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m}+\frac{p_3^2}{2m} + V(q_1,q_2,q_3)
ocsenave@5	2733 \end{equation}
ocsenave@5	2734
ocsenave@5	2735 and discuss the physical meaning of $p_1$, $p_2$, $p_3$.
ocsenave@5	2736 ------
ocsenave@5	2737
ocsenave@5	2738
ocsenave@5	2739
ocsenave@5	2740 Problem (5.2). Find the Hamiltonian for the same particle, in the spherical
ocsenave@5	2741 coordinate system $(q_1,q_2,q_3) = (r,\theta,\phi)$ related to the
ocsenave@5	2742 Cartesian by
ocsenave@5	2743 $x + iy = r\,\sin{\theta}\,e^{i\phi}$, $z=r\,\cos{\theta}$., and again
ocsenave@5	2744 discuss the physical meaning of $p_1$, $p_2$, $p_3$ .
ocsenave@5	2745 ------
ocsenave@5	2746 #+end_quote
ocsenave@5	2747
ocsenave@5	2748 In terms of the Hamiltonian, the equations of motion assume a more
ocsenave@5	2749 symmetrical form:
ocsenave@5	2750
ocsenave@5	2751 \begin{equation}
ocsenave@5	2752 \cdot{q}_i = \frac{\partial H}{\partial p_i}\qquad \dot{p}_i =
ocsenave@5	2753 -\frac{\partial H}{\partial q_i}
ocsenave@5	2754 \end{equation}
ocsenave@5	2755
ocsenave@5	2756 of which the first follows from the definition (5-5) , while the second is
ocsenave@5	2757 equivalent to (5-2).
ocsenave@5	2758
ocsenave@5	2759 The above formulation of mechanics holds only when all forces are
ocsenave@5	2760 conservative; i.e. derivable from a potential energy function
ocsenave@5	2761 $V(q_1,\ldots q_n)$ , and
ocsenave@5	2762 in this case the Hamiltonian is numerically equal to the total energy $(T + V)$.
ocsenave@5	2763 Often, in addition to the conservative forces we have non-conservative ones
ocsenave@5	2764 which depend on the velocities as well as the coordinates. The Lagrangian
ocsenave@5	2765 and Hamiltonian form of the equations of motion can be preserved if there
ocsenave@5	2766 exists a new potential function $M(q_i,\dot{q}_i)$ such that the non-conservative force
ocsenave@5	2767 acting on coordinate $q_i$ is
ocsenave@5	2768
ocsenave@5	2769 \begin{equation}
ocsenave@5	2770 F_i = \frac{d}{dt}\frac{\partial M}{\partial \dot{q}_i} -
ocsenave@5	2771 \frac{\partial M}{\partial q_i}
ocsenave@5	2772 \end{equation}
ocsenave@5	2773
ocsenave@5	2774 We then define the Lagrangian as $L \equiv T - V - M$.
ocsenave@5	2775
ocsenave@5	2776 #+begin_quote
ocsenave@5	2777 ------
ocsenave@5	2778 Problem (5.3). Show that the Lagrangian equations of motion (5-2)
ocsenave@5	2779 are correct with this modified Lagrangian. Find the new momenta and
ocsenave@5	2780 Hamiltonian. Carry this through explicitly for the case of a charged particle moving in a
ocsenave@5	2781 time-varying electromagnetic field $\vec{E}(x,y,z,t),
ocsenave@5	2782 \vec{H}(x,y,z,t)$, for which the
ocsenave@5	2783 non-conservative force is given by the Lorentz force law,
ocsenave@5	2784
ocsenave@5	2785 $\vec{F} = e\left(\vec{E} + \frac{1}{c}\vec{v} \times \vec{B}\right)$
ocsenave@5	2786
ocsenave@5	2787 # Jaynes wrote \dot{A}. typo?
ocsenave@5	2788 /Hint:/ Express the potential $M$ in terms of the vector and scalar
ocsenave@5	2789 potentials of the field $\vec{A},\phi,$ defined by
ocsenave@5	2790 \(\vec{B}=\vec{\nabla}\times\vec{A},
ocsenave@5	2791 \vec{E}=-\vec{\nabla}{\phi}-\frac{1}{c}\vec{A}\).
ocsenave@5	2792 Notice that, since the potentials are not uniquely determined by $E$, $H$, there is no longer any
ocsenave@5	2793 unique connection between momentum and velocity; or between the Hamiltonian
ocsenave@5	2794 and the energy. Nevertheless, the Lagrangian and Hamiltonian equations of
ocsenave@5	2795 motion still describe the correct physical laws.
