1.1 --- a/org/stat-mech.org	Sat Apr 28 22:03:39 2012 -0500
     1.2 +++ b/org/stat-mech.org	Sat Apr 28 23:06:48 2012 -0500
     1.3 @@ -628,9 +628,287 @@
     1.4  ** Heat
     1.5  We are now in a position to consider the results and interpretation of
     1.6  a number of elementary experiments involving
     1.7 +thermal interaction, which can be carried out as soon as a primitive
     1.8 +thermometer is at hand. In fact these experiments, which we summarize
     1.9 +so quickly, required a very long time for their first performance, and
    1.10 +the essential conclusions of this Section were first arrived at only
    1.11 +about 1760---more than 160 years after Galileo's invention of the
    1.12 +thermometer---by Joseph Black, who was Professor of Chemistry at
    1.13 +Glasgow University. Black's analysis of calorimetric experiments
    1.14 +initiated by G. D. Fahrenheit before 1736 led to the first recognition
    1.15 +of the distinction between temperature and heat, and prepared the way
    1.16 +for the work of his better-known pupil, James Watt.
    1.17  
    1.18 +We first observe that if two bodies at different temperatures are
    1.19 +separated by walls of various materials, they sometimes maintain their
    1.20 +temperature difference for a long time, and sometimes reach thermal
    1.21 +equilibrium very quickly. The differences in behavior observed must be
    1.22 +ascribed to the different properties of the separating walls, since
    1.23 +nothing else is changed. Materials such as wood, asbestos, porous
    1.24 +ceramics (and most of all, modern porous plastics like styrofoam), are
    1.25 +able to sustain a temperature difference for a long time; a wall of an
    1.26 +imaginary material with this property idealized to the point where a
    1.27 +temperature difference is maintained indefinitely is called an
    1.28 +/adiabatic wall/. A very close approximation to a perfect adiabatic
    1.29 +wall is realized by the Dewar flask (thermos bottle), of which the
    1.30 +walls consist of two layers of glass separated by a vacuum, with the
    1.31 +surfaces silvered like a mirror. In such a container, as we all know,
    1.32 +liquids may be maintained hot or cold for days.
    1.33  
    1.34 -* Appendix
    1.35 +On the other hand, a thin wall of copper or silver is hardly able to
    1.36 +sustain any temperature difference at all; two bodies separated by
    1.37 +such a partition come to thermal equilibrium very quickly. Such a wall
    1.38 +is called /diathermic/. It is found in general that the best
    1.39 +diathermic materials are the metals and good electrical conductors,
    1.40 +while electrical insulators make fairly good adiabatic walls. There
    1.41 +are good theoretical reasons for this rule; a particular case of it is
    1.42 +given by the [[http://en.wikipedia.org/wiki/Wiedemann_franz_law][Wiedemann-Franz law]] of solid-state theory.
    1.43 +
    1.44 +Since a body surrounded by an adiabatic wall is able to maintain its
    1.45 +temperature independently of the temperature of its surroundings, an
    1.46 +adiabatic wall provides a means of thermally /isolating/ a system from
    1.47 +the rest of the universe; it is to be expected, therefore, that the
    1.48 +laws of thermal interaction between two systems will assume the
    1.49 +simplest form if they are enclosed in a common adiabatic container,
    1.50 +and that the best way of carrying out experiments on thermal
    1.51 +peroperties of substances is to so enclose them. Such an apparatus, in
    1.52 +which systems are made to interact inside an adiabatic container
    1.53 +supplied with a thermometer, is called a /calorimeter/.
    1.54 +
    1.55 +Let us imagine that we have a calorimeter in which there is initially
    1.56 +a volume $V_W$ of water at a temperature $t_1$, and suspended above it
    1.57 +a volume $V_I$ of some other substance (say, iron) at temperature
    1.58 +$t_2$. When we drop the iron into the water, they interact thermally
    1.59 +(and the exact nature of this interaction is one of the things we hope
    1.60 +to learn now), the temperature of both changing until they are in
    1.61 +thermal equilibrium at a final temperature $t_0$.
