Use the link below to share a full-text version of this article with your friends and colleagues. Learn more. Statistical thermodynamics of a system usually begins with the calculation of the partition function. If partition function can be obtained in a convenient analytic form, then all thermodynamic quantities can be calculated from it. This is the basic framework of equilibrium statistical thermodynamics. The chapter develops this formalism and presents illustrative applications.

The formalism of statistical thermodynamics can also be used to relate equilibrium constants of chemical reactions to partitions functions. Classical theories of solids could not explain the behavior at low temperatures. Only the use of quantum theory was able to explain why the molar heat capacity of a solid decreased when the temperature decreased. The chapter explains Einstein's theory of solids and Debye's theory of solids.

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Chapter Tools Request permission Export citation Add to favorites Track citation. This brings us to the same result as was derived before; the two arguments are equivalent. It is because of the element of choice in the labelling ofthe isotherms ofthe first fluid to be selected the thermometric body that the quantity 0 is referred to as the empirical temperature.

It is usual to choose as the thermometric body a fluid whose properties make a rational choice of o particularly simple. For example, in a mercury-in-glass thermo- meter there is effectively only one variable, the volume ofthe mercury, and 0 is taken to be a linear function of the volume. The particular straight line selected depends on the choice of scale; according to the Celsius scale, 0 is put equal to 0 at the temperature of melting ice, and at the temperature of water boiling at standard atmospheric pressure.

Two fixed points are sufficient to determine the linear rela- tion. Consider now the perfect gas scale of temperature. It happens that if the empirical scale is fixed by a mercury-in-glass thermometer, f O is very nearly a linear function over a wide range of temperature. Zeroth and fir8t laW8 13 For the purpose of the foregoing analysis we have considered only the simplest type of system, a fluid, whose state is definable by two parameters. The argument may easily be extended to more com- plicated systems, such as solids under the influence of more than simple hydrostatic stresses, or bodies acted upon by electric or mag- netic fields, in which cases more than two parameters must be specified in order to determine the state uniquely.

The only change involved by this extension is that instead of isothermal lines the body possesses isothermal surfaces in three or more dimensions, and the equation of state may be formally expressed! The existence of temperature may be proved in exactly the same way as before. It should be noted that our knowledge of temperature at this stage is insufficient to correlate the empirical temperature of a body with its hotness or coldness.

There is no reason why a body having a high value of J should necessarily be hotter in the subjective sense, or any other than one having a low value, since the choice of a temperature scale is entirely arbitrary. It is in fact possible, as in the perfect gas scale, to arrange that the' degree of hotness' of a body is a monotonic function of its temperature, but we cannot demonstrate this without first inquiring into the meaning of hotness and coldness, and finding a definition which is based on something less subjective than physio- logical sensation.

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This involves an investigation of the significance of the term heat; only when we have placed this concept on a secure experimental basis can we resolve objectively the relationship between temperature and hotness. Internal energy and heat In pursuance of our plan of developing thermodynamics as a pheno- menological science, we shall pass over any consideration of the molecular interpretation of heat, and the historical controversies between the followers of the caloric and the kinetic theories. The work of Rumford, Joule and innulnerable other experimenters has in truth firmly established the kinetic theory so firmly, indeed, that it is hard to realize that the matter was extremely controversial little more than a century ago , but although the work itself is of fundamental import- ance to the phenomenological aspect of thermodynamics, its molecular interpretation is entirely irrelevant.

A mass of water is enclosed in a calori- meter whose walls are made as nearly adiabatic as possible, and through these walls is inserted a spindle carrying paddles, so that mechanical work can be performed on the system consisting of water, calorimeter, paddles and spindle. A measured amount of ,vork, Ga, is done by applying a known couple, G, to the spindle and rotating it through a known angle, a. As a result of this work the temperature of the water is found to change.

The experiment is repeated with a different amount of water in the same calorimeter, and it can then be deduced, after corrections have been applied to allow for the walls being imperfectly adiabatic, what change of temperature a given isolated mass of water would suffer if a given amount of mechanical work were performed on it. An alternative experiment of the same nature may be carried out by replacing the paddle-wheels and spindle by a resistive coil of wire.

Work is then performed by passing a measured current, i, through the wire for a measured time, t. If the potential difference across the resistance is E, the work done is Eit in Joules if practical units are used, in ergs if absolute electromagnetic or electrostatic units. The observed outcome of this experiment is that the same temperature change is produced as in the paddle-wheel experiment by the performance of the same amount of work.

In many other similar experiments, using different kinds of mechanical work, the same result is obtained. It should be most particularly noted that in none of these experi- ments is any process carried out which can be legitimately called , adding heat to the system'. It would be a purely infei"ential, and phenomenologically quite unjustifiable, interpretation of the experi- ments to regard the mechanical work as transformed into heat, which then raises the temperature of the water.

