# HG changeset patch
# User Dylan Holmes <ocsenave@gmail.com>
# Date 1335685102 18000
# Node ID 8f3b6dcb9addf1a1eff79540ea3c4a7b7b6d27d6
# Parent  afbe1fe19b365863974150fd318b51e0b6f50a6c
Transcribed up to section 1.9, Entropy of an Ideal Boltzmann Gas

diff -r afbe1fe19b36 -r 8f3b6dcb9add org/stat-mech.org
--- a/org/stat-mech.org	Sat Apr 28 23:06:48 2012 -0500
+++ b/org/stat-mech.org	Sun Apr 29 02:38:22 2012 -0500
@@ -91,7 +91,7 @@
 of our present review of the logic underlying classical 
 thermodynamics.
 
-** The Primitive Thermometer. 
+** The Primitive Thermometer
 
 The earliest stages of our
 story are necessarily speculative, since they took place long
@@ -145,7 +145,7 @@
 present.
 
 
-** Thermodynamic Systems.
+** Thermodynamic Systems
 
 The \ldquo{}thermodynamic systems\rdquo{} which
 are the objects of our study may be, physically, almost any
@@ -226,7 +226,7 @@
 of thermodynamics . The most fundamental one is a qualitative
 rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
 
-** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
+** Equilibrium; the Zeroth Law
 
  It is a common experience
 that when objects are placed in contact with each other but
@@ -390,7 +390,7 @@
 by specifying any two of the variables arbitrarily, whereupon the
 third, and all others we may introduce, are determined. 
 Mathematically, this is expressed by the existence of a functional
-relationship of the form[fn:: /Edit./: The set of solutions to an equation
+relationship of the form[fn:: The set of solutions to an equation
 like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
 saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
 rule\rdquo{}, so the set of physically allowed combinations of /X/,
@@ -753,7 +753,7 @@
 \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
+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
@@ -767,18 +767,18 @@
 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
+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} -
+\(i\)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 \(\sum_i \Delta
-A_i = 0$. Thus, the law of conservation of matter in a chemical
+A_i = 0\). 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.
+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 
+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
@@ -883,7 +883,7 @@
 
 # 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
+Another failure of the conservation law (1-13) was [[http://web.lemoyne.edu/~giunta/mayer.html][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
@@ -905,6 +905,940 @@
 conservation law broke down.
 
 ** The First Law
+Corresponding to the partially valid law of \ldquo{}conservation of
+heat\rdquo{}, there had long been known another partially valid
+conservation law in mechanics. The principle of conservation of
+mechanical energy had been given by Leibnitz in 1693 in noting that,
+according to the laws of Newtonian mechanics, one could define
+potential and kinetic energy so that in mechanical processes they were
+interconverted into each other, the total energy remaining
+constant. But this too was not universally valid---the mechanical
+energy was conserved only in the absence of frictional forces. In
+processes involving friction, the mechanical energy seemed to
+disappear.
+
+So we had a law of conservation of heat, which broke down whenever
+mechanical work was done; and a law of conservation of mechanical
+energy, which broke down when frictional forces were present. If, as
+Mayer had suggested, heat was itself a form of energy, then one had
+the possibility of accounting for both of these failures in a new law
+of conservation of /total/ (mechanical + heat) energy. On one hand,
+the difference $C_p-C_v$ of heat capacities of gases would be
+accounted for by the mechanical work done in expansion; on the other
+hand, the disappearance of mechanical energy would be accounted for by
+the heat produced by friction.
+
+But to establish this requires more than just suggesting the idea and
+illustrating its application in one or two cases --- if this is really
+a new conservation law adequate to replace the two old ones, it must
+be shown to be valid for /all/ substances and /all/ kinds of
+interaction. For example, if one calorie of heat corresponded to $E$
+ergs of mechanical energy in the gas experiments, but to a different
+amoun $E^\prime$ in heat produced by friction, then there would be no
+universal conservation law. This \ldquo{}first law\rdquo{} of
+thermodynamics must therefore take the form:
+#+begin_quote
+There exists a /universal/ mechanical equivalent of heat, so that the
+total (mechanical energy) + (heat energy) remeains constant in all
+physical processes.
+#+end_quote
+
+It was James Prescott Joule who provided the [[http://www.chemteam.info/Chem-History/Joule-Heat-1845.html][first experimental data]]
+indicating this universality, and providing the first accurate
+numerical value of this mechanical equivalent. The calorie had been
+defined as the amount of heat required to raise the temperature of one
+gram of water by one degree Centigrade (more precisely, to raise it
+from 14.5 to 15.5$^\circ C$). Joule measured the heating of a number
+of different liquids due to mechanical stirring and electrical
+heating, and established that, within the experimental accuracy (about
+one percent) a /calorie/ of heat always corresponded to the same
+amount of energy. Modern measurements give this numerical value as: 1
+calorie = 4.184 \times 10^7 ergs = 4.184 joules.
+# capitalize Joules? I think the convention is to spell them out in lowercase.
+
+The circumstances of this important work are worth noting. Joule was
+in frail health as a child, and was educated by private tutors,
+including the chemist, John Dalton, who had formulated the atomic
+hypothesis in the early nineteenth century. In 1839, when Joule was
+nineteen, his father (a wealthy brewer) built a private laboratory for
+him in Manchester, England; and the good use he made of it is shown by
+the fact that, within a few months of the opening of this laboratory
+(1840), he had completed his first important piece of work, at the
+age of twenty. This was his establishment of the law of \ldquo{}Joule
+heating,\rdquo{} $P=I^2 R$, due to the electric current in a
+resistor. He then used this effect to determine the universality and
+numerical value of the mechanical equivalent of heat, reported
+in 1843. His mechanical stirring experiments reported in 1849 yielded
+the value 1 calorie = 4.154 \times 10^7 ergs, amount 0.7% too low;
+this determination was not improved upon for several decades.
