摘要:

DISCUSSIONS OF THE FARADAY SOCIETY No. 22, 1956 THE PHYSICAL CHEMISTRY OF PROCESSES AT HIGH PRESSURES THE FARADAY SOCIETY Agents for the Society’s Publications : The Aberdeen University Press Ltd. 6 Upper Kirkgate AberdeenThe Faraday Society reserves the copyright of all Communications published in the " Discussions " PUBLISHED . . 1957 PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS A B E R D E E NA GENERAL DISCUSSION ON THE PHYSICAL CHEMISTRY OF PRQCESSES AT HIGH PRESSURES 20 -21st SEPTEMBER, 1956 A GENERAL DISCUSSION on the Physical Chemistry of Processes at High Pressures was held in the Chemistry Department of the University of Glasgow (by kind permission of the Vice-chancellor) on the 20th and 21st September, 1956. The President, Mr. R. P. Bell, F.R.S., was in the Chair.Among the distinguished members and guests welcomed by the President were the following : Dr. B. J. Alder (California), Mr. R. Bergeon (France), Dr. Rolf Buchdahl (Springfield, Mass.), Dr. J. A. Christiansen (Copenhagen), Prof. and Mrs. M. A. Cook (Utah), Dr. L. Deffet (Brussels), Dr. Russell E. Duff (New Mexico), Mr. V. H. Fraenckel (Schenectady), Mr. L. Galatry (France), Mr. F. C. Gibson (Pittsburgh), Dr. E. F. Greene (Brown University, Pro- vidence), Mr. M. s. de Groot (Amsterdam), Mr. J. Jeener (Brussels), Dr. C. G. Jennergren (Stockholm), Dr., Mrs. and Miss J. V. R. Kaufman (New Jersey), Dr. A. I. M. Keulemans (Amsterdam), Dr. and Mrs. J. Kieffer (France), Mr. H. de Kluiver (Amsterdam), Mr. R. Kolinski (Poland), Prof. K. J. Laidler (Ottawa), Mr. H.Lehmann (Leuna), Dr. E. W. Lindeijer (Holland), Mr. L. Lundberg (Stockholm), Mr. L. McKenzie (Washington), Prof. H. Margenau (Yale University, Conn.), Dr. V. Mathot !(Brussels), Mr. H. K. Nason (St. Louis), Mr. I. Raitzyn (France), Dr. M. W. Rigg (Pasadena), Dr. and Mrs. N. C. Robertson (Cambridge, Mass.), Mr. J. Robin (France), Prof. and Mrs. Runge (Germany), Mr. W. Schuurman (Amsterdam), Dr. F. Sebba (Cape Town), Dr. T. Slebodzinski (Warszawa), Dr. J. R. Sutton Wisconsin), Dr. G. Szasz (Zurich), Dr. G. Theilig (Frankfurt Main), Prof. Orof T. Urbanski (Warszawa), Dr. and Mrs. G. Vidale (Philadelphia), Dr. B. Vodar (France), Dr. J. H. van der Waals (Amsterdam), Dr. G. Wetroff (France), Dr. and Mrs. Allan Wetterholm (Sweden), Dr. E. Whalley (Ottawa), Dr.F. Wolf (Universitat Halle), Mr. and Mrs. H. S. Young (Delaware), Mr. A. P. Zeelenberg (Amsterdam), Dr. Walter G. Zinman (Maryland). 3A GENERAL DISCUSSION ON THE PHYSICAL CHEMISTRY OF PRQCESSES AT HIGH PRESSURES 20 -21st SEPTEMBER, 1956 A GENERAL DISCUSSION on the Physical Chemistry of Processes at High Pressures was held in the Chemistry Department of the University of Glasgow (by kind permission of the Vice-chancellor) on the 20th and 21st September, 1956. The President, Mr. R. P. Bell, F.R.S., was in the Chair. Among the distinguished members and guests welcomed by the President were the following : Dr. B. J. Alder (California), Mr. R. Bergeon (France), Dr. Rolf Buchdahl (Springfield, Mass.), Dr. J. A. Christiansen (Copenhagen), Prof. and Mrs. M. A. Cook (Utah), Dr.L. Deffet (Brussels), Dr. Russell E. Duff (New Mexico), Mr. V. H. Fraenckel (Schenectady), Mr. L. Galatry (France), Mr. F. C. Gibson (Pittsburgh), Dr. E. F. Greene (Brown University, Pro- vidence), Mr. M. s. de Groot (Amsterdam), Mr. J. Jeener (Brussels), Dr. C. G. Jennergren (Stockholm), Dr., Mrs. and Miss J. V. R. Kaufman (New Jersey), Dr. A. I. M. Keulemans (Amsterdam), Dr. and Mrs. J. Kieffer (France), Mr. H. de Kluiver (Amsterdam), Mr. R. Kolinski (Poland), Prof. K. J. Laidler (Ottawa), Mr. H. Lehmann (Leuna), Dr. E. W. Lindeijer (Holland), Mr. L. Lundberg (Stockholm), Mr. L. McKenzie (Washington), Prof. H. Margenau (Yale University, Conn.), Dr. V. Mathot !(Brussels), Mr. H. K. Nason (St. Louis), Mr. I. Raitzyn (France), Dr. M. W. Rigg (Pasadena), Dr.and Mrs. N. C. Robertson (Cambridge, Mass.), Mr. J. Robin (France), Prof. and Mrs. Runge (Germany), Mr. W. Schuurman (Amsterdam), Dr. F. Sebba (Cape Town), Dr. T. Slebodzinski (Warszawa), Dr. J. R. Sutton Wisconsin), Dr. G. Szasz (Zurich), Dr. G. Theilig (Frankfurt Main), Prof. Orof T. Urbanski (Warszawa), Dr. and Mrs. G. Vidale (Philadelphia), Dr. B. Vodar (France), Dr. J. H. van der Waals (Amsterdam), Dr. G. Wetroff (France), Dr. and Mrs. Allan Wetterholm (Sweden), Dr. E. Whalley (Ottawa), Dr. F. Wolf (Universitat Halle), Mr. and Mrs. H. S. Young (Delaware), Mr. A. P. Zeelenberg (Amsterdam), Dr. Walter G. Zinman (Maryland). 3CONTENTS PAGE GENERAL INTRODUCTION. By A. R. Ubbelohde . . 7 A. EQUATIONS OF STATE, PHYSICAL PROPERTIES AND THERMODYNAMIC TRANSFORMATION- Intermolecular Repulsive Forces.By T. L. Cottrell . * 10 Buckingham and J. A. Pople . . 17 Electromagnetic Properties of Compressed Gases. By A. D. Infra-red Spectra of Gaseous Mixtures under High Pressure. By R. Coulon, L. Galatry, J. Robin and B. Vodar . . 22 Perturbation of Electronic Resonance Spectra by Foreign Gases under High Pressure. By J. Robin, R. Bergeon, L. Galatry and B. Vodar The Effect of Pressure on the Vibrational Frequency of Bonds con- taining Hydrogen. By A. M. Benson and H. G. Drickamer . 39 Pressure-Induced Metallic Transitions in Insulators. By B. J. Alder and R. H. Christian . 4 4 30 Transport of Energy and Momentum in a Dense Fluid of Hard Spheres. By H. C. Longuet-Higgins and J. P. Valleau . . 47 The Thermal Conductivity of Associating Gases.By E. Whalley . 54 Equation of State and Thermal Conductivity of Gases at High Pressures and Elevated Temperatures. By J. Saurel, R. Bergeon, P. Johannin, J. Dapoigny, J. Kieffer and B. Vodar . . . 