So long as we take account only of what is observed, the deduction to be drawn from the experi- ments is one which may be stated in the following generalized form: If the state ofan otherwise isolated system is changed by the performance of work, the amount of work needed depends solely on the change accom- plished, and not on the means by which the work is performed, nor on the intermediate stages through which the system passes between its initial and final states.

This statement contains rather more than can be justified by the experimental evidence presented, particularly in its last clause. One might, for instance, change the state of an isolated mass of gas from that represented by the point A in fig. In the former the gas is expanded from A to C, and caused to do external work in the process, and is then changed to B at constant volume by means of work supplied, say, electrically.

In the latter the electrical work is performed first, to bring the state to D, and the work of expansion second. According to A the statement made above the total p work performed in each process should be the same. Unfortunately, it does not seem that experiments B of this kind have ever been carried out carefully.

This is historically C merely a consequence of the rapid and universal acceptance of the V first law of thernl0dynamics, and Fig. Different ways of achieving of the kinetic theory of heat, which the same change of state. Its manifold con- sequences, ho,,"ever, are so ,veIl verified in practice that it may be regarded as being established beyond any reasonable doubt.

As a consequence of the first law we may define an important property of a. This nleans that. Nevertheless, ho","ever roundabout the journey, a suitable path can ahvays be devised in principle for getting froln one given state to another, or vice versa. We have therefore shown the existence of a function ofstate, U, tnat is to say, a function which is determined apart from an additive constant Uo by the parameters defining the state of the system.

For any system, in an adiatkermal process, i. If the system is not so contained, it is found possible to effect a given change of state in different ways which involve different quantities of work. U with W is only correct under adiathermal conditions.

### Elements of Statistical Thermodynamics

For any change between given end- states aU can always be uniquely determined by carrying out an adiathermal experiment, for which 6. Such a manner of introducing and defining heat may appear some- what arbitrary, and in justification it is necessary to show that the quantity Q exhibits those properties that are habitually associated with heat.

In point of fact the properties concerned are not many in number, and may be summarized as follows: I The addition of heat to a body changes its state. Zeroth and first laws 17 It hardly requires proof that the quantity Q exhibits properties 1 and 2 ; from the known existence of diathermal walls it follows that changes of state may be produced not solely by the performance of work, and that the change ofstate is not necessarily inhibited by the intervention of solid or fluid barriers, or even of evacuated spaces.

That Q possesses property 3 is readily demonstrated by considera- tion of a typical calorimetric experiment, in which two bodies at different temperatures are brought into thermal contact within a vessel composed of adiabatic walls. We see then that Q possesses all the properties habitually associated with heat, and the use of the term is justified. Let us now return to the question of what is meant by the terms hotter and colder, which, as we have seen earlier, do not necessarily bear any relation to higher and lower temperatures on an empirical scale.

In the experiment just analysed, we found that the gain of heat by one body equalled the loss by the other, and this behaviour, although it does not necessarily imply that heat is a physical entity whose movement can be followed from one body to another, is called, for the sake of convenience, the transfer ofheat from one body to the other. The argument may readily be extended to include this effect, without modification of the conclusion.

The rate at which the transfer occurs may usually be varied over a wide range by altering the nature of the diathermal wall separating the bodies. The rate is a measure of the thermal conductance of the wall.

The body which loses heat Q negative is said to be hotter than that which gains heat Q positive , and the latter is said to be colder. It now remains to demonstrate that the scale of hotness, so defined, may be consistently linked with a scale of temperature, in the sense that all bodies at a temperature Jl shall be hotter than all bodies at a temperature 2' if l is greater than 2. To prove that this is possible we consider the consequences of presuming it to be impossible.

We may now vary the temperature of B slightly so that it is hotter than C while still colder than A. If the three bodies are then placed in thermal contact, heat will be transferred from A to B, from B to C, and from C to A; by adjustment of the thermal conduct- ances the rate of transfer may be made the same at each contact, and a state of dynamic equilibrium will be established.

But if any two of the bodies are taken away they will be found not to be in equilibrium, since all are at different temperatures, and this is in conflict with the converse to the zeroth law p. We conclude then that if a body at 1 is hotter than anyone body at 2 it is hotter than all bodies at 2. Therefore if we take a suitable thermometric substance and label its isotherms in such a way that successively hotter isotherms are ascribed successively higher empirical temperatures, we have achieved a corre- lation between temperature and hotness which is equally valid lfor all substances.

Henceforth we shall take the term' higher tempetature' to imply' of greater degree of hotness'. In the above argument it has been assumed that no performance of work accompanied the transfer of heat from one body to another. This is unnecessarily restrictive; all that is necessary is that such parameters as are needed, together with the temperature, to define the states of the bodies shall remain constant when the thermal con- tact is made.