ocsenave@5	2796 -----
ocsenave@5	2797 #+end_quote
ocsenave@5	2798 ** Liouville's Theorem
ocsenave@5	2799 The Hamiltonian form (5-7) is of particular value because of the following
ocsenave@5	2800 property. Let the coordinates and momenta $(q_1,p_1;\ldots;q_n,p_n)$
ocsenave@5	2801 be regarded as coordinates of a single point in a $2n$-dimensional /phase space/. This point moves,
ocsenave@5	2802 by virtue of the equations of motion, with a velocity $v$ whose
ocsenave@5	2803 components are $\langle \dot{q}_1, \dot{p}_1; \ldots; \dot{q}_n,\dot{p}_n\rangle$.
ocsenave@5	2804 At each point of phase space there is specified in this way a
ocsenave@5	2805 particular velocity, and the equations of motion thus generate a continuous
ocsenave@5	2806 flow pattern in phase space, much like the flow pattern of a fluid in ordinary
ocsenave@5	2807 space. The divergence of the velocity of this flow pattern is
ocsenave@5	2808
ocsenave@5	2809 \begin{eqnarray}
ocsenave@5	2810 \vec{\nabla}\cdot {v}&=&\sum_{i=1}^n \left[\frac{\partial \dot{q}_i}{\partial q_i} +
ocsenave@5	2811 \frac{\partial \dot{p}_i}{\partial p_i}\right]\\
ocsenave@5	2812 &=& \sum_{i=1}^n \left[\frac{\partial^2 H}{\partial q_i \partial
ocsenave@5	2813 p_i}-\frac{\partial^2 H}{\partial p_i \partial q_i}\right]\\
ocsenave@5	2814 &=& 0
ocsenave@5	2815 \end{eqnarray}
ocsenave@5	2816
ocsenave@5	2817 # note: this is a sort of Jacobian determinant/commutator\|((d_q q_p)(d_q d_p))\|
ocsenave@5	2818
ocsenave@5	2819 so that the flow in phase space corresponds to that of an [[http://en.wikipedia.org/wiki/Incompressible_flow][incompressible fluid]].
ocsenave@5	2820 In an incompressible flow, the volume occupied by any given mass of the
ocsenave@5	2821 fluid remains constant as time goes on and the mass of fluid is carried into
ocsenave@5	2822 various regions. An exactly analogous property holds in phase space by virtue
ocsenave@5	2823 of (5-9). Consider at time $t = 0$ any $2n$-dimensional region
ocsenave@5	2824 $\Gamma_0$ consisting of some possible range of initial conditions
ocsenave@5	2825 $q_i(O), p_i(O)$ for a mechanical system, as shown in Fig. (5.1). This region has a total phase volume
ocsenave@5	2826
ocsenave@5	2827 \begin{equation}
ocsenave@5	2828 \Omega(0) = \int_{\Gamma_{0}} dq_1\ldots dp_n
ocsenave@5	2829 \end{equation}
ocsenave@5	2830
ocsenave@5	2831 In time t, each point $\langle q_1(O) \ldots p_n(O)\rangle$ of
ocsenave@5	2832 $\Gamma_0$ is carried, by the equations of
ocsenave@5	2833 motion, into a new point $\langle q_1(t),\ldots,p_n(t)\rangle$. The totality of all points which
ocsenave@5	2834 were originally in $\Gamma_0$ now defines a new region $\Gamma_t$ with phase volume
ocsenave@5	2835
ocsenave@5	2836 $\Omega(t) = \int_{\Gamma_{t}} dq_1\ldots dp_n$
ocsenave@5	2837
ocsenave@5	2838 and from (5-9) it can be shown that
ocsenave@5	2839
ocsenave@5	2840 \begin{equation}
ocsenave@5	2841 \Omega(t) = \Omega(0)
ocsenave@5	2842 \end{equation}
ocsenave@5	2843
ocsenave@5	2844 #+caption: Figure 5.1: Volume-conserving flow in phase space.
ocsenave@5	2845 [[../images/volume-conserved.jpg]]
ocsenave@5	2846
ocsenave@5	2847
ocsenave@5	2848 An equivalent statement is that the Jacobian determinant of the
ocsenave@5	2849 transformation \( \langle q_1(0), \ldots, p_n(0)\rangle \mapsto
ocsenave@5	2850 \langle q_1(t), \ldots , p_n(t)\rangle \) is identically equal to
ocsenave@5	2851 unity:
ocsenave@5	2852
ocsenave@5	2853 \begin{equation}
ocsenave@5	2854 \frac{\partial(q_{1t},\ldots p_{nt})}{\partial(q_{10}\ldots q_{n0})} =
ocsenave@5	2855 \left\|
ocsenave@5	2856 \begin{matrix}
ocsenave@5	2857 \frac{\partial q_{1t}}{\partial q_{10}}&\cdots &
ocsenave@5	2858 \frac{\partial p_{nt}}{\partial q_{10}}\\
ocsenave@5	2859 \vdots&\ddots&\vdots\\
ocsenave@5	2860 \frac{\partial q_{1t}}{\partial p_{n0}}&\cdots &
ocsenave@5	2861 \frac{\partial p_{nt}}{\partial p_{n0}}\\
ocsenave@5	2862 \end{matrix}\right\| = 1
ocsenave@5	2863 \end{equation}
ocsenave@5	2864
ocsenave@5	2865 #+begin_quote
ocsenave@5	2866 ------
ocsenave@5	2867 Problem (5.4). Prove that (5-9), (5-11), and (5-12) are equivalent statements.