    1.62 +
    1.63 +Now we repeat the experiment with different initial temperatures
    1.64 +$t_1^\prime$ and $t_2^\prime$, so that a new equilibrium is reached at
    1.65 +temperature $t_0^\prime$. It is found that, if the temperature
    1.66 +differences are sufficiently small (and in practice this is not a
    1.67 +serious limitation if we use a mercury thermometer calibrated with
    1.68 +uniformly spaced degree marks on a capillary of uniform bore), then
    1.69 +whatever the values of $t_1^\prime$, $t_2^\prime$, $t_1$, $t_2$, the
    1.70 +final temperatures $t_0^\prime$, $t_0$ will adjust themselves so that
    1.71 +the following relation holds:
    1.72 +
    1.73 +\begin{equation}
    1.74 +\frac{t_2 - t_0}{t_0 - t_1} = \frac{t_2^\prime -
    1.75 +t_0^\prime}{t_0^\prime - t_1^\prime}
    1.76 +\end{equation} 
    1.77 +
    1.78 +in other words, the /ratio/ of the temperature changes of the iron and
    1.79 +water is independent of the initial temperatures used.
    1.80 +
    1.81 +We now vary the amounts of iron and water used in the calorimeter. It
    1.82 +is found that the ratio (1-8), although always independent of the
    1.83 +starting temperatures, does depend on the relative amounts of iron and
    1.84 +water. It is, in fact, proportional to the mass $M_W$ of water and
    1.85 +inversely proportional to the mass $M_I$ of iron, so that 
    1.86 +
    1.87 +\begin{equation}
    1.88 +\frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_I M_I}
    1.89 +\end{equation}
    1.90 +
    1.91 +where $K_I$ is a constant.
    1.92 +
    1.93 +We next repeat the above experiments using a different material in
    1.94 +place of the iron (say, copper). We find again a relation
    1.95 +
    1.96 +\begin{equation}
    1.97 +\frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_C \cdot M_C}
    1.98 +\end{equation}
    1.99 +
   1.100 +where $M_C$ is the mass of copper; but the constant $K_C$ is different
   1.101 +from the previous $K_I$. In fact, we see that the constant $K_I$ is a
   1.102 +new physical property of the substance iron, while $K_C$ is a physical
   1.103 +property of copper. The number $K$ is called the /specific heat/ of a
   1.104 +substance, and it is seen that according to this definition, the
   1.105 +specific heat of water is unity.
   1.106 +
   1.107 +We now have enough experimental facts to begin speculating about their
   1.108 +interpretation, as was first done in the 18th century. First, note
   1.109 +that equation (1-9) can be put into a neater form that is symmetrical
   1.110 +between the two substances. We write $\Delta t_I = t_0 - t_2$, $\Delta
   1.111 +t_W = t_0 - t_1$ for the temperature changes of iron and water
   1.112 +respectively, and define $K_W \equiv 1$ for water. Equation (1-9) then
   1.113 +becomes
   1.114 +
   1.115 +\begin{equation}
   1.116 +K_W M_W \Delta t_W + K_I M_I \Delta t_I = 0
   1.117 +\end{equation}
   1.118 +
   1.119 +The form of this equation suggests a new experiment; we go back into
   1.120 +the laboratory, and find $n$ substances for which the specific heats
   1.121 +\(\{K_1,\ldots K_n\}\) have been measured previously. Taking masses
   1.122 +\(\{M_1, \ldots, M_n\}\) of these substances, we heat them to $n$
   1.123 +different temperatures \(\{t_1,\ldots, t_n\}\) and throw them all into
   1.124 +the calorimeter at once. After they have all come to thermal
   1.125 +equilibrium at temperature $t_0$, we find the differences $\Delta t_j
   1.126 += t_0 - t_j$. Just as we suspected, it turns out that regardless of
   1.127 +the $K$'s, $M$'s, and $t$'s chosen, the relation
   1.128 +\begin{equation}
   1.129 +\sum_{j=0}^n K_j M_j \Delta t_j = 0
   1.130 +\end{equation}
   1.131 +is always satisfied. This sort of process is an old story in
   1.132 +scientific investigations; although the great theoretician Boltzmann
   1.133 +is said to have remarked: \ldquo{}Elegance is for tailors \rdquo{}, it
   1.134 +remains true that the attempt to reduce equations to the most
   1.135 +symmetrical form has often suggested important generalizations of
   1.136 +physical laws, and is a great aid to memory. Witness Maxwell's
   1.137 +\ldquo{}displacement current\rdquo{}, which was needed to fill in a
   1.138 +gap and restore the symmetry of the electromagnetic equations; as soon
   1.139 +as it was put in, the equations predicted the existence of
   1.140 +electromagnetic waves. In the present case, the search for a rather
   1.141 +rudimentary form of \ldquo{}elegance\rdquo{} has also been fruitful,
   1.142 +for we recognize that (1-12) has the standard form of a /conservation
   1.143 +law/; it defines a new quantity which is conserved in thermal
   1.144 +interactions of the type just studied.