64 Ionization of Piperidine in Methanol to 12,000 atm. By S. D. Hamann and W. Strauss . . 70 GENERAL DIscussxoN.-Dr. R. E. Duff, Dr. R. A. Bucleingham, Prof. A. R. Ubbelohde, Dr. S. D. Hamann, Dr. K. M. Guggenheimer, Dr. J. S. Rowlinson, Dr. B. Vodar, Mr. R. P. Bell, Dr. L. Galatry, Mr. H. de Kluiver, Dr. Berni J. Alder, Dr. E. V. Franck, Dr. E. Whalley, Dr. J. Jeener and Dr. M. Lambert, Dr. L. E. Sutton . 75 B. CHEMICAL REACTIONS AND TRANSFORMATIONS AT HIGH PRESSURES- The Role of the Solvent in Chemical Reactions, as Revealed by High The Decomposition of Benzoyl Peroxide in Solution at High Pressures.Polymerization of Styrene at High Pressures using the Sector Technique. Pressure Studies. By K. J. Laidler . . 88 By A. E. Nicholson and R. G. W. Norrish . . . 97 By A. E. Nicholson and R. G. W. Norrish . . . 104 56 CONTENTS PAGE The Kinetics of Some Organic Reactions under Pressure. By S. D. High Pressures and Steric Hindrance in Liquid-Phase Reactions. By Hamann and D. R. Teplitzky . . 114 K. E. Weale . . 122 The Velocity of Ethylene Polymerization at High Pressures. By Liquid-phase Free-radical Dissociations at High Pressure. By A. H. R. K. Laird, A. G. Morrell and L. Seed . . 126 Ewald . , 138 GENERALDI~c~ss~~N.-P~~~. J. K. Laidler, Dr. K. E. Weale, Dr. S. D. Harnann, Prof. G. M. Burnett, Dr. A.H. Ewald, Dr. R. K. Laird, Prof. A. R. Ubbelohde, Dr. S. K. Lachowicz, Prof. A. Runge, Dr. R. Buchdahl, Mr. A. G. Morrell, Mr. R. P. Bell, Dr. E. Whalley . . 144 C. DETONATION AND OTHER HIGH TEMPERATURE PI-IENOWNA AT HIGH PRESSURES- Introduction. By S. Paterson . . The Velocity of Detonation in Compressed Acetylene. By E. Penny . The Rate of Formation of Carbon from the Pyrolysis of Acetylene in Shock Waves. By C. F. Aten and E. F. Greene. A Study of Detonation Waves by X-ray Flashes. By L. Deffet and J. . Boucart . The Reaction Mechanism in the Detonation of Solid High Explosives. By W. Taylor . The Influence of High Pressure on Thermal Explosion and the Decom- position and Detonation of Single Crystals. By F. P. Bowden, B. L. Evans, A. D. Yoffe, and A. M. Yuill . Hugoniot Curves for Inert Solids and Condensed Explosives. By H. Lawton and I. C. Skidmore . The Effect of Inert Components in the Detonation of Gelatinous Explosives. By P. B. Dempster . . Compressibilities of Solids and the Influence of Inert Additives on Detonation Velocity in Solid Explosives. By M. A. Cook . GENERAL D1scussroN.-Dr. R. E. Duff, Dr. H. A. Mayes, Dr. W. G. Zinman, Dr. B. Vodar, Dr. A. R. Fraser, Dr. J. Kieffer, Dr. S. Paterson, Dr. E. F. Greene, Dr. W. E. Soper, Mr. T. L. Cottrell, Mr. E. G. Whitbread, Dr. L. Deffet, Dr. F. P. Bowden, Mr. B. L. Evans, Dr. A. D. Yoffe, Dr. A. M. Yuill, Mr. R. A. W. Hill, Dr. E. W. Lindeijer, Mr. 3. D. Huffington. . 155 157 162 167 172 182 188 196 203 212 Author Index. . . 226

ISSN:0366-9033    

DOI:10.1039/DF9562200001

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

THE PHYSICAL CHEMISTRY OF PROCESSES AT HIGH PRESSURES GENERAL INTRODUCTION BY A. R. UBBELOHDE Dept. of Chemical Engineering, Imperial College, S.W. 7 For condensed states of matter, the pressure variable has been much less fully explored than the temperature variable. However, high-pressure techniques are becoming easier'to develop, at any rate up to about 5000 atm. Increased acces- sibility of operations at high pressures has resulted in a broadening of the range of physico-chemical measurements available for interpretation. For still higher static pressures equipment is at present too specialised and costly to favour its widespread dissemination in many laboratories, but results important for the thermodynamics of condensed phases are being obtained in a few centres. The use of transient high pressures up to 105 atm or more, as developed in shock and detonation waves in solids, was first applied to the measurement of thermodynamic parameters during the war.Increased attention is being paid to the development of such techniques at the present time, though there are still serious difficulties of interpretation particularly in systems containing a mixture of solids. Industrial interest in high pressures adds to the timeliness of the present Discussion ; the highly successful polymerizatiom of ethylene and the suggestive results in the polymerization of acetylene are but two examples of possibilities of applying high pressures industrially. Systematic theory has not wholly kept pacewith all these developments, but some of the papers in the present Discussion clarify important theoretical problems.The widespread influence of repulsive forces in almost all high-pressure phenomena makes the survey by Cottrell particularly timely. This contribution brings out the difficulties in finding high-pressure parameters that are at the same time simple to calculate theoretically and are experimentally sensitive to repulsive force fields. It seems quite definite that for close molecular overlap a simple inverse power function of molecular separations fails to represent repulsions successfully ; an exponential function may be more hopeful. However, the anisotropy of repulsive force fields, and additional complications where repulsions play a part in a many-body problem, have hitherto blocked theoretical progress. One promise of progress in theory stems from the much greater diversity of experimental phenomena where measure- ments are becoming available from which repulsive force fields might in principle be calculated, as is illustrated by many of the experimental contributions to this Discussion.Another broad group of theoretical problems arises from the interpretation of the transport properties of dense gases ; these are of very considerable practical interest. The outcome of many statistical-mechanical calculations which can be formulated is much too complicated to be applied conveniently to experimental examples. For this reason the devising and testing of simple models which permit some theoretical insight into transport processes in dense gases offers special opportunities for progress at the present time.In this direction, there are significant contributions to the present Discussion, by Longuet-Higgins and Valleau, and by Whalley. 