jaynes: org/stat-mech.org comparison

comparison org/stat-mech.org @ 4:299a098a30da

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author	Dylan Holmes <ocsenave@gmail.com>
date	Sun, 29 Apr 2012 17:49:18 -0500
parents	8f3b6dcb9add
children	e7185b523c80

comparison

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+:299a098a30da
 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:: The set of solutions to an equation
+relationship of the form[fn:: Edit: 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.
 thermal interaction, which can be carried out as soon as a primitive
 thermometer is at hand. In fact these experiments, which we summarize
 so quickly, required a very long time for their first performance, and
 the essential conclusions of this Section were first arrived at only
 about 1760---more than 160 years after Galileo's invention of the
-thermometer---by Joseph Black, who was Professor of Chemistry at
+thermometer---[[http://web.lemoyne.edu/~giunta/blackheat.html][by Joseph Black]], who was Professor of Chemistry at
 Glasgow University. Black's analysis of calorimetric experiments
 initiated by G. D. Fahrenheit before 1736 led to the first recognition
 of the distinction between temperature and heat, and prepared the way
 for the work of his better-known pupil, James Watt.
 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 $(V-t)$
+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.
 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)
+/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.
 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 .
+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
+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
+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
 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
+to exchange of volume) between two systems; ie., 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
 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
+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.
 (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
+to another, the first law requires that it satisfy[fn:: Edit: 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 +
+an exact differential [[fn::Edit: 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
 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
+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}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
+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}
 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
+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
+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.
 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
+** 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
+meaning of the function $S$ defined by (1-19); but we can investigate
-form and numerical values. Let us do this for a system con
+its  mathematical form and numerical values. Let us do this for a
-sisting cf n moles of a substance which obeys the ideal gas
+system
+consisting of $n$ 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)
+\begin{equation}PV = nRT\end{equation}
+and for which the heat capacity at constant volume
+$C_V$ is a constant. The difference in entropy between any two states (1)
 and (2) is from (1-19),
+\begin{equation}
+S_2 - S_1 = \int_1^2 \frac{dQ}{T} = \int_1^2
+\left[\left(\frac{\partial S}{\partial V}\right)+\left(\frac{\partial S}{\partial T}\right)_V dT\right]
+\end{equation}
 where we integrate over any reversible path connecting the two
-states. From the manner in which S was defined, this integral
+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
+path consisting of a reversible expansion at constant
-ture to a state 3 which has the initial temperature T, and the
+temperature to a state 3 which has the initial temperature $T_1$, and the
-.L ' "'1 final volume V2; followed by heating at constant volume to the final temperature T2. Then (1-47) becomes
+the final volume $V_2$; followed by heating at constant volume to the
-3 2 I If r85 - on - db — = d — -4 S2 51 J V [aT]v M (1 8)
+final temperature $T_2$.
-1 3
+Then (1-47) becomes
-To evaluate the integral over (1 +3) , note that since
-dU = T :15 — P dV, the Helmholtz free energy function F E U — TS
+\begin{equation}
-has the property dF = --S - P 61V; and of course is an exact
+S_2 - S_1 = \int_1^3 \left(\frac{\partial S}{\partial V}\right)_T dV +
-differential since F is a definite state function. The condition
+\int_3^2 \left(\frac{\partial S}{\partial T}\right)_V dT
-that dF be exact is, analogous to (1-22),
+\end{equation}
+To evaluate the integral over $(1\rightarrow 3)$, note that since $dU
+= TdS - PdV$, the Helmholtz free energy function $F \equiv U -TS$ has
+the property $dF = -SdT - PdV$; and of course $dF$ is an exact
+differential since $F$ is a definite state function. The condition
+that $dF$ be exact is, analogous to (1-22),
+\begin{equation}
+\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial
+P}{\partial T}\right)_V
+\end{equation}
 which is one of the Maxwell relations, discussed further in
-where CV is the molar heat capacity at constant volume. Collec
+Chapter 2. But [the value of this expression] is determined by the equation of state
-ting these results, we have
+(1-46):
-3
-l 3
+\begin{equation}
-1 nR log(V2/V1) + nCV log(T2/Tl) (1-52)
+\left(\frac{\partial S}{\partial V}\right)_T = \frac{nR}{V}
-since CV was assumed independent of T. Thus the entropy function
+\end{equation}
+Likewise, along the path $(3\rightarrow 2)$, we have
+\begin{equation}
+\left(\frac{\partial S}{\partial T}\right)_V = \frac{n C_V}{T}
+\end{equation}
+where $C_V$ is the molar heat capacity at constant volume.