The correlation of temperature and hotness then follows exactly as above. This has the important consequence that addition of heat to a body whose independent parameters of state, apart from the temperature, are maintained constant always causes a rise of temperature, and therefore the principal specific heatst of a body are always positive.

If the specific heat is me88ured under such conditions that the independent parameters of state other than the temperature are mantained constant, the me88ured property is one of the principal specific heats e. Reversibility and irreversibility In this chapter we shall discuss the significance and some applica- tions of the first law of thermodynamics:. However satisfactory this equation may be as an expression of a physical law, from an analytical point of view it leaves much to be desired.

There do not exist functions of state, of which Q and Ware finite differences, as may be illustrated by a simple example. The temperature of a vessel of water may be raised either by performing work as in Joule's experiments or by supplying heat; thus for any given change the values of Q and W may be altered at will, only their sum remaining constant.

All that can be said along these lines is that the internal energy U is well defined apart from a certain arbitrariness in fixing the zero of energy , and that U may be altered by means of work or of heat, the two contributions being usually subject to some measure of arbitrary variation, according to the method by which the change is effected.

Nevertheless, if certain restrictions are imposed on the way in which the change occurs, it is possible to treat q and w as well-behaved differential coefficients, dQ And d W, without becoming involved in mathematical absurdities. We must now inquire what are the restrictive conditions which validate this procedure. A more interesting state of affairs comes about, however, if neither q nor w vanishes, but w is expressible as a function of the parameters of state. This may conveniently be illustrated by reference to a simple fluid.

So that although dQ is a well-defined quantity for any infinitesimal change, it is not an exact differential, that is, the differential coefficient of a function of state. In order that we may equate w to - P d V in this case, two conditions must be satisfied, first that the change must be performed slowly, and secondly that there must be no friction. It is easy to understand the reason for these conditions if we imagine experiments in which they are violated.

Suppose the fluid to be a gas, and the piston to be with- drawn a small distance suddenly; then a rarefaction of the gas will be produced just inside the piston, and the work done by the gas will be less than if the pressure were uniform. An extreme example of this behaviour occurs in Joule's experiment, shown schematically iIi fig. Reversible changes 21 Gas is contained at uniform pressure within a vessel, of which one wall separates it from an evacuated chamber.

If this wall is pierced the gas expands to fill the whole space available, but it does so without performing any external work; w is zero even though Pd V does not vanish. If we are to equate w to - P d V it is essential to perform the expansion so slowly that the term -Pd V has a meaning. During the course of a rapid expansion the gas becomes non-uniform, so that there is no unique pressure P.

In fact the expansion must take place sufficiently slowly that the gas is changed from its initial to its final state by the process of passing through all intermediate states of equilibrium, so that at any time its state may be represented by a point on the in- Vacuum dicator diagram. Such a change is called quasistatic. In the experiment shown in fig. Strietly, of course, if the fluid is to remain in Fig.

Expansion of gas equilibrium at all times during a change into a vacuum. Usually in practice, however, a fluid requires verylittle time to equalize its pressure, and the change can 'be carried out quite quickly without provoking any significant departure from equilibrium. Consider now the effect of friction in the expansion or compression of a gas.

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When the gas is being compressed the pressure, P', to be exerted by the external agency must exceed the pressure, P, of the gas itself if there is friction between the cylinder and piston, and when the gas is being expanded P' must be less than P. In a cycle of compression and expansion, then, the relation between P' and V is as illustrated in fig.

Now in any infinitesimal change of volume the work done by the external agency is - P' d V, and clearly for a change between given initial and final states, the value of this quantity depends on the magnitude of the frictional forces and on the direction in which the change is carried out.

Indeed, during the course of the cycle, at the end of which the gas has been returned to its initial state, the work done by the external agency is - fP'd V, which is positive and equal to the area of the cycle in fig. We may say then that in this process work has been converted into heat through the effect of the frictional forces. By this we do not intend to imply any particular microscopic mechanism, although an analysis of the frictional process from an atomic point of vie"r would surely reveal the degradation of ordered mechanical energy into disorderly motion of the atoms composing the system.

From the point of view of classical thermodynamics this is merely a special example of a very common type of process which we call for convenience conversion of. V Fig. Effect of friction on P -V relation. All determinations of the mechanical equivalent of heat are examples of the conversion of work into heat in this sense. This, however, is somewhat of a digression from our main argument, which is that friction prevents the equating of w to - P d V and hence prevents both wand q in 3'1 from being treated as well-behaved differential coefficients.

This analysis of a special example should enable us to see the importance of that particular class of idealized processes which are designated reversible. In the same way it may be seen that compressions and expansions must be performed slowly, in order to avoid non-uniformities of pressure in the fluid which would prevent the process from being reversible in the sense defined above.