78 GENERAL INTRODUCTION A third type of theoretical calculation refers to properties of compressed gases whose variation with pressure can be attributed to attractive forces. Measurement of such properties as a function of pressure, and particularly a combination of several types of measurements, promises to give information about details of inter- molecular attractions which it would be difficult to obtain in other ways. The contribution by Pople and Buckingham permits a useful extension of the range of properties whose dependence on pressure can be made to yield information about molecular force fields. Advances in techniques of working at high pressures are not specifically included in this Discussion, but the increasing range of phenomena which can now be studied conveniently is illustrated by the diversity of experimental studies presented.Thus effects of pressure on the infra-red and electronic resonance spectra of gases studied by Dr. Vodar and his colleagues contribute novel additions to our general knowledge of molecular force fields. Benson and Drickamer make the interesting suggestion that the pressure coefficient of a vibrational frequency should be em- ployed systematically to supplement the actual numerical value of the frequency itself, in correlating spectral energy levels with molecular structure. When chemical reactivity at higher pressures is considered, it is usual to separate the influence of pressure on the structure of the solvent, and its influence on various activation complexes.From observed changes in the dielectric constant of metha- nol, discussed by Hamann and Strauss, there appears to be a striking influence of pressure on the structure of this solvent. Presumably this is connected with the open net-work structure of hydrogen bonds in liquid methanol. It would be interesting and important to know if such an influence is observed generally in hydrogen-bonded solvents. So far as the activation complex is concerned, relatively simple concepts as discussed by Laidler suffice to account for much of the observed influence of pressure on the free energy of activation.The influence of pressure on reaction rate is quite striking for certain fairly complex molecules as in the cases reported by Hamann and Teplitzky where a 500-fold increase in rate accompanies the increase in pressure up to 15,OOO atm. However, expectations that the steric hindrance in specific reaction mechanisms might be reduced or even eliminated as a result of bond deformation at high pressures proved not to be fuKlled, at any rate in the examples studied by Weale. Other striking influences of increases of pressure arise in polymerization reactions. In the polymerization of styrene, rates of propagation and of chain termination have been shown by Nicholson and Norrish to depend in a marked way on the degree of compression of the solvent. In consequence the mean molecular weight is affected by pressure. It would be interesting to determine how far chain branching for a given size of macromolecule is affected by the pressure at which it has been formed.In the polymerization of ethylene studied by Laird, Morrell and Seed the reaction rate likewise depends in a complex way on the pressure, owing to the various ways in which nearest neighbours influence chain reactions as the packing becomes closer. The group of investigations on physico-chemical process in detonation and shock waves, presented for this Discussion, deals with systems of widely differing com- plexity. In heterogeneous polycrystalline mixtures, a complex diversity of pheno- mena can occur which is by no means fully cleared up. This applies, for example, to the studies reported by Deffet and Boucart, by Dempster, and by Taylor.The mixtures of explosive substances they describe can be termed " solid " only by convention; phenomena such as grain erosion in the reaction zone are of dominant importance and depend on the intercrystalline free space in a complex way. When single crystals are studied individually, as in the work reported by Bowden and his colleagues, high initial pressures are found to have only a slight effect on the " runaway temperature " in the self-heating of expolsives. Experiments on the homogeneous detonation of acetylene should in principle be simpler to interpret ; some interesting experimental abnormalities are presented by Penny. The kinetics of pyrolysis of acetylene in shock waves to give carbon asA . R. UBBELOHDE 9 reported by Atin, Tonnies and Greene offers a further instance of reactions at transient high pressures, which presents complications due to the formation of solid products of variable physical structure and chemical composition. Extensions of such kinetics studies promise to cover ranges of high pressure and high tempera- ture not conveniently obtainable in other ways. However, as in other studies of transient phenomena, relaxation effects may complicate any interpretation of the results on the basis of the thermodynamics of high pressure.