+Collecting these results, we have
+\begin{equation}
+S_2 - S_1 = \int_1^3 \frac{nR}{V} dV + \int_2^3 \frac{n C_V}{T} dT =
+nR\log{(V_2/V_1)} + nC_V \log{(T_2/T_1)}
+\end{equation}
+since $C_V$ 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)
+\begin{equation}
+S(n,V,T) = nR \log{V} + n C_V \log{T} + (\text{const.})
+\end{equation}
 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
+\begin{equation}
-the entropy be an extensive quantity; i .e . , two identical systems
+S(n,V,T) = n\left[R \log{V} + C_V \log{T}\right] + f(n)
-placed together should have twice the entropy of a single system;
+\end{equation}
-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
+where $f(n)$ is a function not determined by the definition (1-47).
-q = 1; we then obtain the differential equation
+The form of $f(n)$ is, however, restricted by the condition that
-n f ' (n) — f (n) + Rn = 0 (1-57)
+the entropy be an extensive quantity; i.e., two identical systems
-which is readily solved; alternatively, just set n = 1 in (1-56)
+placed together should have twice the entropy of a single system; or
-and replace q by n . By either procedure we find
+more generally,
-f (n) = n f (1) — Rn log n . (1-58)
-As a check, it is easily verified that this is the solution of
+\begin{equation}
-where A E f (l) is still an arbitrary constant, not determined
+S(qn, qV, T) = q\cdot S(n,v,T),\qquad 0<q<\infty
-by the definition (l—l9) , or by the condition (l-55) that S be
+\end{equation}
-extensive. However, A is not without physical meaning; we will
-see in the next Section that the vapor pressure of this sub
+Substituting (1-54) into (1-55), we find that $f(n)$ must satisfy
-stance (and more generally, its chemical potential) depends on
+the functional equation
-A. Later, it will appear that the numerical value of A involves
+\begin{equation}
+f(q\cdot n) = q\cdot f(n) - R\cdot n\cdot q\log{q}\end{equation}
+To solve this, one can differentiate with respect to $q$ and set
+$q = 1$; we then obtain the differential equation
+\begin{equation}
+n\cdot f^\prime(n) - f(n) + R\cdot n = 0
+\end{equation}
+# xy' - y + rx = 0
+which is readily solved; alternatively, just set $n = 1$ in (1-56)
+and replace $q$ by $n$ By either procedure we find
+\begin{equation}
+f(n) = n\cdot f(1) - R\cdot n \log{n} (1-58)
+\end{equation}
+As a check, it is easily verified that this is the solution of (1-56)
+and (1-57). We then have finally,
+\begin{equation}
+S(n,V,t) = n\left[C_v\cdot\log{t} + R\cdot \log{\left(\frac{V}{n}\right)} +
+A\right]
+\end{equation}
+where $A\equiv f(1)$ is still an arbitrary constant, not determined
+by the definition (1-19), or by the condition (1-55) that $S$ be
+extensive. However, $A$ is not without physical meaning; we will
+see in the next Section that the vapor pressure of this
+substance (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 .
+requires quantum statistics.