ocsenave@5	2868 (/Hint:/ See A. I. Khinchin, /Mathematical Foundations of Statistical
ocsenave@5	2869 Mechanics/, Chapter II.)
ocsenave@5	2870 ------
ocsenave@5	2871 #+end_quote
ocsenave@5	2872
ocsenave@5	2873 This result was termed by Gibbs the \ldquo{}Principle of conservation
ocsenave@5	2874 of extension-in—phase\rdquo{}, and is usually referred to nowadays as /Liouville's theorem/.
ocsenave@5	2875 An important advantage of considering the motion of a system referred to phase
ocsenave@5	2876 space (coordinates and momenta) instead of the coordinate—velocity space of
ocsenave@5	2877 the Lagrangian is that in general no such conservation law holds in the latter
ocsenave@5	2878 space (although they amount to the same thing in the special case where all
ocsenave@5	2879 the $q_i$ are cartesian coordinates of particles and all forces are conservative
ocsenave@5	2880 in the sense of Problem 5.3).
ocsenave@5	2881
ocsenave@5	2882 #+begin_quote
ocsenave@5	2883 ------
ocsenave@5	2884 Problem (5.5). Liouville's theorem holds only because of the special form of
ocsenave@5	2885 the Hamiltonian equations of motion, which makes the divergence (5-9)
ocsenave@5	2886 identically zero. Generalize it to a mechanical system whose state is defined by a
ocsenave@5	2887 set of variables $\{x_1,x_2,\ldots,x_n\}$ with equations of motion for
ocsenave@5	2888 $x_i(t)$:
ocsenave@5	2889 \begin{equation}
ocsenave@5	2890 \dot{x}_i(t) = f_i(x_1,\ldots,x_n),\qquad i=1,2,\ldots,n
ocsenave@5	2891 \end{equation}
ocsenave@5	2892
ocsenave@5	2893 The jacobian (5-12) then corresponds to
ocsenave@5	2894
ocsenave@5	2895 \begin{equation}
ocsenave@5	2896 J(x_1(0),\ldots,x_n(0);t) \equiv \frac{\partial[x_1(t),\ldots, x_n(t)]}{\partial[x_1(0),\ldots,x_n(0)]}
ocsenave@5	2897 \end{equation}
ocsenave@5	2898
ocsenave@5	2899 Prove that in place of Liouville's theorem $J=1=\text{const.}$, we now
ocsenave@5	2900 have
ocsenave@5	2901
ocsenave@5	2902 \begin{equation}
ocsenave@5	2903 $J(t) = J(0)\,\exp\left[\int_0^t \sum_{i=1}^n \frac{\partial
ocsenave@5	2904 f[x_1(t),\ldots, x_n(t)]}{\partial x_i(t)}
ocsenave@5	2905 dt\right].
ocsenave@5	2906 \end{equation}
ocsenave@5	2907 ------
ocsenave@5	2908 #+end_quote
ocsenave@5	2909
ocsenave@5	2910 ** The Structure Function
ocsenave@5	2911
ocsenave@5	2912 One of the essential dynamical properties of a system, which controls its
ocsenave@5	2913 thermodynamic properties, is the total phase volume compatible with various
ocsenave@5	2914 experimentally observable conditions. In particular, for a system in which
ocsenave@5	2915 the Hamiltonian and the energy are the same, the total phase volume below a
ocsenave@5	2916 certain energy $E$ is
ocsenave@5	2917
ocsenave@5	2918 \begin{equation}
ocsenave@5	2919 \Omega(E) = \int \vartheta[E-H(q_i,p_i)] dq_i\ldots dp_n
ocsenave@5	2920 \end{equation}
ocsenave@5	2921 (When limits of integration are unspecified, we understand integration over
ocsenave@5	2922 all possible values of qi, pi.) In (5-16) , $\vartheta(x)$ is the unit
ocsenave@5	2923 step function
ocsenave@5	2924
ocsenave@5	2925 \begin{equation}
ocsenave@5	2926 \vartheta(x) \equiv \begin{cases}1,&x>0\\ 0,&x<0\end{cases}
ocsenave@5	2927 \end{equation}
ocsenave@5	2928
ocsenave@5	2929 The differential phase volume, called the /structure function/, is
ocsenave@5	2930 given by
ocsenave@5	2931 \begin{equation}
ocsenave@5	2932 \rho(E) = \frac{d\Omega}{dE} = \int \delta[E-H(q_i,p_i)] dq_1\ldots dp_n
ocsenave@5	2933 \end{equation}
ocsenave@5	2934
ocsenave@5	2935 and it will appear presently that essentially all thermodynamic properties of
ocsenave@5	2936 the system are known if $\rho(E)$ is known, in its dependence on such parameters
ocsenave@5	2937 as volume and mole numbers.