In fact, the condition of reversibility in a change is such as to ensure first that the system passes from one state to another without appre- ciably deviating from a state of equilibrium, and secondly that the external forces are uniquely related to the internal forces. It should also be noted, though the significance of this will not appear until later, that in order to transfer heat reversibly from one body to another the hotter of the two bodies should be only infinitesimally hotter, so that by an infinitesimal reduction of its temperature it may become the cooler, and the direction of heat transfer may be reversed.

Different types of work So far we have only considered in any detail the reversible perform- ance of work by means of pressure acting on a fluid. Let us now see how 3'3 may be extended so as to apply to situations in which other types of work are involved.

## [] Elements of sub-quantum thermodynamics: quantum motion as ballistic diffusion

There are a number of simple extensions which may be written down immediately, such as the work done by a tensile force in stretching a wire, or the work done in enlarging a film against the forces of surface tension. In the former case, if a wire is held in tension by a force! The corresponding result for a film having surface tension y and area. This condition is normally satisfied by y being practically independent ofd. There has been in the past a certain amount of confusion concerning the correct formulation of the expression for work per- formed by means of a magnetic field, but no difficulty need arise if we specify exactly what the experimental arrangement is which we are seeking to analyse.

This is, in any case, a highly proper attitude, for thermodynamics is above all a systematic formulation ofthe results of experiment, and we need feel no shame at keeping in mind a concrete physical example ofthe problem under consideration. Let us therefore suppose that we produce a magnetic field by means of a solenoid which is excited by a battery, as in fig. In practice the solenoid would normally possess electrical resistance, but as the result we shall obtain is found by more detailed analysis to be independent of the resistance, we may simplify the argument by taking it to be zero.

Then the function of the battery is to provide the work needed to create or change the field in the solenoid. It may be regarded as being capable of giving an e. The thermodynamic system is now taken to consist of everything within the enclosure marked by broken lines in fig.

Heat may be transmitted through the walls of the enclosure direct to the body we may neglect any thermal effects in the solenoid itself , and work may be done on the system by the battery. Consider first the empty solenoid fig. Jf'2d V. We shall not assume. Now consider the effect pro- duced by the body. Ifits magnetization changes for any reason, a back e. Jf'is varied by a change of i. The rate at which work is being done by the battery on the system is just this e. Reversible changes 25 In order to calculate the rate of working due to changes in the mag- netization ofthe body, we make use oftwo simple principles: first, that the law of superposition holds, so that each element of the body may be treated independently of the rest, and the total effect found by integration over the whole body; and secondly, that the back e.

We may therefore calculate the work done in creating an elementary dipole dm by use of any simple model dipole, and for our purpose a small current loop, as in fig. Suppose then. Empty Fig. Solenoid and Fig. Solenoid and solenoid.

## Elements Thermodynamics Heat Transfer

Calculation of work in reversible magnetization. If then i' changes, the e. This may also be written in the form. For this reason we must exclude ferromagnetic and other substances which exhibit hysteresis from our future discussion, just as we had to exclude plastic deformation in considering the stretching of a wire.

Considerable attention has been paid to the question of how to include hysteresis effects in the framework of classical thermo- dynamics, but it is doubtful whether any really satisfactory treatment of such problems can be given without a more detailed microscopic treatment, which falls outside our scope. Heine, Proc. Reversible changes The corresponding case ofelectric fields and dielectric bodies may be treated in a quite analogous manner, taking, for example, a condenser as the source of the electric field. Then it is found that, expressed in the electrostatic system of units,.

Nevertheless, we have the inexact differential dQ still not expressed in terms of functions of state. The problem of. With its aid we may do for dQ what we have in this chapter done for dW, and at the same time remove the restriction on the application of our results solely to reversible changes. The first law of thermodynamics expresses a generalization of the law of conservation of energy to include heat, and thereby imposes a formidable restriction on the changes of state which a system may undergo, only those being permitted which conserve energy. However, out of all conceivable changes ,vhich satisfy this law there are many which do not occur in practice.

We have already implicitly noted the fact that there is a certain tendency for changes to occur preferentially in one direction rather than for either direction to be equally probable. For example, we have taken it as a basic assumption, in accord with observation, that systems left to themselves tend towards a well- defined state of equilibrium.

It is not observed that a reversion to the original non-equilibrium state occurs; indeed, ifit did it would be very doubtful whether the term equilibrium would have any meaning. Again we have found it valuable to make a distinction between reversible and irreversible changes; the former are as readily accom- plished in the backward as in the forward direction, the latter have not this property. It is the irreversible change which should be regarded as the normal type of behaviour. In order that a change shall occur reversibly very stringent conditions must be imposed which clearly make it a limiting case of an irreversible change.

Strictly speaking the reversible change is an abstract idealization-all changes which occur in nature are more or less irreversible, and exhibit therefore a preferen- tial tendency. However, the idea that there is a preferred direction for a given change has been perhaps most clearly expressed in our discus- sion of the terms hotter and colder.