-We conclude from this that, in any region where experi
-mentally CV const. , and the ideal gas equation of state is
+#edit: "is constant"
+We conclude from this that, in any region where experimentally
+$C_V$ is constant, and the ideal gas equation of state is
-obeyed, the entropy must have the form (1-59) . The fact that
+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
+the term $n\cdot R \cdot \log{(1/n)}$ being missing (Gibbs paradox),
-torically one of the earliest clues indicating the need for the
+was historically 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
+In the case of a liquid, the volume does not change
-dent of temperature, we would have in place of (1-59) ,
+appreciably on heating, and so $dS = n\cdot C_V\cdot dT/T$, and if
-where Ag is an integration constant, which also has physical
+$C_V$ is independent of temperature, we would have in place of (1-59),
+\begin{equation}
+S = n\left[C_V\ln{T}+A_\ell\right]
+\end{equation}
+where $A_\ell$ 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
+** The Second Law: Definition
-than the second law of thermodynamics . It is not in the province
+Probably no proposition in physics has been the subject of more deep
+and sustained 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.
+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
+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
+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.
+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.
+equipped with a piston, containing $N$ moles of carbon dioxide.
+#changed V_f to V_1
-The system is initially at thermal equilibrium at temperature To, volume V0 and pressure PO; and under these conditions it contains
+The system is initially at thermal equilibrium at temperature $T_0$,
-n moles of CO2 in the vapor phase and moles in the liquid
+volume $V_0$ and pressure $P_O$; and under these conditions it contains
-phase . The system is now thermally insulated from its surround
+$n$ moles of CO_2 in the vapor phase and $N-n$ moles in the liquid
-ings, and the piston is moved rapidly (i.e. , so that n does not
+phase The system is now thermally insulated from its
+surroundings, 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
+new volume $V_1$; and immediately after the motion, a new pressure
-PI . The piston is now held fixed in its new position , and the
+$P_1$ 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?
+process, will the CO_2 tend to evaporate further, or condense further?
-It is clear that the firs t law alone is incapable of answering
+What will be the final equilibrium temperature $T_{eq}$,
+the final pressure $P_eq$, and final value of $n_{eq}$?
+It is clear that the first 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
+energy, then the CO_2 might condense, giving up its heat of
-ization and raising the temperature of the system; or it might
+vaporization 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
+of $n_{eq}$ in $O \leq n_{eq} \leq N$ would be possible without any
-to go in only one direction and the sys term will reach a definite
+violation of the first law. In practice, however, this process will be found
+to go in only one direction and the system 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
+the same information has equal right to be called \ldquo{}the second
-law." However, not all of them are equally direct statements of
+law.\rdquo{} 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 .
+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 ;
+Some of the mos t popular statements of the second law
-i.e. , it is impossible to transfer heat from a lower to a higher
+belong to the class of the well-known \ldquo{}impossibility\rdquo{}
+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
+the universe, it is impossible 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
+single experiment would suffice to demonstrate that fact
-sively. It is good to have our principles stated in such a
+conclusively. 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
+However, impossibility statements also have some
-nature cannot be, statements of eiperimental fact. Indeed, we
+disadvantages In the first place, /they are not, and by 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
+succeeded. In the experimental realization of negative spin
-temperatures , one can transfer heat from a lower to a higher
+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) ;
+discussion of this, see the [[../sources/Ramsey.pdf][article of N. F. Ramsey (1956)]];
 experimental details of calorimetry with negative temperature
-spin systems are given by Abragam and Proctor (1958) ] .
+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
+/applications/ of thermodynamics; the assertion that a certain kind
-of machine cannot be built, or that a -certain laboratory feat
+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)
+\ldquo{}correct\rdquo{}, but more useful in practice. Now both Clausius (1875)
 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 \ldquo{}weak form\rdquo{} 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
+first question posed above; thus the carbon dioxide will
-rate further if , and only if , this leads to an increase in the
+evaporate further if, and only if, this leads to an increase in the
-total entropy of the system . This alone , however , is not enough
+total entropy of the system This alone, however, is not enough
-to answer the second question; to predict the exact final equili
+to answer the second question; to predict the exact final
-brium state, we need one more fact.
