# HG changeset patch # User Dylan Holmes # Date 1335672408 18000 # Node ID afbe1fe19b365863974150fd318b51e0b6f50a6c # Parent 4da2176e489001485f21c312036d83400995e486 Transcribed up to section 1.6, the first law. diff -r 4da2176e4890 -r afbe1fe19b36 org/stat-mech.org --- a/org/stat-mech.org Sat Apr 28 22:03:39 2012 -0500 +++ b/org/stat-mech.org Sat Apr 28 23:06:48 2012 -0500 @@ -628,9 +628,287 @@ ** Heat We are now in a position to consider the results and interpretation of a number of elementary experiments involving +thermal interaction, which can be carried out as soon as a primitive +thermometer is at hand. In fact these experiments, which we summarize +so quickly, required a very long time for their first performance, and +the essential conclusions of this Section were first arrived at only +about 1760---more than 160 years after Galileo's invention of the +thermometer---by Joseph Black, who was Professor of Chemistry at +Glasgow University. Black's analysis of calorimetric experiments +initiated by G. D. Fahrenheit before 1736 led to the first recognition +of the distinction between temperature and heat, and prepared the way +for the work of his better-known pupil, James Watt. +We first observe that if two bodies at different temperatures are +separated by walls of various materials, they sometimes maintain their +temperature difference for a long time, and sometimes reach thermal +equilibrium very quickly. The differences in behavior observed must be +ascribed to the different properties of the separating walls, since +nothing else is changed. Materials such as wood, asbestos, porous +ceramics (and most of all, modern porous plastics like styrofoam), are +able to sustain a temperature difference for a long time; a wall of an +imaginary material with this property idealized to the point where a +temperature difference is maintained indefinitely is called an +/adiabatic wall/. A very close approximation to a perfect adiabatic +wall is realized by the Dewar flask (thermos bottle), of which the +walls consist of two layers of glass separated by a vacuum, with the +surfaces silvered like a mirror. In such a container, as we all know, +liquids may be maintained hot or cold for days. -* Appendix +On the other hand, a thin wall of copper or silver is hardly able to +sustain any temperature difference at all; two bodies separated by +such a partition come to thermal equilibrium very quickly. Such a wall +is called /diathermic/. It is found in general that the best +diathermic materials are the metals and good electrical conductors, +while electrical insulators make fairly good adiabatic walls. There +are good theoretical reasons for this rule; a particular case of it is +given by the [[http://en.wikipedia.org/wiki/Wiedemann_franz_law][Wiedemann-Franz law]] of solid-state theory. + +Since a body surrounded by an adiabatic wall is able to maintain its +temperature independently of the temperature of its surroundings, an +adiabatic wall provides a means of thermally /isolating/ a system from +the rest of the universe; it is to be expected, therefore, that the +laws of thermal interaction between two systems will assume the +simplest form if they are enclosed in a common adiabatic container, +and that the best way of carrying out experiments on thermal +peroperties of substances is to so enclose them. Such an apparatus, in +which systems are made to interact inside an adiabatic container +supplied with a thermometer, is called a /calorimeter/. + +Let us imagine that we have a calorimeter in which there is initially +a volume $V_W$ of water at a temperature $t_1$, and suspended above it +a volume $V_I$ of some other substance (say, iron) at temperature +$t_2$. When we drop the iron into the water, they interact thermally +(and the exact nature of this interaction is one of the things we hope +to learn now), the temperature of both changing until they are in +thermal equilibrium at a final temperature $t_0$. + +Now we repeat the experiment with different initial temperatures +$t_1^\prime$ and $t_2^\prime$, so that a new equilibrium is reached at +temperature $t_0^\prime$. It is found that, if the temperature +differences are sufficiently small (and in practice this is not a +serious limitation if we use a mercury thermometer calibrated with +uniformly spaced degree marks on a capillary of uniform bore), then +whatever the values of $t_1^\prime$, $t_2^\prime$, $t_1$, $t_2$, the +final temperatures $t_0^\prime$, $t_0$ will adjust themselves so that +the following relation holds: + +\begin{equation} +\frac{t_2 - t_0}{t_0 - t_1} = \frac{t_2^\prime - +t_0^\prime}{t_0^\prime - t_1^\prime} +\end{equation} + +in other words, the /ratio/ of the temperature changes of the iron and +water is independent of the initial temperatures used. + +We now vary the amounts of iron and water used in the calorimeter. It +is found that the ratio (1-8), although always independent of the +starting temperatures, does depend on the relative amounts of iron and +water. It is, in fact, proportional to the mass $M_W$ of water and +inversely proportional to the mass $M_I$ of iron, so that + +\begin{equation} +\frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_I M_I} +\end{equation} + +where $K_I$ is a constant. + +We next repeat the above experiments using a different material in +place of the iron (say, copper). We find again a relation + +\begin{equation} +\frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_C \cdot M_C} +\end{equation} + +where $M_C$ is the mass of copper; but the constant $K_C$ is different +from the previous $K_I$. In fact, we see that the constant $K_I$ is a +new physical property of the substance iron, while $K_C$ is a physical +property of copper. The number $K$ is called the /specific heat/ of a +substance, and it is seen that according to this definition, the +specific heat of water is unity. + +We now have enough experimental facts to begin speculating about their +interpretation, as was first done in the 18th century. First, note +that equation (1-9) can be put into a neater form that is symmetrical +between the two substances. We write $\Delta t_I = t_0 - t_2$, $\Delta +t_W = t_0 - t_1$ for the temperature changes of iron and water +respectively, and define $K_W \equiv 1$ for water. Equation (1-9) then +becomes + +\begin{equation} +K_W M_W \Delta t_W + K_I M_I \Delta t_I = 0 +\end{equation} + +The form of this equation suggests a new experiment; we go back into +the laboratory, and find $n$ substances for which the specific heats +\(\{K_1,\ldots K_n\}\) have been measured previously. Taking masses +\(\{M_1, \ldots, M_n\}\) of these substances, we heat them to $n$ +different temperatures \(\{t_1,\ldots, t_n\}\) and throw them all into +the calorimeter at once. After they have all come to thermal +equilibrium at temperature $t_0$, we find the differences $\Delta t_j += t_0 - t_j$. Just as we suspected, it turns out that regardless of +the $K$'s, $M$'s, and $t$'s chosen, the relation +\begin{equation} +\sum_{j=0}^n K_j M_j \Delta t_j = 0 +\end{equation} +is always satisfied. This sort of process is an old story in +scientific investigations; although the great theoretician Boltzmann +is said to have remarked: \ldquo{}Elegance is for tailors \rdquo{}, it +remains true that the attempt to reduce equations to the most +symmetrical form has often suggested important generalizations of +physical laws, and is a great aid to memory. Witness Maxwell's +\ldquo{}displacement current\rdquo{}, which was needed to fill in a +gap and restore the symmetry of the electromagnetic equations; as soon +as it was put in, the equations predicted the existence of +electromagnetic waves. In the present case, the search for a rather +rudimentary form of \ldquo{}elegance\rdquo{} has also been fruitful, +for we recognize that (1-12) has the standard form of a /conservation +law/; it defines a new quantity which is conserved in thermal +interactions of the type just studied. + +The similarity of (1-12) to conservation laws in general may be seen +as follows. Let $A$ be some quantity that is conserved; the $i$th +system has an amount of it $A_i$. Now when the systems interact such +that some $A$ is transferred between them, the amount of $A$ in the +$i$th system is changed by a net amount \(\Delta A_i = (A_i)_{final} - +(A_i)_{initial}\); and the fact that there is no net change in the +total amount of $A$ is expressed by the equation \(\sum_i \Delta +A_i = 0$. Thus, the law of conservation of matter in a chemical +reaction is expressed by \(\sum_i \Delta M_i = 0\), where $M_i$ is the +mass of the $i$th chemical component. + +what is this new conserved quantity? Mathematically, it can be defined +as $Q_i = K_i\cdot M_i cdot t_i; whereupon (1-12) becomes + +\begin{equation} +\sum_i \Delta Q_i = 0 +\end{equation} + +and at this point we can correct a slight quantitative inaccuracy. As +noted, the above relations hold accurately only when the temperature +differences are sufficiently small; i.e., they are really only +differential laws. On sufficiently accurate measurements one find that +the specific heats $K_i$ depend on temperature; if we then adopt the +integral definition of $\Delta Q_i$, +\begin{equation} +\Delta Q_i = \int_{t_{i}}^{t_0} K_i(t) M_i dt +\end{equation} + +the conservation law (1-13) will be found to hold in calorimetric +experiments with liquids and solids, to any accuracy now feasible. And +of course, from the manner in which the $K_i(t)$ are defined, this +relation will hold however our thermometers are calibrated. + +Evidently, the stage is now set for a \ldquo{}new\rdquo{} physical +theory to account for these facts. In the 17th century, both Francis +Bacon and Isaac Newton had expressed their opinions that heat was a +form of motion; but they had no supporting factual evidence. By the +latter part of the 18th century, one had definite factual evidence +which seemed to make this view untenable; by the calorimetric +\ldquo{}mixing\rdquo{} experiments just described, Joseph Black had +recognized the distinction between temperature $t$ as a measure of +\ldquo{}hotness\rdquo{}, and heat $Q$ as a measure of /quantity/ of +something, and introduced the notion of heat capacity. He also +recognized the latent heats of freezing and vaporization. To account +for the conservation laws thus discovered, the theory then suggested +itself, naturally and almost inevitably, that heat was /fluid/, +indestructable and uncreatable, which had no appreciable weight and +was attracted differently by different kinds of matter. In 1787, +Lavoisier invented the name \ldquo{}caloric\rdquo{} for this fluid. + +Looking down today from our position of superior knowledge (i.e., +hindsight) we perhaps need to be reminded that the caloric theory was +a perfectly respectable scientific theory, fully deserving of serious +consideration; for it accounted quantitatively for a large body of +experimental fact, and made new predictions capable of being tested by +experiment. + +One of these predictions was the possibility of accounting for the +thermal expansion of bodies when heated; perhaps the increase in +volume was just a measure of the volume of caloric fluid +absorbed. This view met with some disappointment as a result of +experiments which showed that different materials, on absorbing the +same quantity of heat, expanded by different amounts. Of course, this +in itself was not enough to overthrow the caloric theory, because one +could suppose that the caloric fluid was compressible, and was held +under different pressure in different media. + +Another difficulty that seemed increasingly serious by the end of the +18th century was the failure of all attempts to weigh this fluid. Many +careful experiments were carried out, by Boyle, Fordyce, Rumford and +others (and continued by Landolt almost into the 20th century), with +balances capable of detecting a change of weight of one part in a +million; and no change could be detected on the melting of ice, +heating of substances, or carrying out of chemical reactions. But even +this is not really a conclusive argument against the caloric theory, +since there is no /a priori/ reason why the fluid should be dense +enough to weigh with balances (of course, we know today from +Einstein's $E=mc^2$ that small changes in weight should indeed exist +in these experiments; but to measure them would require balances about +10^7 times more sensitive than were available). + +Since the caloric theory derives entirely from the empirical +conservation law (1-33), it can be refuted conclusively only by +exhibiting new experimental facts revealing situations in which (1-13) +is /not/ valid. The first such case was [[http://www.chemteam.info/Chem-History/Rumford-1798.html][found by Count Rumford (1798)]], +who was in charge of boring cannon in the Munich arsenal, and noted +that the cannon and chips became hot as a result of the cutting. He +found that heat could be produced indefinitely, as long as the boring +was continued, without any compensating cooling of any other part of +the system. Here, then, was a clear case in which caloric was /not/ +conserved, as in (1-13); but could be created at will. Rumford wrote +that he could not conceive of anything that could be produced +indefinitely by the expenditure of work, \ldquo{}except it be /motion/\rdquo{}. + +But even this was not enough to cause abandonment of the caloric +theory; for while Rumford's observations accomplished the negative +purpose of showing that the conservation law (1-13) is not universally +valid, they failed to accomplish the positive one of showing what +specific law should replace it (although he produced a good hint, not +sufficiently appreciated at the time, in his crude measurements of the +rate of heat production due to the work of one horse). Within the +range of the original calorimetric experiments, (1-13) was still +valid, and a theory successful in a restricted domain is better than +no theory at all; so Rumford's work had very little impact on the +actual development of thermodynamics. + +(This situation is a recurrent one in science, and today physics offers +another good example. It is recognized by all that our present quantum +field theory is unsatisfactory on logical, conceptual, and +mathematical grounds; yet it also contains some important truth, and +no responsible person has suggested that it be abandoned. Once again, +a semi-satisfactory theory is better than none at all, and we will +continue to teach it and to use it until we have something better to +put in its place.) + +# what is "the specific heat of a gas at constant pressure/volume"? +# changed t for temperature below from capital T to lowercase t. +Another failure of the conservation law (1-13) was noted in 1842 by +R. Mayer, a German physician, who pointed out that the data already +available showed that the specific heat of a gas at constant pressure, +C_p, was greater than at constant volume $C_v$. He surmised that the +difference was due to the work done in expansion of the gas against +atmospheric pressure, when measuring $C_p$. Supposing that the +difference $\Delta Q = (C_p - C_v)\Delta t$ calories, in the heat +required to raise the temperature by $\Delta t$ was actually a +measure of amount of energy, he could estimate from the amount +$P\Delta V$ ergs of work done the amount of mechanical energy (number +of ergs) corresponding to a calorie of heat; but again his work had +very little impact on the development of thermodynamics, because he +merely offered this notion as an interpretation of the data without +performing or suggesting any new experiments to check his hypothesis +further. + +Up to the point, then, one has the experimental fact that a +conservation law (1-13) exists whenever purely thermal interactions +were involved; but in processes involving mechanical work, the +conservation law broke down. + +** The First Law + + + +* COMMENT Appendix | Generalized Force | Generalized Displacement | |--------------------+--------------------------|