ISSN:0366-9033    

DOI:10.1039/DF9562200007

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

A. EQUATIONS OF STATE, PHYSICAL PROPERTLES AND THERMODYNAMIC TRANSFORMATION INTERMOLECULAR REPWIVE FORCES BY T. L. COTTRELL Imperial Chemical Industries, Ltd., Nobel Division, Research Dept. , Stevenston, Ayrshire Received 2nd July, 1956 The theoretical and experimental evidence on intermolecular repulsive forces is summarized and discussed. It is shown that at close range an inverse power law is in- adequate to represent the repulsive potential, whereas an exponential fits the results tolerably well. Information about the equilibrium properties of gases at ordinary tem- peratures and pressures is almost irrelevant to the discussion of intermolecular forces and the discussion of matter at high density is obscured by the need to take into account many-body forces, so that the study of high-energy elastic collisions gives most information about the repulsive potential.Valuable empirical information may, however, be obtained from the study of matter at very high pressure, using new techniques of measurement. Molecular interactions, in the absence of chemical reactions, are usually con- sidered in terms of relatively long-range attractive forces and of short-range repulsive forces. Under normal conditions in gases the attractive forces are of most importance in determining the equilibrium properties, but at high pressures and temperatures repulsive forces become important : indeed, even at quite low temperatures and pressures up to six or seven hundred Amagat it is possible to treat the equation of state of nitrogen and argon by considering attractive forces as perturbations on the repulsive forces.1 This paper reviews some of our know- ledge of these repulsive forces.There are two theoretical ways of discussing intermolecular forces : the direct calculation of the potential between a pair of molecules, and the calculation of the energy of a molecular model when constraints, chosen to represent the average effect of the environment, are applied. The results are discussed in the following sections, and thereafter the experimental evidence is considered. DIRECT CALCULATION OF INTERMOLECULAR INTEUCTION The simplest system to discuss is the 32C state of the hydrogen molecule; the energy, which is accurately known,2* 3 depends approximately exponentially on internuclear distance. If an expression for the energy as an inverse power of the distance is fitted to the results, the repulsive index n varies nearly linearly with distance, from about 2 at 1 A to about 5 at 2A.The more complicated interaction between helium atoms has been treated approximately in various ways : 4-7 all the results for energy against distance fit an exponential with the same exponent (- 2*4/ao), although the absolute magnitude appears to be less certain. Neon has also been treated,s though the errors here may be greater. Direct calculation for two hydrogen molecules 9 gave a minimum potential similar to that calculated from viscosity data using the Lennard-Jones 12 : 6 potentia1,lo although the theoretical potential was appreciably less steep than the empirical at short distances.Calculation of the potential for more complicated molecules is not at present feasible: the information about simpler systems must be extrapolated to more 10T. L. COTTRELL 11 complicated ones. One method of taking account of the theoretical information is to use an exponential rather than an inverse power for the repulsive term in an empirical potential function.11-13 A dficulty arises in the application of simple formulae to polyatomic molecules. Here the main interactions are probably between the peripheral atoms, so that the form of the interaction as a function of the intermolecular distance may well be different from that required to fit mon- atomic gases. Chloroform, for example, deviates little from the principle of corresponding states 14 but when molecular centres are over 4 A apart, the chlorine nuclei may be about 1 A apart.Thus the expected potential, as a function of intermolecular distance, will be steeper than that associated with monatomic gases. The point was noted by Hamann, McManamey and Pearse15 who used an expression for the potential due to a spherical shell, each element of which has a potential given by the inverse power law,16 to derive the intermolecular potential. They found little improvement in the fit to experimental virial coefficient data. However, the correct potential to use on this approximation is rather the potential between two uniform spherical shells each element of which interacts with each element of the other, according to a known potential function.17 In this approximation, one should not use for the peripheral atoms repulsive potentials obtained empirically from the bulk properties of the monatomic gases, because they are known to be inadequate at short range.The method was applied to the repulsive potential between methane molecules, assuming the repulsion between the peripheral atoms to lie between that calculated for hydrogen and for helium.17 The resulting potential was fitted to an exponential, and used to calculate the vibrational relaxation time of methane. Accurate discussion of polyatomic molecules is complicated by the loss 01 even approximate spherical symmetry in the close-range encounters during which repulsive forces predominate. Averaging is adequate for bulk propertiesp but more questionable for close approaches.At greater density, interaction between two molecules only is less relevant, and one must either treat many-body interactions explicitly or treat the interaction of a molecule with its environment described by an averaged potential. Little progress has been made with the former programme; the latter is discussed in the next section. MODEL CALCULATION OF MOLECULE-ENVIRONMENT INTERACTION Cell theories of liquids and gases at high pressures have aroused much interest recently, rather from the point of view of the combinatorial problem than from that of the intermolecular potential.18-23 A different approach to the cell model in which the interaction between molecule and environment is directly calculated arose from a phenomenological treatment of the energy of compressed gases by Michels and his colleagues.24. 