+equilibrium 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;
+further assertion that the entropy not only \ldquo{}tends\rdquo{} to increase;
-in fact it will increase, to the maximum value permitted E2 the
+in fact it /will/ increase, /to the maximum value permitted by the
-constraints imposed.* In the case of the carbon dioxide, these
+constraints imposed[fn::Note, however, that the second law has
-constraints are: fixed total energy (first law) , fixed total
+nothing to say about how rapidly this approach to equilibrium takes place.]/. In the case of the carbon dioxide, these
-amount of carbon dioxide , and fixed position of the piston . The
+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
+compatible with these constraints, and it can be predicted
-titatively from the strong form of the second law if we know,
+quantitatively 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
+dioxide (ie, heat capacity, equation of state, heat of vapor
-ization) .
+ization)
-To illus trate this , we set up the problem in a crude ap
-proximation which supposes that (l) in the range of conditions
+To illustrate this, we set up the problem in a crude
-of interest, the molar heat capacity CV of the vapor, and C2 of
+approximation which supposes that (l) in the range of conditions
-the liquid, and the molar heat of vaporization L, are all con
+of interest, the molar heat capacity $C_v$ of the vapor, and $C_\ell$ of
-stants, and the heat capacities of cylinder and piston are neg
+the liquid, and the molar heat of vaporization $L$, are all con
-ligible; (2) the liquid volume is always a small fraction of the
+stants, and the heat capacities of cylinder and piston are
-total V, so that changes in vapor volume may be neglected; (3) the
+negligible; (2) the liquid volume is always a small fraction of the
-vapor obeys the ideal gas equation of state PV = nRT. The in
+total $V$, so that changes in vapor volume may be neglected; (3) the
-ternal energy functions of liquid and vapor then have the form
+vapor obeys the ideal gas equation of state $PV = nRT$. The
-UPb = + A} (1-61)
+internal energy functions of liquid and vapor then have the form
-T T U = n‘ C '1‘ A + L] (1-62)
-v , v
+\begin{equation}
-where A is a constant which plays no role in the problem. The
+U_\ell = (N-n)\left[C_\ell\cdot T + A\right]
-appearance of L in (1-62) recognizes that the zero from which we
+\end{equation}
-*Note , however , that the second law has nothing to say about how rapidly this approach to equilibrium takes place.
+\begin{equation}
+U_v = n\left[C_v\cdot T + A + L\right]
+\end{equation}
+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
 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
+the energy $L$ necessary to form the vapor. On evaporation of $dn$
-moles of liquid, the total energy increment is (ill = + dUV= O,
+moles of liquid, the total energy increment is $dU = dU_\ell + dU_v =
-or
+0$; or
-[n CV [(CV — CQ)T + = O (l—63)
+\begin{equation}
+\left[n\cdot C_v + (N-n)C_\ell\right] dT + \left[(C_v-C_\ell)T + L\right]dn = 0
+\end{equation}
 which is the constraint imposed by the first law. As we found
-previously (l~59) , (1-60) the entropies of vapor and liquid are
+previously (1-59), (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
+\begin{equation}
-where AV, ASL are the constants of integration discussed in the
+S_v = n\left[C_v\cdot\ln{T} + R\cdot \ln{\left(V/n\right)} + A_v\right]
-Si
+\end{equation}
+\begin{equation}
+S_\ell = (N-n)\left[C_\ell\cdot \ln{T}+A_\ell\right]
+\end{equation}
+where $A_v$, $A_\ell$ are the constants of integration discussed in the
 last Section.
 We leave it as an exercise for the reader to complete the
-derivation from this point , and show that the total entropy
+derivation from this point, and show that the total entropy
-S = 82 + SV is maximized subject to the constraint (1-6 3) , when
+$S = S_\ell + S_v$ is maximized subject to the constraint (1-63), when
-R
+the values $n_{eq}$, $T_{eq}$ are related by
-the values 11 , T are related by
-eq eq
+\begin{equation}
+\frac{n_{eq}}{V}= B\cdot T_{eq}^a\cdot \exp{\left(-\frac{L}{RT_{eq}}\right)}
+\end{equation}
+where $B\equiv \exp{(-1-a-\frac{A_\ell-A_v}{R})}$ and $a\equiv
+(C_v-C_\ell)/R$ are constants.
 Equation (1-66) is recognized as an approximate form of the Vapor
-pressure formula .
