jaynes: org/stat-mech.org comparison

comparison org/stat-mech.org @ 3:8f3b6dcb9add

Transcribed up to section 1.9, Entropy of an Ideal Boltzmann Gas

author	Dylan Holmes <ocsenave@gmail.com>
date	Sun, 29 Apr 2012 02:38:22 -0500
parents	afbe1fe19b36
children	299a098a30da

comparison

equal deleted inserted replaced

-:afbe1fe19b36
+:8f3b6dcb9add
 (probability). Our later development of probability theory in
 Chapter 6,7 will be, to a considerable degree, a paraphrase
 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
 before the beginnings of recorded history. But we can hardly
 doubt that primitive man learned quickly that objects exposed
 is not specified), any monotonic increasing function
 \(t‘ = f(t)\) provides an equally good temperature scale for the
 present.
-** Thermodynamic Systems.
+** Thermodynamic Systems
 The \ldquo{}thermodynamic systems\rdquo{} which
 are the objects of our study may be, physically, almost any
 collections of objects. The traditional simplest system with
 which to begin a study of thermodynamics is a volume of gas.
 special status, these relations have become known as the
 \ldquo{}laws\rdquo{}
 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
 isolated from their surroundings, they may undergo observable
 changes for a time as a result; one body may become warmer,
 parameters with one constraint) are said to possess two
 /degrees of freedom/; for the range of possible equilibrium states is defined
 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/,
 /x/, and /t/ in equilibrium states can be
 expressed as the level set of a function.
 \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
+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
 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
+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
+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
+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}
 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
+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
 difference was due to the work done in expansion of the gas against
 atmospheric pressure, when measuring $C_p$. Supposing that the
 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
+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
 * COMMENT  Appendix

Mercurial > jaynes

comparison org/stat-mech.org @ 3:8f3b6dcb9add