25 Applying the virial theorem to a substance at pressure p and volume v, where Eis the total energy and Tthe average kinetic energy of electrons and nuclei.26 A h ) can be measured and ATcalculated in this way for the compression of some simple gases up to 3000 atm, showed a large increase amounting to 5 kcal mole-1 for nitrogen.This kinetic energy must be electronic and is thus an indication of the effect of pressure in distorting the electron distribution. It was therefore suggested 25 that at high densities the effect of environment on a molecule might be calculated by replacing it by the impenetrable wall of a small sphere of radius ro. From the virial theorem25.27 the average kinetic energy of electrons and nuclei is 3E 3r0 AT = 3AW) - AE, T = - r o - - E.12 INTERMOLECULAR REPULSIVE FORCES Comparison of the kinetic energy calculated in this way with the experimental values should check the validity of this approach.De Groot and ten Seldam?* extending earlier work 24 discussed the hydrogen atom in a spherical box in terms of the radius of the box, and found that the change in kinetic energy on compression was similar to that found experimentally. Such calculations have been made for helium,29 argon,30 and the hydrogen molecule-ion.31 In the last, results were obtained to 106 atm, to throw some light on the behaviour of gases at very high pressures. For molecular volumes less than 60 A3, the energy-volume curve calculated in this way for the hydrogen molecule-ion is approximated by an inverse square,32 which implies an inverse sixth power of distance.This confirms the view that the inverse power most suitable at long range is too high for short range. Further, the inverse sixth power is becoming too steep at volumes less than 25A3. Thus the evidence suggests that for close approach, the potential varies with quite low powers of r. EXPERIMENTAL EVIDENCE EQUATION OF STATE OF GASES At low pressures the equation of state of gases is insensitive to the repulsive force. For example, the second virial coefficient of methane can be represented equally well by two Morse functions,33 in which the exponents in the one differ from those in the other by a factor of 2. Hamann et aZ.l5,34,35 have tried to use virial coefficient data to discuss the shape of the potential, but were not able to to reach very firm conclusions, except that the Lennard-Jones function does not fit polyatomic molecules very well.The equation of state for gases at very high densities and temperatures above the critical is expected to depend significantly on intermolecular repulsive forces. Gaseous products from condensed detonating explosives are formed at densities between 1 and 2 g cm-3 and temperatures of a few thousand OK. The hydro- dynamic theory of detonation 36 relates the thermodynamic properties of the system, including the equation of state of the products, to the properties of the detonation wave : that is, to the detonation pressure, temperature, density, streaming velocity and wave velocity. Until recently only the last of these could be measured, and it was at one time thought that comparison of observed detonation velocities with the calculated would allow the determination of the equation of state.371 38 This, however, is not ~0,39140 the problem not being determinate.Moreover, the calculated velocity is insensitive to the form of equation used; for example an equation of the form p v = RTf(v), which implies (3E/3 V)T f 0 has been successfully used in calculating the velocity of detonation of a wide range of explosives,37~ 41 although the assumed form is probably incorrect. On the other hand, good agreement with experiment for a high-energy explosive has also been obtained with an equation which may exaggerate intermolecular repulsive forces.32 A possibility of progress stems from the recent measurement at the Los Alamos laboratory of a second detonation wave property, the detonation pressure (Chapman-Jouguet pressure).42 This should allow closer definition of the equa- tion of state, and hence of intermolecular forces, but the results so far published 43 are concerned only with the determination of the parameters in the Kistiakowsky- Wilson equation of state which cannot be readily interpreted in terms of inter- molecular forces and does not even fit the data.However, the Lennard-Jones and Devonshire equation of state is being similarly examined: the results have still (May, 1956) to be published. Measurement of the temperature would be more useful in determining inter- molecular repulsive forces, because this quantity is most sensitive to the assumptions made.For example, an equation making no allowance for repulsive forces reproduces the detonation velocity against density curve for penta-erythritolT. L . COTTRELL 13 tetranitrate fairly well with no adjustable parameters,4lP 36 and predicts that the temperature at the Chapman-Jouguet point should increase with density. On the other hand, the equation based on the molecule-in-a-box calculation reproduces the detonation velocity curve better than the other equation, though with one adjustable parameter, and predicts that the temperature should decrease with increasing density.32 This is because much of the energy liberated is used in overcoming intermolecular forces at high density. The relevant figures are given in table 1, where the sensitiveness of the temperature is clearly shown.TABLE 1 .