+
+The first law of thermodynamics may then be stated mathematically as
+follows:  
+
+#+begin_quote
+There exists a state function (i.e., a definite function of the
+thermodynamic state) $U$, representing the total energy of any system,
+such that in any process in which we change from one equilibrium to
+another, the net change in $U$ is given by the difference of the heat
+$Q$ supplied to the system, and the mechanical work $W$ done by the
+system.
+#+end_quote
+On an infinitesimal change of state, this becomes
+
+\begin{equation}
+dU = dQ - dW.
+\end{equation}
+
+For a system of two degrees of freedom, defined by pressure $P$,
+volume $V$, and temperature $t$, we have $dW = PdV$. Then if we regard
+$U$ as a function $U(V,t)$ of volume and temperature, the fact that
+$U$ is a state function means that $dU$ must be an exact differential;
+i.e., the integral
+
+\begin{equation}
+\int_1^2 dU = U(V_2,t_2) - U(V_1,t_1)
+\end{equation}
+between any two thermodynamic states must be independent of the
+path. Equivalently, the integral $\oint dU$ over any closed cyclic
+path  (for example, integrate from state 1 to state 2 along path A,
+then back to state 1 by a different path B) must be zero. From (1-15),
+this gives for any cyclic integral,
+
+\begin{equation}
+\oint dQ = \oint P dV
+\end{equation}
+
+another form of the first law, which states that in any process in
+which the system ends in the same thermodynamic state as the initial
+one, the total heat absorbed by the system must be equal to the total
+work done.
+
+Although the equations (1-15)-(1-17) are rather trivial
+mathematically, it is important to avoid later conclusions that we
+understand their exact meaning. In the first place, we have to
+understand that we are now measuring heat energy and mechanical energy
+in the same units; i.e. if we measured $Q$ in calories and $W$ in
+ergs, then (1-15)  would of course not be correct. It does
+not matter whether we apply Joule's mechanical equivalent of heat
+to express $Q$ in ergs, or whether we apply it in the opposite way
+to express $U$ and $W$ in calories; each procedure will be useful in
+various problems. We can develop the general equations of
+thermodynamics 
+without committing ourselves to any particular units,
+but of course all terms in a given equation must be expressed
+in the same units.
+
+Secondly, we have already stressed that the theory being
+developed must, strictly speaking, be a theory only of 
+equilibrium states, since otherwise we have no operational definition
+of temperature . When we integrate over any \ldquo{}path\rdquo{} in the $(V-t)$
+plane, therefore, it must be understood that the path of 
+integration is, strictly speaking, just a /locus of equilibrium
+states/; nonequilibrium states cannot be represented by points
+in the $(V-t)$ plane.
+
+But then, what is the relation between path of equilibrium
+states appearing in our equations, and the sequence of conditions
+produced experimentally when we change the state of a system in
+the laboratory? With any change of state (heating, compression,
+etc.) proceeding at a finite rate we do not have equilibrium in
+termediate states; and so there is no corresponding \ldquo{}path\rdquo{} in
+the $(V-t)$ plane ; only the initial and final equilibrium states
+correspond to definite points. But if we carry out the change
+of state more and more slowly, the physical states produced are
+nearer and nearer to equilibrium state. Therefore, we interpret
+a path of integration in the $(V-t)$ plane, not as representing
+the intermediate states of any real experiment carried out at
+a finite rate, but as the /limit/ of this sequence of states, in
+the limit where the change of state takes place arbitrarily
+slowly.
+
+An arbitrarily slow process, so that we remain arbitrarily
+near to equilibrium at all times, has another important property.
+If heat is flowing at an arbitrarily small rate, the temperature
+difference producing it must be arbitrarily small, and therefore
+an arbitrarily small temperature change would be able to reverse
+the direction of heat flow. If the Volume is changing very
+slowly, the pressure difference responsible for it must be very
+small; so a small change in pressure would be able to reverse
+the direction of motion. In other words, a process carried out
+arbitrarily slowly is /reversible/; if a system is arbitrarily
+close to equilibrium, then an arbitrarily small change in its
+environment can reverse the direction of the process.
+Recognizing this, we can then say that the paths of integra
+tion in our equations are to be interpreted physically as
+/reversible paths/ . In practice, some systems (such as gases)
+come to equilibrium so rapidly that rather fast changes of
+state (on the time scale of our own perceptions) may be quite
+good approximations to reversible changes; thus the change of
+state of water vapor in a steam engine may be considered 
+reversible to a useful engineering approximation.
+
+
+** Intensive and Extensive Parameters
+
+The literature of thermodynamics has long recognized a distinction between two
+kinds of quantities that may be used to define the thermodynamic
+state. If we imagine a given system as composed of smaller 
+subsystems, we usually find that some of the thermodynamic variables
+have the same values in each subsystem, while others are additive,
+the total amount being the sum of the values of each subsystem.
+These are called /intensive/ and /extensive/ variables, respectively.
+According to this definition, evidently, the mass of a system is
+always an extensive quantity, and at equilibrium the temperature
+is an intensive ‘quantity. Likewise, the energy will be extensive
+provided that the interaction energy between the subsystems can
+be neglected.
+
+It is important to note, however, that in general the terms
+\ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{} 
+so defined cannot be regarded as
+establishing a real physical distinction between the variables.