There is an unmistakable tendency for heat to flow from a body of higher temperature to one of lower temperature rather than for either direction of flow to occur spon- taneously. The second law of thermodynamics is little more than a generalization of these elementary observations.

In essence it states that there is no process devisable whereby the natural tendency of heat to flow from higher to lower temperatures may be systclna- tically reversed. It is impossible to devise an engine which, working in a cycle, shall produce no effect other than the transfer of heat froln a colder to a hotter body. Only by the use of a cyclical process can it be guaranteed that the process is exactly repeatable so that the amount of heat which can be transferred is unlimited. It is easy to devise non-cyclical processes which transfer heat from a colder to a hotter body.

Consider, for instance, a quantity of gas contained in a cylinder and in thermal contact with a cold body. The gas may be expanded to extract heat from the body; if it is then isolated and compressed it will become hotter, and may be brought into equilibrium with a hot body; further compression will enable heat to be transferred to the hot body. But at the end of this process the gas is not in its originalstate, and no violation ofthe second law has occurred yet, even though the total amount of work done by or on the gas may have been made to vanish.

Only if the gas can be brought back to its original state without undoing the heat transfer already effected can any violation ofthe second law be claimed. In fact, no violation can be brought about in this case, nor with any of the ingenious and often subtle engines which have been devised with the object of circum- venting the law. Moreover, the consequences of the law are so unfailingly verified by experiment that it has come to be regarded as among the most firmly established of all the laws of nature.

Kelvin's formulation of the second law is very similar to that of Clausius, with'its emphasis rather more on the practical engineering aspect of heat engines: It is impossible to devise an engine which, working in a cycle, shall produce no effect other than the extraction of heat from a reservoir and the performance of an equal amount of mechanical work.

Kelvin's law denies the possibility of constructing, for example, an engine which takes heat from the atmosphere and does useful work, while at the same time giving out liquid air as a 'waste product'. It is not difficult to show that a violation of Clausius's law enables Kelvin's law to be violated, and vice versa, so that the two laws are entirely equivalent; a proof of this assertion is left as an exercise for the reader.

A third formulation, due to Caratheodory, is not so clearly related: In the neighbourhood of any equilibrium state of a system there are states which are inaccessible by an adiathermal process. Each formulation has its enthusiastic supporters. For the engineer or the practically minded physicist Clausius's or Kelvin's formulations are more directly meaningful, and, moreover, the derivation from them of the important consequences of the law may be made without a great deal of mathematics.

A typically impossible process is the cooling of the water in Joule's paddle-wheel experiment; this is a very simple example since there is only one means whereby work may be per- formed, and the impossibility of cooling arises from the obvious fact that the paddle-wheel can only do work on the water and not extract energy. But Caratheodory asserts that however complex the system, however many the different forms of work involved, which may be both positive and negative, it is still true that not all changes may be accomplished adiathermally.

Caratheodory's law appears to differ in outlook from the others. The average physicist is prepared to take Clausius's and Kelvin's laws as reasonable generalizations of common experience, but Caratheodory's law at any rate in the author's opinion is not immediately acceptable except in the trivial cases, of which Joule's experiment is one; it is neither intuitively obvious nor supported by a mass of experimental evidence. It may be argued therefore that the further development of thermodynamics should not be made to rest on this basis, but that Caratheodory's law should be regarded, in view of the fact that it leads to the same conclusions as the others, as a statement of the minimal postulate which is needed in order to achieve the desired end.

It bears somewhat the same relation to the other statements as Hamilton's principle bears to Newton's laws of motion. In view of the wealth of discussion which has centred on the development of the consequences of the second law, no harm will arise from giving several different approaches, not always in complete detail. The reader may then select which he prefers, or, if none is to his taste, consult other texts for further varieties of what is basically the same argument.

For a simple fluid, takes the form. Thus represents a family of adiabatic lines covering the whole indicator diagram. As soon, however, as we go to more elaborate systems, having more than two independent parameters of state, the situation alters. This example will serve to show that we must not, in what follows, take for granted the existence of adiabatic surfaces, except in the special case of simple two-parameter fluids for which the existence of adiabatic lines has been proved.

As CarathOOdory pointed out, the assumption that adiabatic surfaces exist enables one of the most important consequences of the second law to be deduced purely mathematically. We shall therefore allow ourselves to consider adiabatic changes, but shall not assume that they constitute lines on well-defined adiabatic surfaces until, by means of the second law, we have proved the existence of such surfaces. For the first development of the consequences of the second law we shall start from Kelvin's formulation, and proceed along the conven- tionallines, making use of Oarnot cycle8.

Consider a system of any degree of complexity, and in particular consider two isotherms of the system which correspond to temperatures Jl and J2. These isotherms will be lines for simple fluids or 'surfaces' of two or more dimensions for more complex systems. Let us suppose that in the isothermal change AB the system takes in an amount of heat Q1 from a reservoir at Jl' and along OD an amount Q2 from another reservoir at J2.