-DETONATION OF PENTA-ERYTHIUTOL TETRANITRATE loading velocity velocity velocity temp. temp. pressure pressure density obs. calc. calc. c alc. calc. CalC. calc. g.cm-3 m,sec (4 (6) (a) OK (b) O K (a)atm (4 a m 1.727 8,360 9,500* 8,200 5,500 1,950 2-20 X lO5* 2.85 x 105 1-40 7,000 7,600 7,010 5,290 2,630 1.61 1.86 1-00 5,520 5,560 5,530 5,150 3,650 0.74 0.88 0.75 4,700 4,520 4,760 5,060 4,060 0.42 053 0.40 3,710 3,350 3,660 4,990 4,525 0.15 0.19 (u) pv = RTf(v),41. 36 (t7) E 1 p . 3 2 * extrapolated. EQUATION OF STATE OF LIQUIDS For liquids the statistical problems are so difficult that it is doubtful whether any important evidence about intermolecular repulsive forces can be derived.EQUATION OF STATE OF SOLIDS There are two approaches to deducing intermolecular forces from information on solids. One is to accept the observed structure of the solid, and calculate from the experimental data on interatomic distances and sublimation energy the values of the parameters in an assumed force function, checking the results with those obtained for the gas. An investigation of the values of the indices in the double reciprocal potential for a large number of face-centred cubic crystals 44 showed a wide variation from substance to substance, the repulsive index ranging from - 6 to - 12, and the attractive from - 2 to - 6. Corner 45 has determined the intermolecular potentials in neon and argon in this way, and obtained slightly better agreement with an exponential than with a reciprocal repulsion term, A more fundamental approach is to deduce the most stable crystal lattice from the theory and compare with experiment.It has been shown 4% 47 that the lattice energy arising from pure London forces is less for hexagonal than for cubic close-packing; the conclusion is not upset by the addition of central exchange forces.48 However, apart from helium, the rare gases adopt cubic close-packing. Four explanations have been suggested : (i) zero-point vibrational energy, (ii) thermal vibrational energy, (iii) the potential is additive but of a different shape, (iv) the potential is not strictly additive. The third possibility was considered by Kihara and Koba,49 who suggested that the actual potential well is much wider than that given by the 12 : 6 law,50 but a wider potential gives disagreement in other respects.All the possibilities except the last have been eliminated by the work of Banon and Domb.51 Thus, in order to obtain exact information about intermolecular potentials from crystals, account has to be taken of many-body interactions.14 INTERMOLECULAR REPULSIVE FORCES However, it ought to be possible to deduce some semi-empirical information about repulsive forces from the equation of state of solids at very high pressures. For example, Guggenheimersz has shown that Bridgman's high pressure data lead to an exponential repulsion. The available experimental information for such studies has recently been increased by the measurement of the Hugoniot for metals up to 4 x lo5 atm, using the shock wave from explosives.S3 TRANSPORT PROPERTIES In the theory of transport properties the coefficients are expressed in terms of integrals which involve the dynamics of molecular collisions and hence the inter- molecular potential.Detailed analysis of the viscosity results for helium, com- pared with other evidence, gives results in accord with those obtained theoreti- cally.54 Hirschfelder and his colleagues 1 0 ~ 5 5 have used the transport properties to derive the values of the parameters in the Lennard-Jones potential for various gases, the agreement with those from second virial coefficients not being very close. because the two properties depend on different parts of the potential. More recently, an exponential repulsive term has been used in discussing transport properties.56-58 For spherical molecules, good agreement between transport properties and second virial coefficients is obtained, but results are less satisfactory for polyatomic molecules.HIGH-VELOCITY ELASTIC COLLISIONS Study of the scattering of high-velocity atoms by room temperature atoms is the only theoretically straightforward method for deducing interactions at close range. In matter at high temperatures and pressures many-body interactions must be considered, whereas in this process only two atoms are involved. Un- fortunately it has been applied to very few systems. For helium-helium inter- actions,sg the scattering results, together with viscosity results, can be fitted by a potential function including an exponential repulsion with an exponent the same as that obtained theoretically.4.7 INELASTIC COLLISIONS Calculation of the probability of energy transfer from external to internal modes during a molecular encounter requires knowledge of the potential in the highly repulsive region.Experimental results have usually been interpreted on the Landau-Teller theory,ao in which an exponential repulsion