ocsenave@5	2938
ocsenave@5	2939
ocsenave@5	2940 Calculation of $\rho(E)$ directly from the definition (5-18) is generally
ocsenave@5	2941 very difficult. It is much easier to calculate first its [[http://en.wikipedia.org/wiki/Laplace_transform][Laplace transform]],
ocsenave@5	2942 known as the /partition function/:
ocsenave@5	2943
ocsenave@5	2944 \begin{equation}
ocsenave@5	2945 Z(\beta) = \int_0^\infty e^{-\beta E} \rho(E)\, dE
ocsenave@5	2946 \end{equation}
ocsenave@5	2947
ocsenave@5	2948 where we have assumed that all possible values of energy are positive; this
ocsenave@5	2949 can always be accomplished for the systems of interest by
ocsenave@5	2950 appropriately choosing the zero from which we measure energy. In addition, it will develop that
ocsenave@5	2951 full thermodynamic information is easily extracted directly from the partition
ocsenave@5	2952 function $Z(\beta)$ , so that calculation of the structure function
ocsenave@5	2953 $\rho(E)$ is
ocsenave@5	2954 unnecessary for some purposes.
ocsenave@5	2955
ocsenave@5	2956 * COMMENT
ocsenave@5	2957 Using (1-18) , the partition function can be written as
ocsenave@5	2958 which is the form most useful for calculation. If the structure function p (E)
ocsenave@5	2959 is needed, it is then found by the usual rule for inverting a Laplace trans
ocsenave@5	2960 form:
ocsenave@5	2961 the path of integration passing to the right of all singularities of Z(B) , as
ocsenave@5	2962 in Fig. (5.2) -
ocsenave@5	2963
ocsenave@5	2964
ocsenave@5	2965 Figure 5.2. Path of integration in Equation (5-21) .
ocsenave@5	2966 To illustrate the above relations, we now compute the partition function
ocsenave@5	2967 and structure function in two simple examples.
ocsenave@5	2968 Example 1. Perfect monatomic gas. We have N atoms, located by cartesian co
ocsenave@5	2969 ordinates ql...qN, and denote a particular component (direction in space) by
ocsenave@5	2970 an index oz, 0: = l, 2, 3. Thus, qia denotes the component of the position
ocsenave@5	2971 vector of the particle. Similarly, the vector momenta of the particles
ocsenave@5	2972 are denoted by pl.. .pN, and the individual components by pig. The Hamiltonian
ocsenave@5	2973 and the potential function u(q) defines the box of volume V containing the
ocsenave@5	2974
ocsenave@5	2975
ocsenave@5	2976 otherwise
ocsenave@5	2977 The arbitrary additive constant uo, representing the zero from which we
ocsenave@5	2978 measure our energies, will prove convenient later. The partition function is
ocsenave@5	2979 then
ocsenave@5	2980
ocsenave@5	2981 If N is an even number, the integrand is analytic everywhere in the com
ocsenave@5	2982 plex except for the pole of order 3N/2 at the origin. If E > Nuo,
ocsenave@5	2983 the integrand tends to zero very rapidly as GO in the left half—plane
ocsenave@5	2984 Re(;,%) 5 O. The path of integration may then be extended to a closed one by
ocsenave@5	2985 addition of an infinite semicircle to the left, as in Fig. (5.3), the integral
ocsenave@5	2986 over the semicircle vanishing. We can then apply the Cauchy residue theorem
ocsenave@5	2987 where the closed contour C, illustrated in Fig. (5.4) , encloses the point
ocsenave@5	2988 z = a once in a counter—clockwise direction, and f(z) is analytic everywhere
ocsenave@5	2989 on and within C.
ocsenave@5	2990
ocsenave@5	2991
ocsenave@5	2992
ocsenave@5	2993
ocsenave@5	2994
ocsenave@5	2995
ocsenave@5	2996
ocsenave@5	2997
ocsenave@2	2998 * COMMENT Appendix
ocsenave@1	2999
ocsenave@1	3000 \| Generalized Force \| Generalized Displacement \|
ocsenave@1	3001 \|--------------------+--------------------------\|
ocsenave@1	3002 \| force \| displacement \|
ocsenave@1	3003 \| pressure \| volume \|
ocsenave@1	3004 \| electric potential \| charge \|

Mercurial > jaynes

annotate org/stat-mech.org @ 5:e7185b523c80 tip