We use a negative sign because Ql and Q2 are necessarily of opposite sign, as follows from Kelvin's law. By Kelvin's law this cannot be positive, Q Q' 0 4. Hence which proves the proposition. Further, we may readily show that! Ol' 2 is a universal function it is only necessary to prove the decomposition in anyone special case in order to establish its generality, and the special case which we shall choose is a simple fluid, for which isotherms and unique adiabatics may be dra""n, as in fig.

Let the heat absorbed in going from left to right along the three isotherms be Ql' Q2and Qa respectively. Isotherms and adiabatics of a simple fluid. The symbol K for Kelvin is used to designate the absolute scale of temperature. We shall defer a proof of this result until the analytical development of the second law has proceeded far enough to make it a very straightforward matter. Consider a system, u, which executes a cyclical process of any degree of complexity. In the course of traversing one element of this cycle work will be done in general on or by the system, and it will also be necessary to transfer heat to, or abstract heat from, the system.

It is convenient to imagine that each element of heat q is transferred to the system from a subsidiary body, u', at temperature T, which may be caused to execute Camot cycles between the temperature T and the fixed temperature, To, of a heat reservoir. The traversal of the element of the main cycle may then be considered as involving the following procedure: I u is in its initial state; u' is at temperature To.

Now the intemal energies of both I' and 1" are the same at the end as at the beginning of the cycle, so that this heat, if positive, is equal to the work done by I' and 1" during the course of the cycle. It follows, then, from Kelvin's law that since To is constant ,. If the cycle of u is reversible we may imagine the whole process carned out backwards, when for every element q in the original cycle we have now -q.

If, however, the cycle of J' is not reversible then we can only establish the inequality 4'14 , which is known as Olausius's inequality.

And in these circumstances we must be careful to interpret T as the temperature of the body which supplies the heat. This distinction will be of importance when we come to consider cyclical changes in which the system J' may not be in equilibrium at all times, and may not have a single definable temperature. We shall not at present discuss the consequences of Clausius's inequality, but consider first the important consequences arising from 4'15 , which is applicable to reversible processes only.

For if we consider two such paths we may traverse one in the opposite sense from the other and thus construct a reversible cycle. If then we introduce a function S, the entropy, by means of the definition. This result enables us to infer imme- diately that adiabatic surfaces exist for all systems, however complex. Let us now see how we can arrive at the same result by a more economical use of physical principles and a more lavish use of mathematics. We shall still employ Kelvin's formulation of the second law, but shall first establish the existence of adiabatic surfaces, without employing Carnot cycles, and thence deduce the existence of entropy and of an absolute scale of tempera- ture.

The argument is more readily visualized if we consider a system defined by three parameters, for which the isothermals are two- dimensional surfaces. Let us construct an adiabatic surface by the following procedure. Draw any adiabatic line, Le. To do this we first consider two points, P and P', lying in the surface and infinitesimally close to one another.

The surface we have constructed is thus entirely surrounded by adiabatically inaccessible points, and 80 is a unique adiabatic surface. The argument may readily be extended to systems of any number of parameters. Let us regard Yl as the dependent variable, and choose A such that. For this purpose consider acornposite system consisting oftwo separate systems in thermal equilibrium, at a common empirical temperature J, the first system having n coordinates Yl Yn' and the second m coordinates Zl Therefore since A is independent of the z-coordinates, A also must be independent of the z-coordinates; and since A' is independent of the y-coordinates, A also must be indepen- dent of the lI-coordinates.

Finally, let us see in outline how Caratheodory's law may be used instead of Kelvin's to establish the existence of adiabatic surfaces. According to this law there are in the neighbourhood of any state other states which are inaccessible by an adiathermal, and a fortiori by an adiabatic, process. By the word neighbourhood we need not imply that the inaccessible states are infinitesimally close, although that is the conclusion we shall reach eventually.

All we mean is that the inaccessible states are always close at hand, and do not consist of a few singular points or surfaces which are inaccessible from everywhere else. If then the nearest adiabatically inaccessible point Qfrom a given point P is a finite distance from it, we may assume that this holds in general wherever P may be situated. But this is clearly impossible. For if we join P and Q by a line L, there will be on L a point P' which, being nearer than Q, must be accessible from P but inaccessible from Q; since P' may be made as close as we choose to Q it follows that the nearest inaccessible point to P is at a finite distance from P, ,vhile the nearest inaccessible point to Q is infinitesimally close.