is assumed. It has been stated54 that values of the exponent obtained in this way agree with those obtained in other ways, but further examination suggests that the Landau- Teller theory requires a steeper repulsive potential than that indicated theoretically.17 Using energy transfer to determine the potential has been suggested61 but the theory is not sufficiently far advanced for this to be a practicable programme.DISCUSSION Twenty years ago, London presented a paper on " The general theory of mole- cular forces " to a Faraday Society discussion.11 In it the form of the long-range attractive forces was established, but little information was available about re- pulsion which, on the basis of the work of Slater 4 and Bleick and Mayer,* London took to be exponential. The present examination of our knowledge of the repulsive potential does not take us much further than London was able to go, though the range of evidence on which we can draw is much greater. The conclusion that the repulsive potential for monatomic gases is best repre- sented by an exponential has been confirmed, both theoretically and experi- mentally, and it is clear that an inverse power is inadequate at distances of one or two Angstroms. The expected superiority of the exponential repulsion is notT.L. COTTRELL 15 demonstrated by the results on gases at fairly low pressures, either in the virial coefficients or in the transport properties, from which one may conclude that the detailed form of the repulsive potential is not yet relevant in the average collision at normal temperatures and pressures. In matter at high density the detailed form is important, but the situation is obscured by the need to take account of many- body interactions, and it is possible that a more empirical approach, involving the interaction of a molecule with its averaged environment, may be valuable. The repulsive potential may be most accurately mapped by studying high velocity collisions, but this can only be done in face of considerable experimental difEculty for elastic collisions, and theoretical difficulty for inelastic collisions.It seems likely that advance will come from further experimental observation, either of the bulk properties of matter under more extreme conditions, as in the work with explosives, or in the extension of the collision experiments to a wider range of substances. 1 Zwanzig, J. Chem. Physics, 1954, 22, 1420. 2 James Coolidge and Present, J. Chem. Physics, 1936,4, 187. 3 Hirschfelder and Linnett, J. Chem. Physics, 1950, 18, 130. 4 Slater, Physic. Rev., 1928,32,349. 5 Rosen, J. Chem. Physics, 1950, 18, 1182. 6 Margenau and Rosen, J. Chem. Physics, 1953,21, 394. 7 Griffing and Wehner, J.Chem. Physics, 1955,23, 1024. 8 Bleick and Mayer, J. Chem. Physics, 1934, 2, 252. 9 Evett and Margenau, Physic. Rev., 1953,90, 1021. 10 Hirschfelder, Bird and Spotz, J. Chem. Physics, 1948, 16,968. 11 London, Trans. Faraday SOC., 1937,33, 8. 12 Buckingham, Proc. Roy. SOC. A, 1938,168,264. 13 Buckingham and Corner, Proc. Roy. SOC. A, 1947,189, 118. 14 Guggenheim, Rev. Pure Appl. Chem., Australia, 1953, 3, 1. 15 Hamann, McManamey and Pearse, Trans. Faraday SOC., 1953, 49,351. 16 Lennard-Jones and Devonshire, Proc. Roy. SOC. A , 1937,163, 53. 17 Cottrell and Ream, Tram. Faraday SOC., 1955, 51, 1453. 18 Kirkwood, J. Chem. Physics, 1950, 18, 380. 19 Janssens and Prigogine, Physica, 1950, 16, 895. 20 Pople, Phil. Mag., 1951,41, 459. 21 de Boer, Physica, 1954,20, 665. 22 Taylor, J.Chem. Physics, 1956, 24,454. 23 Green, J. Chem. Physics, 1956,24, 732. 24 Michels, de Boer, and Bijl, Physica, 1937, 4, 981. 25 Michels and de Groot, Physica, 1950, 16, 183. 26 Schottky, Physik. Z., 1920,21, 232. 27 Cottrell and Paterson, Phil. Mag., 1951, 42, 391. 28 de Groot and ten Seldam, Physica, 1946,12,669. 29 de Groot and ten Seldam, Physica, 1952,18,905. 30 de Groot and ten Seldam, Physica, 1952, 18,910. 31 Cottrell, Trans. Faraday SOC., 1951,47, 337. 32 Cottrell and Paterson, Proc. Roy. SOC. A, 1952,213,214. 33 Cottrell and Ream, Trans. Faraday SOC., 1955, 51, 159. 34 Hamann, J . Chem. Physics, 1951, 19, 655. 35 Hamann and Pearse, Tram. Faraday SOC., 1952,48, 101. 36 Taylor, Detonation in Condensed Explosives (Oxford, 1952). 37 Caldirola, J. Chem. Physics, 1946, 14, 738. 38 Cook, J. Chem. Physics, 1947, 15, 518. 39 Paterson, J. Chem. Physics, 1948, 16, 159. 40 Jones, 3rd Symp. Combustion, Flame and ExpIosion Phenomena (Baltimore, 1949), 41 Paterson, Research, 1948,1,221. 42 Duff and Houston, J. Chem. Physics, 1955,23, 1268. 43 Cowan and Fickett, J. Chem. Physics, 1956, 24,932. 44 Furth, Proc. Roy. SOC. A, 1945, 183,87. p. 590.16 INTERMOLECULAR REPULSIVE FORCES 45 Corner, Trans. Faraday Soc., 1948,44,914. 46 Axilrod, J. Chem. Physics, 1951, 19,719, 724. 47 Prins, Dumore and Tjoan, Physica, 1952,18, 307. 48 Jansen and Dawson, J. Chem. Physics, 1955,23,482. 49 Kihara and Koba, J. Phys. SOC. Japan, 1952,7, 348. 50 Kihara, J. Phys. SOC. Japan., 1951, 6, 184. 51 Barron and Domb, Proc. Roy. SOC. A, 1955,227,447. 52 Guggenheimer, to be published. 53 Walsh and Christian, Physic. Rew., 1955, 97, 1544. 54 Massey and Burhop, EZectronic and Ionic Impact Phenomena (Oxford, 1952). 55 Hirschfelder, Curtiss and Bird, Molecular Theory of Gases and @ui& (New York, 56 Mason, J. Chem. Physics, 1954,22, 169 ; 1955,23,49. 57 Mason and Rice, J. Chem. Physics, 1954,22,522,843. 58 Madan, J. Chem. Physics, 1955,23,763. 59 Amdur and Harkness, J. Chem. Physics, 1954,22, 664. 60 Landau and Teller, Physik. 2. Sowjetunion, 1936, 10, 34. 61 Bauer, J. Chem. Physics, 1955, 23, 1087. 1954).

