April, 1962
GENERALIZED CO~RDINATES AND FORCES
585
v = Eodr/ro) (1) cal pressure, it is a real test of the correspondence of the repulsive portions of the intermolecular where C$ is a universal function but Eo and ro are potential curves. energy and distance parameters for The application of the principle of corresponding characteristic substance. The London theory yields a genstates to solid inert gases at very low temperature, each sixth power dependence on r at long where quantum effects have to be considered, also eral inverse which is in accordance with the concept of has been made in two recent independent studies by distances function cp. Unfortunately, the theory Bernardes” and ZuckerI2; their results also support aof universal intermolecular interaction at short distances is the general validity of the principle. not precise enough to show whether this assumption Discussion and Conclusions of a universal function is exact or merely a good We note first that the corresponding states prin- approximation. ciple (as extended by the acentric factorls where apThe molecular theory also predicts pairwise addiplicable) still may be recommended as a reliable tivity of potential interactions only as an approxibasis for estimating volumetric data provided ac- mation. It is possible that differences between Ar curate critical temperature and pressure values are and Xe with respect to the importance of triple available, Since critical volume data usually are interaction and still higher terms may be significant. relatively inaccurate, comparisons on the basis of The differences in compressibility factor a t the reduced volume or reduced density should be highest pressures, which were noted in Table I, avoided. may arise from one or more of the sources just From the viewpoint of microscopic theory, exact noted. However, the data on the melting curve, conformity to the corresponding states principle which shows no significant deviation from corhas been shownl4 to follow for the heavier rare responding states, extend to even higher pressures gases if their intermolecular potentials are pairwise and densities than those on the fluid density. Hence additive and are given by an expression of the type further experimental work seems to be indicated before concluding that any deviation of Ar and Xe (11) N. Bernard=, Phys. Reo., 190, 807 (1960). (12) I. J. Zuoker, Proc. Phyu. SOC.(London), ‘77, 889 (1961). from corresponding states behavior exists. (13) K. 9. Pitzer, J. Am. Chem. Soc., 7 7 , 3427 (1955). For a oomplete discussion of the properties of fluids in terms of the acentric fa* tor see G. N. Lewis and hl. Randall, “Thermodynamlcs” revised by K. S. Pitser and L. 13rewer. 2nd Ed., MeGraw-1Iill Book Co., New York, N. Y.,1960. A m . 1, p. 605. (14) K. 9. Pitzer, J. Chem. Phys., 7 , 683 (1939).
Acknowledgment.-This research was carried out under the auspices of the U. S. Atomic Energy Commission. The aid of a Fulbright Travel Grant to one of us (F. D.) is gratefully acknowledged.
GENERALIZED COORDINATES AND FORCES BY OTTO REDLICFI* Shell Development Company,Emetyrille, California Received Septsmbsr 2.9, l B 8 1
The terms “generalized co6rdinate.P and “generalized forces” have replaced what early authors called “exteneitiea” or “capacities” and “intensities.” But the change in names has not yet been accompanied by the necessary clarification of the concepts. An explicit formulation of the meaning of these concepts is presented.
Classical or phenomenological or “pure” thermodynamics always has been praised as the model of a strict science. Yet an attentive observor is able to notice a certain uneasiness showing up time and again in its whole history. CarathCodory’s great achievements have not answered some of the most elementary questions. They can be used rather to illustrate the problem. Indeed, CarathBodory has lucidly shown how the concepts of energy and heat can be derived from the concept of work on the basis of the First Law. Work, of course, is, in modern language, the integral of a generalized force along the conjugate generalized coordinate. But what are these forces and coordinates? Everybody has learned to enumerate certain forces and coordinates. Rut this is not enough. The defect is quite obvious in teaching. No intelligent student can be satisfied by parroting a list of forces and coordinates. Yet no textbook of
*
Department of Chemical Engineering, University of California, Berkeley, California.