+pressure formula
 We note that AQ, AV, which appeared first as integration
-constants for the entropy with no parti cular physical meaning ,
+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
+** The Second Law: Discussion
-because (with the exception of Boltzmann ' s original unsuccessful
-argument based on the H—theorem) , most attempts to deduce the
+We have emphasized the distinction between the weak and strong forms
-second law from statis tical mechanics have considered only the
+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 statistical mechanics have considered only the
 weak form; whereas it is evidently the strong form that leads
 to definite quantitative predictions, and is therefore needed
+for most applications. As we will see later, a demonstration of
+the weak form is today almost trivial---given the Hamiltonian form
+of the equations of motion, the weak form is a necessary
+condition for any experiment to be reproducible. But demonstration
+of the strong form is decidedly nontrivial; and we recognize from
+the start that the job of statistical mechanics is not complete
+until that demonstration is accomplished.
+As we have noted, there are many different forms of the
+seoond law, that have been favored by various authors. With
+regard to the entropy statement of the second law, we note the
+following. In the first place, it is a direct statement of
+experimental fact, verified in many thousands of quantitative
+measurements, /which have actually been performed/. This is worth a
+great deal in an age when theoretical physics tends to draw
+sweeping conclusions from the assumed outcomes of
+\ldquo{}thought-experiments.\rdqquo{} Secondly, it has stood the test
+of time; it is the entropy statement which remained valid in the case
+of negative spin temperatures, where some others failed. Thirdly, it
+is very easy to apply in practice, the weak form leading
+immediately to useful predictions as to which processes will go and
+which will not; the strong form giving quantitative predictions
+of the equilibrium state. At the present time, therefore, we
+cannot understand what motivates the unceasing attempts of many
+textbook authors to state the second law in new and more
+complicated ways.
+One of the most persistent of these attempts involves the
+use of [[http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Principle_of_Carath.C3.A9odory][Carath\eacute{}odory's principle]]. This states that, in the
+neighborhood of any thermodynamic state there are other states which
+cannot be reached by an adiabatic process. After some mathematical
+analysis
+[Margenau and Murphy (1943), pp. 26-31; or Wannier (1966),
+pp. 126-132]
+one infers the existence of a state function (entropy) which tends
+to increase; or at least, cannot decrease. From a /mathematical/
+standpoint there can be no objection at all to this; the analysis
+is quite rigorous. But from a /physical/ standpoint it is subject
+to the same objection that its premise is an impossibility
+statement, and therefore not an experimental fact.
+Indeed, the conclusion of Carath\eacute{}odory's
+argument is a far more direct statement of observed fact than its
+premise; and so it would seem more logical to use the argument
+backwards. Thus, from the experimental fact that the entropy
+tends to increase, we would infer that there must exist
+neighboring states inaccessible in an adiabatic process; but the
+result is then trivial. In a similar way, other impossibility
+statements follow trivially from the entropy statement of the
+second law.
+Finally, we note that all statements of the second law are
+subject to a very important qualification, not always sufficiently
+emphasized. As we stress repeatedly, conventional thermodynamics
+is a theory only of states of thermal equilibrium; such concepts
+as temperature and entropy are not even defined for others.
+Therefore, all the above statements of the second law must be under
+stood as describing only the /net result/ of processes /which begin
+and end in states of complete thermal equilibrium/. Classical
+thermodynamics has nothing to say about processes that do not
+meet this condition, or about intermediate states of processes
+that do. Again, it is nuclear magnetic resonance (NMR)
+experiments which provide the most striking evidence showing how
+essential this qualification is; the spin-echo experiment
+(Hahn, 1950) is, as we will see in detail later, a gross violation of
+any statement of the second law that fails to include it.