   1.145 +
   1.146 +The similarity of (1-12) to conservation laws in general may be seen
   1.147 +as follows. Let $A$ be some quantity that is conserved; the $i$th
   1.148 +system has an amount of it $A_i$. Now when the systems interact such
   1.149 +that some $A$ is transferred between them, the amount of $A$ in the
   1.150 +$i$th system is changed by a net amount \(\Delta A_i = (A_i)_{final} -
   1.151 +(A_i)_{initial}\); and the fact that there is no net change in the
   1.152 +total amount of $A$ is expressed by the equation \(\sum_i \Delta
   1.153 +A_i = 0$. Thus, the law of conservation of matter in a chemical
   1.154 +reaction is expressed by \(\sum_i \Delta M_i = 0\), where $M_i$ is the
   1.155 +mass of the $i$th chemical component.
   1.156 +
   1.157 +what is this new conserved quantity? Mathematically, it can be defined
   1.158 +as $Q_i = K_i\cdot M_i cdot t_i; whereupon (1-12) becomes 
   1.159 +
   1.160 +\begin{equation}
   1.161 +\sum_i \Delta Q_i = 0
   1.162 +\end{equation}
   1.163 +
   1.164 +and at this point we can correct a slight quantitative inaccuracy. As
   1.165 +noted, the above relations hold accurately only when the temperature
   1.166 +differences are sufficiently small; i.e., they are really only
   1.167 +differential laws. On sufficiently accurate measurements one find that
   1.168 +the specific heats $K_i$ depend on temperature; if we then adopt the
   1.169 +integral definition of $\Delta Q_i$,
   1.170 +\begin{equation}
   1.171 +\Delta Q_i = \int_{t_{i}}^{t_0} K_i(t) M_i dt
   1.172 +\end{equation}
   1.173 +
   1.174 +the conservation law (1-13) will be found to hold in calorimetric
   1.175 +experiments with liquids and solids, to any accuracy now feasible. And
   1.176 +of course, from the manner in which the $K_i(t)$ are defined, this
   1.177 +relation will hold however our thermometers are calibrated.
   1.178 +
   1.179 +Evidently, the stage is now set for a \ldquo{}new\rdquo{} physical
   1.180 +theory to account for these facts. In the 17th century, both Francis
   1.181 +Bacon and Isaac Newton had expressed their opinions that heat was a
   1.182 +form of motion; but they had no supporting factual evidence. By the
   1.183 +latter part of the 18th century, one had definite factual evidence
   1.184 +which seemed to make this view untenable; by the calorimetric
   1.185 +\ldquo{}mixing\rdquo{} experiments just described, Joseph Black had
   1.186 +recognized the distinction between temperature $t$ as a measure of
   1.187 +\ldquo{}hotness\rdquo{}, and heat $Q$ as a measure of /quantity/ of
   1.188 +something, and introduced the notion of heat capacity. He also
   1.189 +recognized the latent heats of freezing and vaporization. To account
   1.190 +for the conservation laws thus discovered, the theory then suggested
   1.191 +itself, naturally and almost inevitably, that heat was /fluid/,
   1.192 +indestructable and uncreatable, which had no appreciable weight and
   1.193 +was attracted differently by different kinds of matter. In 1787,
   1.194 +Lavoisier invented the name \ldquo{}caloric\rdquo{} for this fluid.
   1.195 +
   1.196 +Looking down today from our position of superior knowledge (i.e.,
   1.197 +hindsight) we perhaps need to be reminded that the caloric theory was
   1.198 +a perfectly respectable scientific theory, fully deserving of serious
   1.199 +consideration; for it accounted quantitatively for a large body of
   1.200 +experimental fact, and made new predictions capable of being tested by
   1.201 +experiment.
   1.202 +
   1.203 +One of these predictions was the possibility of accounting for the
   1.204 +thermal expansion of bodies when heated; perhaps the increase in
   1.205 +volume was just a measure of the volume of caloric fluid
   1.206 +absorbed. This view met with some disappointment as a result of
   1.207 +experiments which showed that different materials, on absorbing the
   1.208 +same quantity of heat, expanded by different amounts. Of course, this
   1.209 +in itself was not enough to overthrow the caloric theory, because one
   1.210 +could suppose that the caloric fluid was compressible, and was held
   1.211 +under different pressure in different media.
