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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

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 appropriate.
 ** 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
-* Appendix
+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.
+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 |
 |--------------------+--------------------------|
 | force              | displacement             |
 | pressure           | volume                   |

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