+This distinction is, like the notion of number of degrees of
+freedom, in part an anthropomorphic one, because it may depend
+on the particular kind of subdivision we choose to imagine. For
+example, a volume of air may be imagined to consist of a number
+of smaller contiguous volume elements. With this subdivision,
+the pressure is the same in all subsystems, and is therefore in
+tensive; while the volume is additive and therefore extensive .
+But we may equally well regard the volume of air as composed of
+its constituent nitrogen and oxygen subsystems (or we could re
+gard pure hydrogen as composed of two subsystems, in which the
+molecules have odd and even rotational quantum numbers 
+respectively, etc.) . With this kind of subdivision the volume is the
+same in all subsystems, while the pressure is the sum of the
+partial pressures of its constituents; and it appears that the
+roles of \ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{}
+ have been interchanged. Note that this ambiguity cannot be removed by requiring
+that we consider only spatial subdivisions, such that each sub
+system has the same local composi tion . For, consider a s tressed
+elastic solid, such as a stretched rubber band. If we imagine
+the rubber band as divided, conceptually, into small subsystems
+by passing planes through it normal to its axis, then the tension
+is the same in all subsystems, while the elongation is additive.
+But if the dividing planes are parallel to the axis, the elonga
+tion is the same in all subsystems, while the tension is 
+additive; once again, the roles of \ldquo{}extensive\rdquo{} and
+\ldquo{}intensive\rdquo{} are
+interchanged merely by imagining a different kind of subdivision.
+In spite of the fundamental ambiguity of the usual definitions, 
+the notions of extensive and intensive variables are useful, 
+and in practice we seem to have no difficulty in deciding
+which quantities should be considered intensive. Perhaps the
+distinction is better characterized, not by considering 
+subdivisions at all, but by adopting a different definition, in which
+we recognize that some quantities have the nature of a \ldquo{}force\rdquo{}
+or \ldquo{}potential\rdquo{}, or some other local physical property, and are
+therefore called intensive, while others have the nature of a
+\ldquo{}displacement\rdquo{} or a \ldquo{}quantity\rdquo{} of 
+something (i.e. are proportional to the size of the system), 
+and are therefore called extensive. Admittedly, this definition is somewhat vague, in a
+way that can also lead to ambiguities ; in any event, let us agree
+to class pressure, stress tensor, mass density, energy density,
+particle density, temperature, chemical potential, angular
+velocity, as intensive, while volume, mass, energy, particle
+numbers, strain, entropy, angular momentum, will be considered
+extensive.
+
+** The Kelvin Temperature Scale
+The form of the first law,
+$dU = dQ - dW$, expresses the net energy increment of a system as
+the heat energy supplied to it, minus the work done by it. In
+the simplest systems of two degrees of freedom, defined by 
+pressure and volume as the thermodynamic variables, the work done
+in an infinitesimal reversible change of state can be separated
+into a product $dW = PdV$ of an intensive and an extensive quantity.
+Furthermore, we know that the pressure $P$ is not only the 
+intensive factor of the work; it is also the \ldquo{}potential\rdquo{} 
+which governs mechanical equilibrium (in this case, equilibrium with respect
+to exchange of volume) between two systems; i .e., if they are
+separated by a flexible but impermeable membrane, the two systems
+will exchange volume $dV_1 = -dV_2$ in a direction determined by the
+pressure difference, until the pressures are equalized. The
+energy exchanged in this way between the systems is a product
+of the form
+#+begin_quote
+(/intensity/ of something) \times (/quantity/ of something exchanged)
+#+end_quote
+
+Now if heat is merely a particular form of energy that can
+also be exchanged between systems, the question arises whether
+the quantity of heat energy $dQ$ exchanged in an infinitesimal 
+reversible change of state can also be written as a product of one
+factor which measures the \ldquo{}intensity\rdquo{} of the heat, 
+times another that represents the \ldquo{}quantity\rdquo{}
+ of something exchanged between
+the systems, such that the intensity factor governs the 
+conditions of thermal equilibrium and the direction of heat exchange,
+in the same way that pressure does for volume exchange.
+
+
+But we already know that the /temperature/ is the quantity
+that governs the heat flow (i.e., heat flows from the hotter to
+the cooler body until the temperatures are equalized) . So the
+intensive factor in $dQ$ must be essentially the temperature. But
+our temperature scale is at present still arbitrary, and we can
+hardly expect that such a factorization will be possible for all
+calibrations of our thermometers.
+
+The same thing is evidently true of pressure; if instead of
+the pressure $P$ as ordinarily defined, we worked with any mono
+tonic increasing function $P_1 = P_1 (P)$ we would find that $P_1$ is
+just as good as $P$ for determining the direction of volume 
+exchange and the condition of mechanical equilibrium; but the work
+done would not be given by $PdV$; in general, it could not even
+be expressed in the form $P_1 \cdot dF(V)$, where $F(V)$ is some function
+of V.
+
+
+Therefore we ask: out of all the monotonic functions $t_1(t)$ 
+corresponding to different empirical temperature scales, is
+there one (which we denote as $T(t)$) which forms a \ldquo{}natural\rdquo{}
+intensity factor for heat, such that in a reversible change
+$dQ = TdS$, where $S(U,V)$ is a new function of the thermodynamic
+state? If so, then the temperature scale $T$ will have a great
+theoretical advantage, in that the laws of thermodynamics will
+take an especially simple form in terms of this particular scale,
+and the new quantity $S$, which we call the /entropy/, will be a
+kind of \ldquo{}volume\rdquo{} factor for heat.
+
+We recall that $dQ = dU + PdV$ is not an exact differential;
+i.e., on a change from one equilibrium state to another the
+integral
+
+\[\int_1^2 dQ\]
+
+cannot be set equal to the difference $Q_2 - Q_1$ of values of any
+state function $Q(U,V)$, since the integral has different values
+for different paths connecting the same initial and final states.
