annotate org/stat-mech.org @ 2:afbe1fe19b36

Transcribed up to section 1.6, the first law.
author Dylan Holmes <ocsenave@gmail.com>
date Sat, 28 Apr 2012 23:06:48 -0500
parents 4da2176e4890
children 8f3b6dcb9add
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ocsenave@0 13 [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]].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@0 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@0 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@0 148 ** 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@0 229 ** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
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@1 393 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@1 574 We leave it as an exercise for the reader (Problem 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@2 636 thermometer---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@2 756 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@2 770 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@2 773 $i$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@2 776 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@2 778 mass of the $i$th chemical component.
ocsenave@2 779
ocsenave@2 780 what is this new conserved quantity? Mathematically, it can be defined
ocsenave@2 781 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@2 886 Another failure of the conservation law (1-13) was 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@2 908
ocsenave@2 909
ocsenave@2 910
ocsenave@2 911 * COMMENT Appendix
ocsenave@1 912
ocsenave@1 913 | Generalized Force | Generalized Displacement |
ocsenave@1 914 |--------------------+--------------------------|
ocsenave@1 915 | force | displacement |
ocsenave@1 916 | pressure | volume |
ocsenave@1 917 | electric potential | charge |