+This situation has some interesting consequences, in that
+impossibility statements may be misleading if we try to read too
+much into them. From classical thermodynamics alone, we cannot
+logically infer the impossibility of a \ldquo{}perpetual motion machine\rdquo{}
+of the second kind (i.e., a machine which converts heat energy
+into useful work without requiring any low temperature heat sink,
+as does the Carnot engine); we can infer only that such a machine
+cannot operate between equilibrium states. More specifically, if
+the machine operates by carrying out some cyclic process, then
+the states of (machine + environment) at the beginning and end
+of a cycle cannot be states of complete thermal equilibrium, as
+in the reversible Carnot engine. But no real machine operates
+between equilibrium states anyway. Without some further analysis
+involving statistical mechanics, we cannot be at all certain that
+a sufficiently clever inventor could not find a way to convert
+heat energy into useful work on a commercially profitable scale;
+the energy is there, and the only question is whether we could
+persuade it to \ldquo{}organize\rdquo{} itself enough to perform useful work
+against pistons, magnets, gravitational or electric fields,
+chemical activation energy hills, etc.
+It was Maxwell himself who first ([[../sources/Maxwell-Heat.pdf][1871]])[fn::See also, the [[http://openlibrary.org/books/OL7243600M/Theory_of_heat][Open Library
+page]], where you can read and download Maxwell's book in a variety of formats.] suggested such
+possibilities, in his invention of the \ldquo{}Maxwell Demon\rdquo{},
+an imaginary being (or mechanism) which can regulate valves so as to allow
+fast molecules to pass through a partition in one direction only,
+thus heating up one side at the expense of the other. We could
+then allow the heat to flow back from the hot side to the cold
+through a conventional Carnot engine, generating useful work; and
+the whole arrangement would constitute a perpetual motion machine
+of the second kind.
+#http://naca.larc.nasa.gov/search.jsp?R=19760010893&qs=Ns%3DLoaded-Date|1%26N%3D4294709597
+Maxwell did not regard such a device as impossible in principle;
+only very difficult technically. Later authors ([[../sources/Szilard.pdf][Szilard, 1929]];
+Brillouin, 1951, 1956)
+have argued, on the basis of quantum
+theory or connections between entropy and information, that it
+fundamentally impossible. However, all these arguments seem
+to contain just enough in the way of questionable assumptions or
+loopholes in the logic, as to leave the critical reader not quite
+convinced. This is particularly so when we recall the lessons
+of history; clever experimenters have, over and over again, made
+fools of theorists who were too quick to assert that something
+cannot be done.
+A recent example worth recalling concerns the Overhauser
+effect in magnetic resonance (enhancement of the polarization
+of one set of spins by irradiation of another set coupled to them).
+When this effect was first proposed, several well-known
+authorities on thermodynamics and statistical mechanics ridiculed the
+suggestion and asserted that the effect could not possibly exist,
+because it violated the second law of thermodynamics. This
+incident is a valuable reminder of how little we really understand
+the second law, or how to apply it in new situations.
+In this connection, there is a fascinating little gadget
+known as the Hilsch tube or Vortex tube, in which a jet of
+compressed air is injected into a pipe at right angles to its
+axis, but off center so that it sets up a rapid rotational
+motion of the gas. In some manner, this causes a separation of
+the fast and slow molecules, cold air collecting along the axis
+of the tube, and hot air at the walls. On one side of the jet,
+a diaphragm with a small hole at the center allows only the cold
+air to escape, the other side is left open so that the hot air
+can escape. The result is that when compressed air at room
+temperature is injected, one can obtain air from the hot side
+at +100^\circ F from the cold side at -70^\circ F, in sufficient quantities
+to be used for quick-freezing small objects, or for cooling
+photomultiplier tubes [for construction drawings and experi
+mental data, see Stong (1960); for a partial thermodynamic
+analysis, see Hilsch (19-47)].
+Of course, the air could also be cooled by adiabatic expansion
+(i.e., by doing work against a piston); and it appears that
+the amount of cooling achieved in vortex tubes is comparable to,
+but somewhat less than, what could be obtained this way for the
+same pressure drop. However, the operation of the vortex tube
+is manifestly not simple adiabatic since no work is
+done; rather, part of the gas is heated up, at the cost of cooling
+the rest; i.e., fast and slow molecules are separated spatially.