thermodynamics explains the meaning of these terms. If the student goes back to analytical mechanics, where the same words have been used before, he finds that a generalized force is defined as the negative derivative of a potential function with respect to the conjugate generalized coordinate. Obviously there is a serious circle definition involved since the potential function is a special kind of energy. Moreover, terms defined in the narrow field of mechanics are glibly and without any discussion applied in the whole field of physics and chemistry. This confusion is not only a didactic problem since textbook writers may justly plead that they cannot find an answer in the literature. Thermodynamics plainly needs some solid underpinnings. The often heard objection “Defining is a recurrent operation and you must stop somewhere” does not discharge us of the obligation to make clear in plain English what we mean by the words we are using, such as generalized force and generalized coordinate. We may say that a concept is explained
5 86
OTTO R EI)T,ICII
VoI. 66
if we have given siifficient instructions of how to basis of any orderly knowledge of physical nature, measure the quantity involved. R e need not be and at the vsme t,inie that these concepts imply at pedantic about this mattcr, yet we should he artic- every turn some idealization and even limitation, ulate enough to go beyond the enumeration of and that they therefore necessarily fall short of a examples. complete description of nature. Previous Discussions.-Helm’ saw quite clearly Concepts of such a general nature appear only in the need for introducing the general concepts which strictly classical thermodynamics, properly called nowadays are called generalized forces and co- thermostatics by Kohnstam. On the other hand, ordinates. He introduced the terms “Intensitaten” these concepts build the base for the whole field of and “Extensitaten” and tried to explain the mean- physics and chemistry, and the split into the various ing of these terms. Hut his discussion does not branches occurs only when we proceed from statics strike the significant points and is not w e n correct to kinetics arid dynamics, to the description of de(he states wrongly that conservation laws exist for pendent properties, and to molecular, atomic, and the extensities). His concepts (and still more Ost- nuclear theory. wa1d’s2 discussions) are SO vague that no specific Such a discussion of the basic concepts is believed objections need or can be raised to their terms. to be an indispensable precursor of any fundamental Quite a few modern authors, however, replaced or axiomatic theory. Helm’s terms by “intensive” and “extensive” propIsolation and Interaction.-Since we cannot deerties. These terms have been precisely defined scribe the whole universe in a single instant we are by Lewis and Randall. h simple consideration of bound to concentrate our attention to a part of it two galvanic cells shows that forces are not always that me are able to study by itself; that is, we intensive and coordinates not always extensive. examine, at first, an isolated object. By “object” If we switch the cells in series, the voltage is exten- we shall understand any part of the world which sive and the charge intensive though the voltage is we can isolate arbitrarily. The term “isolate” imalways the force and the charge always the co- plies an idealizing assumption, namely, that we are ordinate. Stress arid strain of two rods present the able to establish conditions under which any object same situation. But even if thc classification in- behaves always in thc same way whatever may tensive-extensive were correct, it would be neither happen in the rest of the world, i.e., its surroundcharacteristic nor sufficient : there are many other ings. What such conditions are we leave to the exintensive and extensive quantities! perimenter, ignoring the wistful smile of the man Carathkodory’s fanious paper3 did not change who has attempted to design such a simple thing as this situation. 12hrenfest4 knew this paper well a good calorimeter. But eyen so we notice at this when he wrote with admirablc frankness: “Kine first step that we are not exactly timid with our rnich vollig befriedigende Definition dieser Begriffe idealizing assumptions. The isolation of an atom or habe ich weder in der Litcratur finden konnen noch a galaxy is a quite far-reaching idea. Yet it is obauch selber zuwege gcbracht .” This discussion vious that we cannot dispense with it. Our ability was much later taken up by Planck6 and hIrs. to describe the world cannot go farther than our Ehrenfest-iifanassjc\ya and RIrs. de ITa:is-Lorentx.6 ability to isolate objects. Unquestionably the existence of the problem was When we introduce the term “interaction” we felt by the parlicipants in this discussion but a iniply that we can desigii devices which connect clarification has not been :tchicvcd. two otherwise isolated objects in such a manner In particular, the meaning of the term LLgeneral- that the behavior of one object depends on the state ized coordinate” has never been even approxi- of the other object. Such a device is, for instance, mately outlined. Obviously a coordinate is one a hook and eye which can be used for connecting a variable of a set of indcpendent variables used for weight and spiral spring hanging down from the ceilthe description of the state of an object. But why ing (Fig. 1). Another example is a pair of copper should the path, the surface area, the electric wires and a switch, connecting a storage cell and a eharge be coordinates while the specific resistivity capacitor. Another example is a glass tube and a or the resistance or the dielectric constant are not? stopcock connecting two bulbs containing some gas. Any of these properties can be a member of a set of We require also that these interaction devices are independent variables. so small that their size, shape, and nature have no The Method.-There is a way of building a sys- influence on the behavior of the two objects. We tem of concepts such as we necd: It is simply to leave to the experimenter the chore of realizing all describe in plain words and in a logical arrangement these assumptions, of checking his compliance with what we are really doing in thermodynamic rc- them, of correcting for all shortcomings as fnr search. The objective of this discussion will be as he isand able to do so, Gladly we let him assume reached if the reader will recognize, in a Kantian responsibility for uncorrected shortcomings, which sense, that this system of concepts is inevitable as a wc include in his “experimental errors.” (1) G. Helm. “Die Energetik,” Veit & Co., 1,eipzig. 1898, PP. 253Coordinates.-Establishing interaction between 274, especially p. 267. two otherwise isolated objects means then that a (2) W. Ostwald, 2. phyeok. Chem., 9, 563: 10, 303 (1892). Also condition is imposed upon two otherwise independ“Lehrbuch der allgemeinen Chemie,” 1890. ent properties 2’ and T” of the two objects. In ( 3 ) C. CarathBodory, Math. Ann., 67, 355 (1909). (4) P. Ehronfest, 2. phymk. Chem., 77, 237 (1911). illustrating this definition by an example we shall (5) M. Planck, A n n I’hysik, 19, 739 (1934): Physzca, 2, 1029 not in the least restrict its generality. As an ex(1936). ample we take the spiral spring (Fig. 1); its length (6) T. Ehrenfest-Afanassjewa and G. I,. de ITaas-Lorentz, ibid., 2, is called 5 ’ . A weight is hooked into t,he lower end 743 (lQ35).