+Thus there is no \ldquo{}heat function\rdquo{} $Q(U,V)$, and the notion of
+\ldquo{}amount of heat\rdquo{} $Q$ stored in a body has no meaning 
+(nor does the \ldquo{}amount of work\rdquo{} $W$; 
+only the total energy is a well-defined quantity). 
+But we want the entropy $S(U,V)$ to be a definite quantity,
+like the energy or volume, and so $dS$ must be an exact differential.
+On an infinitesimal reversible change from one equilibrium state
+to another, the first law requires that it satisfy[fn:: The first
+equality comes from our requirement that $dQ = T\,dS$. The second
+equality comes from the fact that $dU = dQ - dW$ (the first law) and
+that $dW = PdV$ in the case where the state has two degrees of
+freedom, pressure and volume.]
+
+\begin{equation}
+dS(U,V) = \frac{dQ}{T} = \frac{dU}{T} + \frac{P}{T}dV
+\end{equation}
+
+Thus $(1/T)$ must be an /integrating factor/ which converts $dQ$ into
+an exact differential [[fn::A differential $M(x,y)dx +
+N(x,y)dy$ is called /exact/ if there is a scalar function
+$\Phi(x,y)$ such that $M = \frac{\partial \Phi}{\partial x}$ and
+$N=\frac{\partial \Phi}{\partial y}$. If there is, \Phi is called the
+/potential function/ of the differential, Conceptually, this means
+that M(x,y)dx + N(x,y) dy is the derivative of a scalar potential and
+so consequently corresponds to a conservative field.
+ 
+Even if there is no such potential function
+\Phi for the given differential, it is possible to coerce an
+inexact differential into an exact one by multiplying by an unknown
+function $\mu(x,y)$ (called an /integrating factor/) and requiring the
+resulting differential $\mu M\, dx + \mu N\, dy$ to be exact.
+
+To complete the analogy, here we have the differential $dQ =
+dU + PdV$ (by the first law) which is not exact---conceptually, there
+is no scalar potential nor conserved quantity corresponding to
+$dQ$. We have introduced a new differential $dS = \frac{1}{T}dQ$, and we
+are searching for the temperature scale $T(U,V)$ which makes $dS$
+exact (i.e. which makes $S$ correspond to a conserved quantity). This means
+that $\frac{1}{T}$ is playing the role of the integrating factor
+\ldquo{}\mu\rdquo{} for the differential $dQ$.]]
+
+Now the question of the existence and properties of 
+integrating factors is a purely mathematical one, which can be 
+investigated independently of the properties of any particular
+substance. Let us denote this integrating factor for the moment
+by $w(U,V) = T^{-1}$; then the first law becomes
+
+\begin{equation}
+dS(U,V) = w dU + w P dV
+\end{equation}
+
+from which the derivatives are
+
+\begin{equation}
+\left(\frac{\partial S}{\partial U}\right)_V = w, \qquad
+\left(\frac{\partial S}{\partial V}\right)_U = wP. 
+\end{equation}
+
+The condition that $dS$ be exact is that the cross-derivatives be
+equal, as in (1-4):
+
+\begin{equation}
+\frac{\partial^2 S}{\partial U \partial V} = \frac{\partial^2
+S}{\partial V \partial U},
+\end{equation}
+
+or 
+
+\begin{equation}
+\left(\frac{\partial w}{\partial V}\right)_U = \left(\frac{\partial
+P}{\partial U}\right)_V + P\cdot \left(\frac{\partial w}{\partial U}\right)_V.
+\end{equation}
+
+Any function $w(U,V)$ satisfying this differential equation is an
+integrating factor for $dQ$.
+
+But if $w(U,V)$ is one such integrating factor, which leads
+to the new state function $S(U,V)$, it is evident that
+$w_1(U,V) \equiv w \cdot f(S)$ is an equally good integrating factor, where
+$f(S)$ is an arbitrary function. Use of $w_1$ will lead to a 
+different state function 
+
+#what's with the variable collision?
+\begin{equation}
+S_1(U,V) = \int^S f(S) dS
+\end{equation}
+
+The mere conversion of into an exact differential is, therefore, 
+not enough to determine any unique entropy function $S(U,V)$.
+However, the derivative
+
+\begin{equation}
+\left(\frac{dU}{dV}\right)_S = -P
+\end{equation}
+
+is evidently uniquely determined; so also, therefore, is the
+family of lines of constant entropy, called /adiabats/, in the
+$(U-V)$ plane. But, as (1-24) shows, the numerical value of $S$ on
+each adiabat is still completely undetermined.
+
+In order to fix the relative values of $S$ on different 
+adiabats we need to add the condition, not yet put into the equations, 
+that the integrating factor $w(U,V) = T^{-1}$ is to define a new
+temperature scale . In other words, we now ask: out of the
+infinite number of different integrating factors allowed by
+the differential equation (1-23), is it possible to find one
+which is a function only of the empirical temperature $t$? If
+$w=w(t)$, we can write
+
+\begin{equation}
+\left(\frac{\partial w}{\partial V}\right)_U = \frac{dw}{dt}\left(\frac{\partial
+t}{\partial V}\right)_U
+\end{equation}
+\begin{equation}
+\left(\frac{\partial w}{\partial U}\right)_V = \frac{dw}{dt}\left(\frac{\partial
+t}{\partial U}\right)_V
+\end{equation}
+
+
+and (1-23) becomes
+\begin{equation}
+\frac{d}{dt}\log{w} = \frac{\left(\frac{\partial P}{\partial
+U}\right)_V}{\left(\frac{\partial t}{\partial V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V}
+\end{equation}
+
+
+which shows that $w$ will be determined to within a multiplicative
+factor.
