April, 1962
CORBESPONDXKG STATESTIIEORS FOR ARGON AND XESON
583
CORRESPONDISG STATES THEORY FOR ARGOX ASTI XESON BY
F. DANONANI> KENNETH s. F’ITZER‘
Department of Chenzistry and Lawrence Radiation Laboratory, University qf Calafornza, Berkeley, Californza Reeaned September 18, 1961
Recent experimental results allow a more precise check than was previously possible on the Conformity of argon and xenon to the principle of corresponding states. The data show agreement within experimental error ex(-ept tlt very high pressures where small differences are found for some but not for other measurements.
Precise check on the theory of corresponding states for the rare gases until recently has been limited by the lack of experimental volumetric data for argon at low reduced temperatures. Consequently the rccerit volumetric measurements of Leveltz are of particular interest. Other relevant data also will be considered. Compressibility Factor.-Unfortunately the compressibility fact or comparisons of Levelt’s results* with values for xenon were made in ways which introduced unnecessary inaccuracies from auxiliary data. Consequently it seemed desirable to present the comparison of the compressibility factor data of Ar and Xe on the basis of reduced temperature and pressure. It was been pointed out3 that the critical temperature and pressure are readily measured with high accuracy, hence the selection of these reduced variables introduces littlc error. By contrast the critical volume frequently is much less accurately known than the volumetric data a t other temperatures and pressures. The critical properties used iii our calculations are
TABLE I COMPARISON OF COMPRESSIBILITY FACTORS FOR ARCONAND XENON
(1) Rice University, Houston, Texas. J. M. H . Levelt, Physica, 26, 361 (1960); Thesis, Amsterdam, 1958. (3) R. P. Curl, Jr., and K. S. Pitzer, I n d . E m . Chem.. 60, 265 (1958); “Thermodynamic and Transport Propertica of Fluids,” Inst. Mech. Engr., London, 1958, p. 1. ( 4 ) H. W. Habgood and W. G . Schneidcr, Can. J . Chem., 32, 98, 164 (1954).
(1951): A. hlichels, Hk. Wijkor, and IIerb. Wijker, Physaea, 16, 027 (1949). (0) K. S. Pitzer and R. ..1 Curl, Jr., J . A m . Chem. SOC.,79, 2309 (1957). (7) E. A. Guggenheim and M. I>. MeGIashan, 1 ’ 7 0 ~ . R o y . SOC. (London), A266, 456 (1900). (8)E. A. Guggenhoirn and M. L. McClashun. M o l . Phys., S, 563 (1960).
Tr Pr
71.05Ar Xe
0 9
0.665
1 5
.308
3 0
.465 .816
6.0 9.0
0.666 ,304 .453 809
-1.20Ar
XP
0 860
0 860 .615
613
521 809 1.095
532 791 1 066
7 1 . 4 5 - 7 Ar Xe
0 906 847 753 ,857 1.071
0.906 .847 .750 .842 1.046
... ... Second Virial Coefficient.-The second virial coefficients of argon aid xenon5 also were compared on the reduced basis B* = BP,/IZTC. The results are shown in Fig. 1 as differences from values calculated by the empirical second virial equation presented by F‘itzer and CurL6 While the xenon \-alucs ayerage a little ‘L ,eve those for argon a t the higher temperatures, the difference appears to be within the cxperirnental uncertainties. Indeed there is a much larger difference between the Ottawa and the Amsterdam measurements a t the lowest temperatures where no Ar Xe Ar-Xe differcnces are found in the data of either laboratory. 150.86 289.74 To,“K. While it was iiot our primary purpose to test 48.34 57.64 P,,atm. further the empirical equation of Pitzer and Curl, 0,2913 0.287 20 we may note that the differcnces from the empirical The values for argon are from Levelt’s work; equation remain below 0.005 except for the Ottawa those for xenon are from Habgood and S ~ h n e i d e r , ~measurements near thc critical temperature. Thus and are the same as Im-elt used. The substantial we find agreement in this respect also within the difference in the zc values is apparent a t once. A range of experimental error. comparison of the compressibility factor of argon Solid State Properties.