GFSER.U,IZEDCOORDINATES
April, 1962
AYD
I~ORCES
of the spring; its height over the floor is x”. As long as the weight is not hooked in, the two variables x’ and x” are indepcndent. After interaction is established a condition is imposed on them, which in the particular example is XI
+-
XI1
= I€
587
x‘
(1)
The special form of this condition is unessential; it will be discussed however in the following section. I n the definition of coordinates x’ and x” the important point is that we have an actual device of imposing a condition upon the previously independent properties x’ and x”. We have such devices for different modes of interaction, for example, for the transfer of electric charges, for changing surface areas or volumes and other properties. Previous independence, however, is not enough to endow a property with the quality of a coordinate. I n the simple example of Fig. 1 we could take some other property, either some property trivially connected with x’ such as the number of coils per cm., or some less trivially connected property such as the electrical resistivity of the spring. Actually it would not matter which of these properties we call coordinate as long as we restrict ourselves entirely to the case of Pig. 1. But the arbitrariness disappears as soon as the range of interest is extended. The extension necessarily proceeds in two directions: The same object (weight) can interact by the same mode of interaction with different other objects (with another weight by means of a balance, with a gas by means of a piston). And the same object can interact simultaneously with other objects by different modes. The composite object “capacitor-weight” shown in Pig. 2 can interact with other objects in two different modes. I n order to forge a useful tool, RC have to restrict the concept of a coordinate to a variable that always is involved in a certain mode of interaction and never in any other mode. In other words, a sct of independent variables is a set of coordinates if for any mode of interaction all members of the set but one are constant. In the capacitor-balance of Fig. 2, for instance, we may arrest the balance and keep the balance position h constant. At the same time we may connect the two capacitor plates by thin wires with a storage cell and add or remove some electric chargc. Conversely we can isolate the capacitor and let the balance move. There are properties, suc’h as the voltage, which change in both modes of interaction. T o be sure, one can keep the voltage constant : hut in this case the object changes in both modes of interaction; the capacitor charge and the balance position change according to a specific relation to be imposed on them. A codrdinate, however, will be defined in such a manner that it is distinguished from ot,her properties by the exclusive connection with a singlc mode of interaction. Again the introduction of the concept of gencralized coordinates is a requirement, not an assumption or an axiom. This is demonstrated easily by the fact that we cannot always satisfy the requirement. Indeed, no coordinate exists for thermal interaction. Neither heat nor temperature is a coordinate. In fact, thermal interaction requires an entirely differcnt consideration, beyond the scope
-
i l Fig. 1.-Spring
and weight.
+I T
m
Fig. 2.--Capacitor
balance.
of the present paper. Thermal interaction thercfore is excluded from the whole of the following discussion. An earlier statement on coordinates can be reformulated in the following may: Each mode of interaction for which a conjugate coordinate exists influences an object, only through this coordinate. The behavior of an object therefore is determined solely by its own properties, and by its cnvironmerit only as far as interacting coordinates are influenced. Interaction Conditions.-It has been mentioned before that the special form of the interaction condition (1) is unessential. Actually we could replace any variable x in the prcseiit discussion by a monotonically increasing function y(x) such as ez or log 3 and so on. Wc shall make certain specific choices for convenience. While the general interaction conditioqwould be G(x’,z’’) = 0
(2)
we shall always choose our coordinates x’ and so that the interaction condition becomes dx’
+ dz”
=
0
2’’
( 3)
Wc did so in eq. 1. If x’ and 5” are extensive, eq. 3 can be interpreted :LS a conservation condition. Hut this intcrprd ation is unessential, particularly siiiw 1’and .I/’ need not be extensive. Equilibrium and Force.-As soon as interaction between two objects is established, they undergo, in general, a change. In sword with condition 3, 2’ may become grrater and 2’’ smaller, or conversely. The exceptional case that no change a t all occurs on establishing interaction is called cquilibrium. By observation we can establish a function F ( d , 2’’) s w h that the three conditions F>O;F=O;Ff”;f’=f”;I’