+
+Is the temperature scale thus defined independent of the
+empirical scale from which we started? To answer this, let
+$t_1 = t_1(t)$ be any monotonic function which defines a different
+empirical temperature scale. In place of (1-28), we then have
+
+\begin{equation}
+\frac{d}{dt_1}\log{w} \quad=\quad \frac{\left(\frac{\partial P}{\partial
+U}\right)_V}{\left(\frac{\partial t_1}{\partial V}\right)_U-P\left(\frac{\partial t_1}{\partial U}\right)_V}
+\quad = \quad 
+ \frac{\left(\frac{\partial P}{\partial
+U}\right)_V}{\frac{dt_1}{dt}\left[ \left(\frac{\partial t}{\partial
+V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V\right]},
+\end{equation}
+or
+\begin{equation}
+\frac{d}{dt_1}\log{w_1} = \frac{dt}{dt_1}\frac{d}{dt}\log{w}
+\end{equation}
+
+which reduces to $d \log{w_1} = d \log{w}$, or
+\begin{equation}
+w_1 = C\cdot w
+\end{equation}
+
+Therefore, integrating factors derived from whatever empirical
+temperature scale can differ among themselves only by a 
+multiplicative factor. For any given substance, therefore, except
+for this factor (which corresponds just to our freedom to choose
+the size of the units in which we measure temperature), there is
+only /one/ temperature scale $T(t) = 1/w$ with the property that
+$dS = dQ/T$ is an exact differential.
+
+To find a feasible way of realizing this temperature scale
+experimentally, multiply numerator and denominator of the right
+hand side of (1-28) by the heat capacity at constant volume,
+$C_V^\prime = (\partial U/\partial t) V$, the prime denoting that 
+it is in terms of the empirical temperature scale $t$. 
+Integrating between any two states denoted 1 and 2, we have
+
+\begin{equation}
+\frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}
+\frac{\left(\frac{\partial P}{\partial t}\right)_V dt}{P - C_V^\prime
+\left(\frac{\partial t}{\partial V}\right)_U} \right\}
+\end{equation}
+
+If the quantities on the right-hand side have been determined
+experimentally, then a numerical integration yields the ratio
+of Kelvin temperatures of the two states.
+
+This process is particularly simple if we choose for our
+system a volume of gas with the property found in Joule's famous
+expansion experiment; when the gas expands freely into a vacuum
+(i.e., without doing work, or $U = \text{const.}$), there is no change in
+temperature. Real gases when sufficiently far from their condensation 
+points are found to obey this rule very accurately.
+But then 
+
+\begin{equation}
+\left(\frac{dt}{dV}\right)_U = 0
+\end{equation}
+
+and on a change of state in which we heat this gas at constant
+volume, (1-31) collapses to
+
+\begin{equation}
+\frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}
+\frac{1}{P}\left(\frac{\partial P}{\partial t}\right)_V dt\right\} = \frac{P_2}{P_1}.
+\end{equation}
+
+Therefore, with a constant-volume ideal gas thermometer, (or more
+generally, a thermometer using any substance obeying (1-32) and
+held at constant volume), the measured pressure is directly 
+proportional to the Kelvin temperature.
+
+For an imperfect gas, if we have measured  $(\partial t /\partial
+V)_U$ and $C_V^\prime$, Eq. (1-31) determines the necessary
+corrections to (1-33). However, an alternative form of (1-31), in
+which the roles of pressure and volume are interchanged, proves to be
+more convenient for experimental determinations. To derive it, introduce the
+enthalpy function
+
+\begin{equation}H = U + PV\end{equation}
+
+with the property
+
+\begin{equation}
+dH = dQ + VdP
+\end{equation}
+
+Equation (1-19) then becomes
+
+\begin{equation}
+dS = \frac{dH}{T} - \frac{V}{T}dP.
+\end{equation}
+
+Repeating the steps (1-20) to (1-31) of the above derivation
+starting from (1-36) instead of from (1-19), we arrive at
+
+\begin{equation}
+\frac{T_2}{T_1} = \exp\left\{\int_{t_1}^{t_2}
+\frac{\left(\frac{dV}{dt}\right)_P dt}{V + C_P^\prime
+\left(\frac{\partial t}{\partial P}\right)_H}\right\}
+\end{equation}
+
+or
+
+\begin{equation}
+\frac{T_2}{T_1} = \exp\left\{\frac{\alpha^\prime
+dt}{1+\left(C_P^\prime \cdot \mu^\prime / V\right)}\right\}
+\end{equation}
+
+where
+\begin{equation}
+\alpha^\prime \equiv \frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P
+\end{equation}
+is the thermal expansion coefficient,
+\begin{equation}
+C_P^\prime \equiv \left(\frac{\partial H}{\partial t}\right)_P
+\end{equation}
+is the heat capacity at constant pressure, and
+\begin{equation}
+\mu^\prime \equiv \left(\frac{dt}{dP}\right)_H
+\end{equation}
+
+is the coefficient measured in the Joule-Thompson porous plug
+experiment, the primes denoting again that all are to be measured
+in terms of the empirical temperature scale $t$.
+Since $\alpha^\prime$, $C_P^\prime$, $\mu^\prime$ are all 
+easily measured in the laboratory, Eq. (1-38) provides a 
+feasible way of realizing the Kelvin temperature scale experimentally, 
+taking account of the imperfections of real gases. 
+For an account of the work of Roebuck and others based on this
+relation, see [[http://books.google.com/books?id=KKJKAAAAMAAJ][Zemansky (1943)]]; pp. 252-255.