   1.212 +
   1.213 +Another difficulty that seemed increasingly serious by the end of the
   1.214 +18th century was the failure of all attempts to weigh this fluid. Many
   1.215 +careful experiments were carried out, by Boyle, Fordyce, Rumford and
   1.216 +others (and continued by Landolt almost into the 20th century), with
   1.217 +balances capable of detecting a change of weight of one part in a
   1.218 +million; and no change could be detected on the melting of ice,
   1.219 +heating of substances, or carrying out of chemical reactions. But even
   1.220 +this is not really a conclusive argument against the caloric theory,
   1.221 +since there is no /a priori/ reason why the fluid should be dense
   1.222 +enough to weigh with balances (of course, we know today from
   1.223 +Einstein's $E=mc^2$ that small changes in weight should indeed exist
   1.224 +in these experiments; but to measure them would require balances about
   1.225 +10^7 times more sensitive than were available).
   1.226 +
   1.227 +Since the caloric theory derives entirely from the empirical
   1.228 +conservation law (1-33), it can be refuted conclusively only by
   1.229 +exhibiting new experimental facts revealing situations in which (1-13)
   1.230 +is /not/ valid. The first such case was [[http://www.chemteam.info/Chem-History/Rumford-1798.html][found by Count Rumford (1798)]],
   1.231 +who was in charge of boring cannon in the Munich arsenal, and noted
   1.232 +that the cannon and chips became hot as a result of the cutting. He
   1.233 +found that heat could be produced indefinitely, as long as the boring
   1.234 +was continued, without any compensating cooling of any other part of
   1.235 +the system. Here, then, was a clear case in which caloric was /not/
   1.236 +conserved, as in (1-13); but could be created at will. Rumford wrote
   1.237 +that he could not conceive of anything that could be produced
   1.238 +indefinitely by the expenditure of work, \ldquo{}except it be /motion/\rdquo{}.
   1.239 +
   1.240 +But even this was not enough to cause abandonment of the caloric
   1.241 +theory; for while Rumford's observations accomplished the negative
   1.242 +purpose of showing that the conservation law (1-13) is not universally
   1.243 +valid, they failed to accomplish the positive one of showing what
   1.244 +specific law should replace it (although he produced a good hint, not
   1.245 +sufficiently appreciated at the time, in his crude measurements of the
   1.246 +rate of heat production due to the work of one horse). Within the
   1.247 +range of the original calorimetric experiments, (1-13) was still
   1.248 +valid, and a theory successful in a restricted domain is better than
   1.249 +no theory at all; so Rumford's work had very little impact on the
   1.250 +actual development of thermodynamics.
   1.251 +
   1.252 +(This situation is a recurrent one in science, and today physics offers
   1.253 +another good example. It is recognized by all that our present quantum
   1.254 +field theory is unsatisfactory on logical, conceptual, and
   1.255 +mathematical grounds; yet it also contains some important truth, and
   1.256 +no responsible person has suggested that it be abandoned. Once again,
   1.257 +a semi-satisfactory theory is better than none at all, and we will
   1.258 +continue to teach it and to use it until we have something better to
   1.259 +put in its place.)
   1.260 +
   1.261 +# what is "the specific heat of a gas at constant pressure/volume"?
   1.262 +# changed t for temperature below from capital T to lowercase t.
   1.263 +Another failure of the conservation law (1-13) was noted in 1842 by
   1.264 +R. Mayer, a German physician, who pointed out that the data already
   1.265 +available showed that the specific heat of a gas at constant pressure, 
   1.266 +C_p, was greater than at constant volume $C_v$. He surmised that the
   1.267 +difference was due to the work done in expansion of the gas against
   1.268 +atmospheric pressure, when measuring $C_p$. Supposing that the
   1.269 +difference $\Delta Q = (C_p - C_v)\Delta t$ calories, in the heat
   1.270 +required to raise the temperature by $\Delta t$ was actually a
   1.271 +measure of amount of energy, he could estimate from the amount
   1.272 +$P\Delta V$ ergs of work done the amount of mechanical energy (number
   1.273 +of ergs) corresponding to a calorie of heat; but again his work had
   1.274 +very little impact on the development of thermodynamics, because he
   1.275 +merely offered this notion as an interpretation of the data without
   1.276 +performing or suggesting any new experiments to check his hypothesis
   1.277 +further.
   1.278 +
   1.279 +Up to the point, then, one has the experimental fact that a
   1.280 +conservation law (1-13) exists whenever purely thermal interactions
   1.281 +were involved; but in processes involving mechanical work, the
   1.282 +conservation law broke down.
   1.283 +
   1.284 +** The First Law
   1.285 +
   1.286 +
   1.287 +
   1.288 +* COMMENT  Appendix
   1.289  
   1.290  | Generalized Force  | Generalized Displacement |
   1.291  |--------------------+--------------------------|