-Further support for with that for xenon a t a variety of reduced tempera- conformity to Ihe principle of corresponding states tures and pressures is shown in Table I. These is obtained from a study of the solid state properties values were obtained by calculating the P, values of the noble gases. Guggenhcim and McClashan’ corresponding to Levelt’s tabulated values of z a t recently have Considered the interaction energy even I: and dr. These z values then were inter- between pairs of argon atoms using experimental polated to even P, values. data both on the equilibrium properties of the crysThe results in Table I show essentially perfect talline solid and on the gas phase. The potcntial conformity to the principle of corresponding states fiinction which they obtained then was used for the at densities substantially less than critical density. calculation of several thermodynamic properties, At higher densities, however, z for argon rises from e.g., entropy and energy of the crystal as a function 2 to 3y0 above that for xenon. These differences of temperature and pressure and the temperature are much smaller than those reported by Levclt on dependence of the second viriul coeficicnt. the basis of comparison in terms of reduced temI n a Inter publication* by the same authors, their perature and density where a difference of about resultson the interactionetiergy betwcenargon atoms 10% is found a t Tr = 1.05 and dr = 2.0. This ( 5 ) J. M. H. Levelt (ref. 1): E. Whalley, Y. Lupien, and W. G. point is near Tr = 1.05, P, = 6.0, where Table I Schneider, Can. J . Chem., Si, 723 (1953); 58, 033 (1955); J . .4. shows less than 1% difference. Besttie, R. J. Darriault, and J. S. Erirrly, J . Chem. I’hys., 19, 1222 (2)
F. DANON AND KENNETH S. PITZER
584
I
I
I
I
Argon
I5-
I
I
Xenon Awholley et 01. Beattie e t 01. v M i c h e l s et al.
o Whalley et 0 1 . a Levelt
a
Vol. GO
-
-
10A 0
-
5*@a
0 0 0
Q e
0
0-
o
bv.0v v
A
a
*
a A
A
A
o
A
1
I
0
I
I
2 .o
1.0
0
O
O
0
0
0
-5-
A
o
I
0
I
3.0
Tr. Fig. 1.-Reduced
lo!
- 0
i
second virial coefficients of argon and xenon compared as differences from the empirical equation of Pitzer and Curl.
I
0.00
0.70
Tr
0.90
I
.
Fig. 2.-Melting curve for argon and xenon compared in terms of reduced prqsure and temperature.
together with the principle of corresponding states are applied to the evaluation of thermodynamic properties of xenon. They obtained an agreement between calculated and measured values of the entropy and the enthalpy within the limits of the experimental error from the triple point temperature down to about 2OOK. For the molar volume the agreement is of the order of 1%. This larger deviation is, however, as these authors point out, not surprising considering that the equilibrium molar volume is determined by minimizing the free energy and a small inaccuracy in the shape of the potential curve would give rise to a much greater inaccuracy in the position of the minimum. The variation of the melting temperature of argon with pressure was measured by several investigators.9 There had not been, however, any similar study made for xenon, until very recently Stryland, et ~ 2 . reported ~ 1 ~ the results of their measurements. This makes possible a comparison of the melting curve for both substances, which is shown in reduced form, Pr = P / P o , Tr = T/T,, in Fig. 2. The curve through the single series of xenon points lies within the range of the several series of results for argon. Hence any deviation from corresponding states must be less than present experimental error. Since this comparison extends to 50 times the criti(9) D. W. Robinson, Proc. R o y . Soc. (London), A226, 393 (1954); P. W. Bridgman, P h y a . Rso., 46, 930 (1934); F. E. Simon, M. Ruhemann, and N. A. M. Edwards, Z . p h y m k . Chern., B6, 62, 331 (1930). (10) J. C. Stryland, J. E. Crawford, and M. A. Mastoor, Can. J . Pliya., 88, 1546 (1960).
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