+
+Note that if $\mu^\prime = O$ and we heat the gas at constant
+pressure, (1-38) reduces to
+
+\begin{equation}
+\frac{T_2}{T_1} = \exp\left\{ \int_{t_1}^{t_2}
+\frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P dt \right\} = \frac{V_2}{V_1}
+\end{equation}
+
+so that, with a constant-pressure gas thermometer using a gas for
+which the Joule-Thomson coefficient is zero, the Kelvin temperature is
+proportional to the measured volume.
+
+Now consider another empirical fact, [[http://en.wikipedia.org/wiki/Boyle%27s_law][Boyle's law]]. For gases
+sufficiently far from their condensation points---which is also
+the condition under which (1-32) is satisfied---Boyle found that
+the product $PV$ is a constant at any fixed temperature. This
+product is, of course proportional to the number of moles $n$
+present, and so Boyle's equation of state takes the form
+
+\begin{equation}PV = n \cdot f(t)\end{equation}
+
+where f(t) is a function that depends on the particular empirical
+temperature scale used. But from (1-33) we must then have 
+$f(t) = RT$, where $R$ is a constant, the universal gas constant whose
+numerical value (1.986 calories per mole per degree K) , depends
+on the size of the units in which we choose to measure the Kelvin
+temperature $T$. In terms of the Kelvin temperature, the ideal gas
+equation of state is therefore simply
+
+\begin{equation}
+PV = nRT
+\end{equation}
+
+
+The relations (1-32) and (1-44) were found empirically, but
+with the development of thermodynamics one could show that they
+are not logically independent. In fact, all the material needed
+for this demonstration is now at hand, and we leave it as an
+exercise for the reader to prove that Joule‘s relation (1-32) is
+a logical consequence of Boyle's equation of state (1-44) and the
+first law.
+
+
+Historically, the advantages of the gas thermometer were
+discovered empirically before the Kelvin temperature scale was
+defined; and the temperature scale \theta defined by
+
+\begin{equation}
+\theta = \lim_{P\rightarrow 0}\left(\frac{PV}{nR}\right)
+\end{equation}
+
+was found to be convenient, easily reproducible, and independent
+of the properties of any particular gas. It was called the
+/absolute/ temperature scale; and from the foregoing it is clear
+that with the same choice of the numerical constant $R$, the 
+absolute and Kelvin scales are identical.
+
+
+For many years the unit of our temperature scale was the
+Centigrade degree, so defined that the difference $T_b - T_f$ of
+boiling and freezing points of water was exactly 100 degrees.
+However, improvements in experimental techniques have made another
+method more reproducible; and the degree was redefined by the
+Tenth General Conference of Weights and Measures in 1954, by
+the condition that the triple point of water is at 273.l6^\circ K,
+this number being exact by definition. The freezing point, 0^\circ C,
+is then 273.15^\circ K. This new degree is called the Celsius degree.
+For further details, see the U.S. National Bureau of Standards
+Technical News Bulletin, October l963.
+
+
+The appearance of such a strange and arbitrary-looking
+number as 273.16 in the /definition/ of a unit is the result of
+the historical development, and is the means by which much
+greater confusion is avoided. Whenever improved techniques make
+possible a new and more precise (i.e., more reproducible) 
+definition of a physical unit, its numerical value is of course chosen
+so as to be well inside the limits of error with which the old
+unit could be defined. Thus the old Centigrade and new Celsius
+scales are the same, within the accuracy with which the 
+Centigrade scale could be realized; so the same notation, ^\circ C, is used
+for both . Only in this way can old measurements retain their
+value and accuracy, without need of corrections every time a
+unit is redefined.
+
+#capitalize Joules?
+Exactly the same thing has happened in the definition of
+the calorie; for a century, beginning with the work of Joule,
+more and more precise experiments were performed to determine
+the mechanical equivalent of heat more and more accurately . But
+eventually mechanical and electrical measurements of energy be
+came far more reproducible than calorimetric measurements; so
+recently the calorie was redefined to be 4.1840 Joules, this
+number now being exact by definition. Further details are given
+in the aforementioned Bureau of Standards Bulletin.
+
+
+The derivations of this section have shown that, for any
+particular substance, there is (except for choice of units) only
+one temperature scale $T$ with the property that $dQ = TdS$ where
+$dS$ is the exact differential of some state function $S$. But this
+in itself provides no reason to suppose that the /same/ Kelvin
+scale will result for all substances; i.e., if we determine a
+\ldquo{}helium Kelvin temperature\rdquo{} and a 
+\ldquo{}carbon dioxide Kelvin temperature\rdquo{} by the measurements 
+indicated in (1-38), and choose the units so that they agree numerically at one point, will they then
+agree at other points? Thus far we have given no reason to 
+expect that the Kelvin scale is /universal/, other than the empirical
+fact that the limit (1-45) is found to be the same for all gases.
+In section 2.0 we will see that this universality is a conse
+quence of the second law of thermodynamics (i.e., if we ever
+find two substances for which the Kelvin scale as defined above
+is different, then we can take advantage of this to make a 
+perpetual motion machine of the second kind).
+
+
+Usually, the second law is introduced before discussing
+entropy or the Kelvin temperature scale. We have chosen this
+unusual order so as to demonstrate that the concepts of entropy
+and Kelvin temperature are logically independent of the second
+law; they can be defined theoretically, and the experimental
+procedures for their measurement can be developed, without any
+appeal to the second law. From the standpoint of logic, there
+fore, the second law serves /only/ to establish that the Kelvin
+temperature scale is the same for all substances.