Hence we conclude that Caratheodory's law can be obeyed only if there are adiabatically inaccessible points infinitesimally close to any given point. We may now proceed as in the earlier analysis and, starting from any point P, draw an adiabatic line through P, an adiabatic surface through this line, and so on, according to the number of parameters involved. If we apply the same procedure to two points, Q and Q', infinitesimally close to P on opposite sides ofthe surface, and inaccessible from P, we may cordon off the surface through P by two inaccessible surfaces as close as we please, and thus demonstrate the uniqueness of the adiabatic surface constructed.

From this point onward the rest of the argument proceeds as before. We have only proved Clausius's inequality by one means; to construct a proof without invoking cyclical processes may be left as a not very easy exercise for the reader. For more careful discussions consult A. Eisenschitz, Sci. This result, derived by a consideration of reversible changes, states a relationship between functions which are all functions of state. This point is illustrated by the experiment pictured in fig.

We should find that it was just equal to Pd V, and that in consequence the entropy increase during the irreversible expansion was Pd VIT. We have now solved the problem which we set ourselves, of finding expressions for q and w in terms of functions of state, so that we should have only exact differentials to handle. It is desirable of course that the combination chosen should be dimension- ally homogeneous, and clearly this condition is satisfied by H, p" and G, which all have the dimensions of energy.

In this conncxion it is worth remarking on three points. First, there is nothing in the way temperature and entropy are introduced which ena bles us to determine their dimensions separately; all we can say is that the product TS has the same dimensions as U. We must expect therefore that S will never appear in any equation without T to render it dimensionally meaningful, unless the equation is in itself hOlllO- geneous in S so that its dilnensions are unimportant. It is clear that this proportionality is usually achieved by having only one member of each pair of variables size-dependent.

For example, if we consider a wire in tension we cannot ascribe the terms intensive or extensive uniquely, since the behaviour of the wire depends not simply on its volume but on its shape as well. We need not concern ourselves with such hair- splitting, however; in any practical application no ambiguity arises. Finally, we may note that if we have a composite system, consisting of several subsystems, each in equilibrium within itself, though, if thermally isolated, not necessarily in equilibrium with the others, the values of the extensive variables V, U and S for the whole system may be taken without inconsistency to be simply the sum of the contribu- tions from all the subsystems.

For V this is obvious; proof for the other variables, based on the definitions of LlU and LlS, is left to the reader. The proviso that each subsystem shall be in equilibrium is of course essential if the entropy is to have any meaning. This additivity rule applies also to the free energy F if all the subsystems have the same temperature, since.

Useful ideas Maxwell's thermodynamic relations Returning now to , we may regard U and all other parameters as functions of two independent variables, say y and z, and in general. This is particularly true t In view of the wide application of Maxwell's relations it is worth while committing them to memory, even though their derivation is 80 simple. The following remarks may help in remembering them.

First we may note that they are dimensionally homogeneous, in that cross-multiplication yields each time the pairs TS and PV the operator ais of course dimensionless. Secondly, the equations may always be written 80 as to exhibit the independent variables in the denominator. Maxwell's relations represent the four possible equations which satisfy these requirements, so that one can write them down immediately, complete except for the signs, of which two are positive and two negative.

The signs can easily be found by inspection if we consider the meaning of the equa- tions when they are applied to a perfect gas. To take M. The necessity for the negative sign is now clear. The reader should satisfy himself that he understands the meaning of the other three relations in the same way. Useful ideas 47 of M. This point should become clear as we proceed, in later chapters, to the applications of thermodynamics. A simple example of the use of Maxwell's relations is, however, conveniently introduced here, to justify an assertion made earlier that the perfect gas scale and the thermodynamic scale of temperature are identical.

We must first define what we mean by a perfect gas without reference to a scale of temperature, and for this purpose we make use of the following experimental facts: I As the pressure of a real gas tends to zero, the product PV at constant temperature tends to a finite limit Boyle's law.

The experimental evidence for the second statement is, first, Joule's observation that a gas expanding into a vacuum under isolated condi- tions experiences no change of temperature, and, secondly, more recent experimentst in which more precise observation has revealed a dependence of U on P which becomes negligible as the pressure is lowered.

We take the laws of Boyle and Joule to define a perfect gas, in confidence that a good approximation to a perfect gas can be found in practice, and that the behaviour of perfect gases may in principle be found experimentally by extrapolating the behaviour of real gases to zero pressure see p. The two laws are sufficient to enable us to deduce the equation of state of a perfect gas thermodynamically. Rossini and M. Frandsen, J.

By choosing the same fixed points the constant can be made to equal unity, and the scales of temperature coincide. This argument does not of course solve the practical problem of calibrating a thermometer according to the absolute scale; it indicates a method whereby it can be done. We shall not enter upon this problem at all here, but refer the interested reader to other treatises for more detailed accounts. By the use of gas liquefiers lower temperatures than this are readily attained, the normal boiling-point of liquid helium, for example, being 4.