+There is, apparently, no violation of the laws of thermodynamics,
+since work must be supplied to compress the air; nevertheless,
+the device resembles the Maxwell Demon so much as to make one
+uncomfortable.. This is so particularly because of our
+embarrassing inability to explain in detail (i.e., in molecular terms)
+how such asimple device works. If we did understand it, would
+we be able to see still more exciting possibilities? No one
+knows.
+It is interesting to note in passing that such considerations
+were very much in Planck's mind also; in his [[http://books.google.com/books?id=kOjy3FQqXPQC&printsec=frontcover][/Treatise on Thermodynamics/]] (Planck, 1897; 116), he begins his discussion
+of the second law in these words (translation of A. Ogg):
+#+begin_quote
+\ldquo{}We
+$\ldots$ put forward the following proposition $\ldots$ :
+/it is impossible to construct an engine which will work a complete cycle,
+and produce no effect except the raising of a weight and the cooling of a heat-reservoir./ Such an engine could be used simultaneously
+as a motor and a refrigerator without any waste of energy or
+material, and would in any case be the most profitable engine
+ever made. It would, it is true, not be equivalent to perpetual
+motion, for it does not produce work from nothing, but from the
+heat which it draws from the reservoir. It would not, therefore,
+like perpetual motion, contradict the principle of energy, but
+would nevertheless possess for man the essential advantage of
+perpetual motion, the supply of work without cost; for the in
+exhaustible supply of heat in the earth, in the atmosphere, and
+in the sea, would, like the oxygen of the atmosphere, be at
+everybody ‘s immediate disposal. For this reason we take the
+above proposition as our starting point. Since we are to deduce
+the second law from it, we expect, at the same time, to make a
+most serviceable application of any natural phenomenon which may
+be discovered to deviate from the second law.\rdquo{}
+#+end_quote
+The ammonia maser ([[../sources/Townes-Maser.pdf][Townes, 1954]]) is another example of an
+experimental device which, at first glance, violates the second
+law by providing \ldquo{}useful work\rdquo{} in the form of coherent microwave
+radiation at the expense of thermal energy. The ammonia molecule
+has two energy levels separated by 24.8 GHz, with a large electric
+dipole moment matrix element connecting them. We cannot obtain
+radiation from ordinary ammonia gas because the lower state
+populations are slightly greater than the upper, as given by
+the usual Boltzmann factors. However, if we release ammonia gas
+slowly from a tank into a vacuum so that a well-collimated jet
+of gas is produced, we can separate the upper state molecules
+from the lower. In an electric field, there is a quadratic
+Stark effect, the levels \ldquo{}repelling\rdquo{} each other according to
+the well-known rule of second-order perturbation theory. Thus,
+the thermally excited upper-state molecules have their energy
+raised further by a strong field; and vice versa for the lower
+state molecules. If the field is inhomogeneous, the result is
+that upper-state molecules experience a force drawing them into
+regions of weak field; and lower-state molecules are deflected
+toward strong field regions. The effect is so large that, in a
+path length of about 15 cm, one can achieve an almost complete
+spatial separation. The upper-state molecules then pass through
+a small hole into a microwave cavity, where they give up their
+energy in the form of coherent radiation.
+Again, we have something very similar to a Maxwell Demon;
+for without performing any work (since no current flows to the
+electrodes producing the deflecting field) we have separated
+the high-energy molecules from the low-energy ones, and obtained
+useful work from the former. This, too, was held to be
+impossible by some theorists before the experiment succeeded!
+Later in this course, when we have learned how to formulate
+a general theory of irreversible processes, we will see that the
+second law can be extended to a new principle that tells us which
+nonequilibrium states can be reached, reproducibly, from others;
+and this will of course have a direct bearing on the question of
+perpetual motion machines of the second kind. However, the full
+implications of this generalized second law have not yet been
+worked out; our understanding has advanced just to the point
+where confident, dogmatic statements on either side now seem
+imprudent. For the present, therefore, we leave it as an open
+question whether such machines can or cannot be made.
 * COMMENT  Appendix
 | Generalized Force  | Generalized Displacement |

Mercurial > jaynes

comparison org/stat-mech.org @ 4:299a098a30da