+
+
+** COMMENT Entropy of an Ideal Boltzmann Gas
+
+At the present stage we are far from understanding the physical 
+meaning of the function $S$ defined by (1-19); but we can investigate its mathematical
+form and numerical values. Let us do this for a system con
+sisting cf n moles of a substance which obeys the ideal gas
+equation of state
+and for which the heat capacity at constant volume CV is a
+constant. The difference in entropy between any two states (1)
+and (2) is from (1-19),
+
+
+where we integrate over any reversible path connecting the two
+states. From the manner in which S was defined, this integral
+must be the same whatever path we choose. Consider, then, a
+path consisting of a reversible expansion at constant tempera
+ture to a state 3 which has the initial temperature T, and the
+.L ' "'1 final volume V2; followed by heating at constant volume to the final temperature T2. Then (1-47) becomes
+3 2 I If r85 - on - db — = d — -4 S2 51 J V [aT]v M (1 8)
+1 3
+To evaluate the integral over (1 +3) , note that since
+dU = T :15 — P dV, the Helmholtz free energy function F E U — TS
+has the property dF = --S - P 61V; and of course is an exact
+differential since F is a definite state function. The condition
+that dF be exact is, analogous to (1-22),
+which is one of the Maxwell relations, discussed further in
+where CV is the molar heat capacity at constant volume. Collec
+ting these results, we have
+3
+l 3
+1 nR log(V2/V1) + nCV log(T2/Tl) (1-52)
+since CV was assumed independent of T. Thus the entropy function
+must have the form
+S(n,V,T) = nR log V + n CV log T + (const.) (l~53)
+
+
+From the derivation, the additive constant must be independent
+of V and T; but it can still depend on n. We indicate this by
+writing
+where f (n) is a function not determined by the definition (1-47).
+The form of f (n) is , however, restricted by the condition that
+the entropy be an extensive quantity; i .e . , two identical systems
+placed together should have twice the entropy of a single system;
+Substituting (l—-54) into (1-55), we find that f(n) must satisfy
+To solve this, one can differentiate with respect to q and set
+q = 1; we then obtain the differential equation
+n f ' (n) — f (n) + Rn = 0 (1-57)
+which is readily solved; alternatively, just set n = 1 in (1-56)
+and replace q by n . By either procedure we find
+f (n) = n f (1) — Rn log n . (1-58)
+As a check, it is easily verified that this is the solution of
+where A E f (l) is still an arbitrary constant, not determined
+by the definition (l—l9) , or by the condition (l-55) that S be
+extensive. However, A is not without physical meaning; we will
+see in the next Section that the vapor pressure of this sub
+stance (and more generally, its chemical potential) depends on
+A. Later, it will appear that the numerical value of A involves
+Planck's constant, and its theoretical determination therefore
+requires quantum statistics .
+We conclude from this that, in any region where experi
+mentally CV const. , and the ideal gas equation of state is
+
+
+obeyed, the entropy must have the form (1-59) . The fact that
+classical statistical mechanics does not lead to this result,
+the term nR log (l/n) being missing (Gibbs paradox) , was his
+torically one of the earliest clues indicating the need for the
+quantum theory.
+In the case of a liquid, the volume does not change appre
+ciably on heating, and so d5 = n CV dT/T, and if CV is indepen
+dent of temperature, we would have in place of (1-59) ,
+where Ag is an integration constant, which also has physical
+meaning in connection with conditions of equilibrium between
+two different phases.
+1.1.0 The Second Law: Definition. Probably no proposition in
+physics has been the subject of more deep and sus tained confusion
+than the second law of thermodynamics . It is not in the province
+of macroscopic thermodynamics to explain the underlying reason
+for the second law; but at this stage we should at least be able
+to state this law in clear and experimentally meaningful terms.
+However, examination of some current textbooks reveals that,
+after more than a century, different authors still disagree as
+to the proper statement of the second law, its physical meaning,
+and its exact range of validity.
+Later on in this book it will be one of our major objectives
+to show, from several different viewpoints , how much clearer and
+simpler these problems now appear in the light of recent develop
+ments in statistical mechanics . For the present, however, our
+aim is only to prepare the way for this by pointing out exactly
+what it is that is to be proved later. As a start on this at
+tempt, we note that the second law conveys a certain piece of
+informations about the direction in which processes take place.
+In application it enables us to predict such things as the final
+equilibrium state of a system, in situations where the first law
+alone is insufficient to do this.
+A concrete example will be helpful. We have a vessel
+equipped with a piston, containing N moles of carbon dioxide.
+
+
+The system is initially at thermal equilibrium at temperature To, volume V0 and pressure PO; and under these conditions it contains
+n moles of CO2 in the vapor phase and moles in the liquid
+phase . The system is now thermally insulated from its surround
+ings, and the piston is moved rapidly (i.e. , so that n does not
+change appreciably during the motion) so that the system has a
+new volume Vf; and immediately after the motion, a new pressure
+PI . The piston is now held fixed in its new position , and the
+system allowed to come once more to equilibrium. During this
+process, will the CO2 tend to evaporate further, or condense further? What will be the final equilibrium temperature Teq, the final pressure PeCE , and final value of n eq?
+It is clear that the firs t law alone is incapable of answering
+these questions; for if the only requirement is conservation of
+energy, then the CO2 might condense , giving up i ts heat of vapor
+ization and raising the temperature of the system; or it might
+evaporate further, lowering the temperature. Indeed, all values
+of neq in O i neq i N would be possible without any violation of the first law. In practice, however, this process will be found
+to go in only one direction and the sys term will reach a definite
+final equilibrium state with a temperature, pressure, and vapor
+density predictable from the second law.