A bath of liquid helium, contained in a Dewar vessel to minimize the leakage of heat from out- side, may be cooled further by pumping away the vapour and causing the liquid to boil at a reduced pressure. To reach still lower temperatures the technique of adiabatic demagnetization was devised p. These experi- ments in themselves suggest that, however low the temperature may be brought, there may be some limitation to all cooling methods which prevents the absolute zero, 0 0 K.

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It was once a rather widely held belief that the second law provided arguments against the possibility of attaining the absolute zero, and we shall now indicate the lines of thought involved, although as we shall see later they cannot be regarded as convincing. The most efficient way of lowering temperature is to employ an adiabatic process, such as expanding an isolated mass of gas reversibly, lowering the pressure over an isolated bath of liquid, or reducing the magnetic field applied to an isolated block of paramagnetic salt.

Henning, Temperaturmu8Ung Barth, For a given change of fJ the greatest cooling is achieved by making the change reversibly, as the reader should be able to prove from the second law. Now suppose that by such an adiabatic process the temperature of the system may be reduced to zero. It might appear that this provides the possibility of operating a Carnot engine between zero and a non- zero temperature, and that this would enable the second law to be violated, since such an engine need discharge no heat at the lo,ver operating temperature and can still do useful work.

It has been argued in this way that the second law requires that any reversible change taking place at the absolute zero shall involve no change in entropy; and this requirement is readily seen to preclude the attainment of the absolute zero. For Boltzmann's law of equipartition of energy would ensure that the specific heat remained at a non- vanishing value at all temperatures.

In other words, the entropy of all substances would tend to - 00 as T tended to zero, and no isentropic process could roduco T to zero. Thus the entropy tends to a finite limit and the question of the attainability of absolute zero is one which must be considered. However, op being a principal specific heat is always positive, as pointed out on p.

T Fig. Hypothetical branched adiabatic. Three criticisms may be directed against the foregoing argument. The first is that we have considered an idealized experiment, in that we have postulated perfect reversibility in the Camot cycle, and perfect isolation in the adiabatic parts. But the third criticism is the most cogent. Certainly this is true in the sense already discussed, that an isothermal change at zero temperature involves no transfer of heat.

But this very circumstance makes the cycle inoperative; for there is no practical way of compelling the isolated system to perform the isothermal change rather than an adiabatic isentropic change. In view of this failure we must, if we are to incorporate the idea in the development of thermodynamics, introduce it as a new postulate, the third law of thermodynamics: By no finite aeries of prOCU8es i8 the absolute zero attainable.

This enables us to give an alternative statement to the third law: A8 the temperature tentlB to zero, tAe magnitude of the e'IIJropy change in any reversible proces8 tentlB to zero. We shall meet, in the following chapters, a few examples which demonstrate the application of the third law, but it has not the same importance in physics as it has in chemistry, where it plays a most valuable role in enabling the equilibrium constants of chemical reac- tions to be calculated from the thermal properties of the reactants.

The success which attends its application in this field leaves no room for doubt of its correctness. In view, however, of the limited use we shall make of the law, and the need for a full development of chemical thermodynamics in order to appreciate its application, we shall not discuss it any further. From the unattainability of the absolute zero and the experimental fact which is a consequence of the validity of classical mechanics in describing the atomic behaviour of matter at high temperatures that the specific heat of a complete system does not tend to zero as T tends to infinity, it follows that the temperatures of all bodies have the same Iign, which is positive by definition.

The specific heat of such a subsystem tends to zero at high temperatures sufficiently rapidly for only a finite amount ofenergy to be needed to raise the temperature to infinity. The thermal conductance between the nuclear subsystem and the rest of the lattice is so low that this may be achieved while the lattice is maintained at ordinary temperatures. It is even possible to add still more energy to the subsystem, and this is equivalent to forcing its temperature into the negative region. This is because from the point of view of statistical thermodynamics it is liT, rather than T, which is the physically significant parameter, and there is no energetic barrier in these par- ticular experiments preventing passage through the origin of liT.

It is legitimate to inquire whether the second law may be violated by working a Carnot engine between two temperature baths of which one is negative and the other positive. It may be shown that no violation is possible, since no isentropic surfaces connect positive and negative temperatures, and therefore no reversible cycle may be constructed. In the experiment the passage from positive to negative temperature was effected by an ingenious trick which does not correspond to any ordinary reversible process.

This experiment has been mentioned solely in order to point out that there are exceptions to the rule that temperatures are positive.

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But these exceptions never occur with complete systems in equili- brium, only for very special isolable subsystems, and for normal purposes we take T to be always a positive quantity. The experi- ment is described and discussed by N. Ramsey, Phys. An elementary graphical method of solving thermodynamic problems We shall now examine briefly a graphical method which is some- times used to solve elementary thermodynamic problems.

The sub- stance considered is imagined taken around a Carnot cycle between two neighbouring temperatures T and T - oT.