+Now there are dozens of possible verbal statements of the
+second law; and from one standpoint, any statement which conveys
+the same information has equal right to be called "the second
+law." However, not all of them are equally direct statements of
+experimental fact, or equally convenient for applications, or
+equally general; and it is on these grounds that we ought to
+choose among them .
+Some of the mos t popular statements of the s econd law be
+long to the class of the well-—known "impossibility" assertions ;
+i.e. , it is impossible to transfer heat from a lower to a higher
+temperature without leaving compensating changes in the rest of
+the universe , it is imposs ible to convert heat into useful work
+without leaving compensating changes, it is impossible to make
+a perpetual motion machine of the second kind, etc.
+
+
+Suoh formulations have one clear logical merit; they are
+stated in such a way that, if the assertion should be false, a
+single experiment would suffice to demonstrate that fact conclu
+sively. It is good to have our principles stated in such a
+clear, unequivocal way.
+However, impossibility statements also have some disadvan
+tages . In the first place, their_ are not, and their very
+nature cannot be, statements of eiperimental fact. Indeed, we
+can put it more strongly; we have no record of anyone having
+seriously tried to do any of the various things which have been
+asserted to be impossible, except for one case which actually
+succeeded‘. In the experimental realization of negative spin
+temperatures , one can transfer heat from a lower to a higher
+temperature without external changes; and so one of the common
+impossibility statements is now known to be false [for a clear
+discussion of this, see the article of N. F . Ramsey (1956) ;
+experimental details of calorimetry with negative temperature
+spin systems are given by Abragam and Proctor (1958) ] .
+Finally, impossibility statements are of very little use in
+applications of thermodynamics; the assertion that a certain kind
+of machine cannot be built, or that a -certain laboratory feat
+cannot be performed, does not tell me very directly whether my
+carbon dioxide will condense or evaporate. For applications,
+such assertions must first be converted into a more explicit
+mathematical form.
+For these reasons, it appears that a different kind of
+statement of the second law will be, not necessarily more
+"correct,” but more useful in practice. Now both Clausius (3.875)
+and Planck (1897) have laid great stress on their conclusion
+that the most general statement, and also the most immediately
+useful in applications, is simply the existence of a state
+function, called the entropy, which tends to increase. More
+precisely: in an adiabatic change of state, the entropy of
+a system may increase or may remain constant, but does not
+decrease. In a process involving heat flow to or from the
+system, the total entropy of all bodies involved may increase
+
+
+or may remain constant; but does not decrease; let us call this
+the “weak form" of the second law.
+The weak form of the second law is capable of answering the
+first question posed above; thus the carbon dioxide will evapo
+rate further if , and only if , this leads to an increase in the
+total entropy of the system . This alone , however , is not enough
+to answer the second question; to predict the exact final equili
+brium state, we need one more fact.
+The strong form of the second law is obtained by adding the
+further assertion that the entropy not only “tends" to increase;
+in fact it will increase, to the maximum value permitted E2 the
+constraints imposed.* In the case of the carbon dioxide, these
+constraints are: fixed total energy (first law) , fixed total
+amount of carbon dioxide , and fixed position of the piston . The
+final equilibrium state is the one which has the maximum entropy
+compatible with these constraints , and it can be predicted quan
+titatively from the strong form of the second law if we know,
+from experiment or theory, the thermodynamic properties of carbon
+dioxide (i .e . , heat capacity , equation of state , heat of vapor
+ization) .
+To illus trate this , we set up the problem in a crude ap
+proximation which supposes that (l) in the range of conditions
+of interest, the molar heat capacity CV of the vapor, and C2 of
+the liquid, and the molar heat of vaporization L, are all con
+stants, and the heat capacities of cylinder and piston are neg
+ligible; (2) the liquid volume is always a small fraction of the
+total V, so that changes in vapor volume may be neglected; (3) the
+vapor obeys the ideal gas equation of state PV = nRT. The in
+ternal energy functions of liquid and vapor then have the form
+UPb = + A} (1-61)
+T T U = n‘ C '1‘ A + L] (1-62)
+v , v
+where A is a constant which plays no role in the problem. The
+appearance of L in (1-62) recognizes that the zero from which we
+*Note , however , that the second law has nothing to say about how rapidly this approach to equilibrium takes place.
+
+
+measure energy of the vapor is higher than that of the liquid by
+the energy L necessary to form the vapor. On evaporation of dn
+moles of liquid, the total energy increment is (ill = + dUV= O,
+or
+[n CV [(CV — CQ)T + = O (l—63)
+which is the constraint imposed by the first law. As we found
+previously (l~59) , (1-60) the entropies of vapor and liquid are
+given by
+S = n [C 1n T + R ln (V/n) + A ] (1-64)
+v v v
+where AV, ASL are the constants of integration discussed in the
+Si
+last Section.
+We leave it as an exercise for the reader to complete the
+derivation from this point , and show that the total entropy
+S = 82 + SV is maximized subject to the constraint (1-6 3) , when
+R
+the values 11 , T are related by
+eq eq
+Equation (1-66) is recognized as an approximate form of the Vapor
+pressure formula .
+We note that AQ, AV, which appeared first as integration
+constants for the entropy with no parti cular physical meaning ,
+now play a role in determining the vapor pressure.
+l.ll The Second Law: Discussion. We have emphasized the dis
+tinction between the weak and strong forms of the second law
+because (with the exception of Boltzmann ' s original unsuccessful
+argument based on the H—theorem) , most attempts to deduce the
+second law from statis tical mechanics have considered only the
+weak form; whereas it is evidently the strong form that leads
+to definite quantitative predictions, and is therefore needed