ISSN:0366-9033    

DOI:10.1039/DF9562200010

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

ELECTROMAGNETIC PROPERTIES OF COMPRESSED GASES BY A. D. BUCKINGHAM * AND J. A. POPLE Received 18th June, 1956 The effects of molecular interactions on several equilibrium electromagnetic properties of compressed gases are discussed. A suitably-chosen measurable property is expanded in inverse powers of the molar volume, and the term representing initial deviations from perfect-gas behaviour, and therefore depending upon pair interactions only, is examined in detail. The particular properties dealt with include the dielectric constant, refractive index, Kerr and Cotton-Mouton constants and the molar paramagnetic susceptibility. The usefulness of each property in yielding information about intermolecular forces is discussed ; the dielectric constant and the Kerr and Cotton-Mouton constants are par- ticularly suitable as aids to the study of orientationally-dependent forces, while it is suggested that observations of the density dependence of the magnetic susceptibility of oxygen might throw some light on the relative magnitudes of the interaction energies of the singlet, triplet and quintuplet dimers.1. INTRODUCTION The simplest theories of the electromagnetic properties of materials, such as the dielectric constant and magnetic susceptibility, are based on the assumption that each molecule can be treated as an independent system and so are really only appropriate to a perfect gas. It is sometimes possible to treat the rest of the material by some simple semi-macroscopic model and so extend the perfect gas results to high densities (as in the Clausius-Mossotti formula for the dielectric constant of a nonpolar substance), but this can only be approximate, for the bulk properties will depend upon the way in which the molecules interact.Ac- curate measurements on electromagnetic properties of compressed gases do reveal deviations from the simple formulae which, if interpreted correctly, give information about the details of interactions. The most systematic way to examine these effects is by means of a virial-type expansion. If Q is a suitably-chosen measurable property, its observed value can (for non-ionic systems) be expanded in inverse powers of the molar volume V Q = AQ + BQIV+ CQIV~ + . ., (1) where the coefficients AQ, BQ, CQ, . . ., are functions of temperature only. AQ is the perfect gas value corresponding to an independent-molecule treatment.BQ represents the correction if the pressure is high enough for pair interactions to be important. Higher terms in the series arise from multiple interactions. The use of an expression of this type in the equation of state is familiar and the ordinary second virial coefficient B(7') has been widely used for determining inter- molecular potentials. In this paper we shall survey some electromagnetic pro- perties from the same point of view, analyshg the coefficients BQ theoretically and discussing the additional information about molecular interactions that can be obtained from measurements. * Physical Chemistry Laboratory, The University of Oxford. f Department of Theoretical Chemistry, The University of Cambridge, 1718 ELECTROMAGNETIC PROPERTIES 2.GENERAL THEORY OF INITIAL DEVIATIONS FROM PERFECT GAS BEHAVIOUR Suppose we are dealing with some macroscopic property Q of a mole of gas, which, for a set of independent molecules, is the sum of mean contributions 4 of individual molecules. The leading term in the virial-type expansion (1) is then AQ = NG. If we are dealing with higher densities, however, the contribution of particle 1 to Q is not always 4, because, for part of the time, particle 1 has to be treated as half of a dimer or interacting pair. Alternatively, it could be said that, at any given instant, some of the particles are interacting in pairs. If particle 1 has a neighbour whose configuration relative to 1 is represented by the collective symbol T, its contribution to Q at such instants should be written 3q12(T), where q1&) is the corresponding contribution of the dimer.Q will therefore be given by e = N 4i + [k12(7) - qT1pOdrjy { S (3) where P(7)dT is the probability of particle 1 having a neighbour in the range (7, T + d7). This is related to the intermolecular potential energy U(T) by N QV P(T) = - exp [- u(-r)/kq, (4) where Q is the integral over the orientational co-ordinates of the neighbouring molecule. Substituting in (3), and comparing with (l), we obtain the following general expression for BQ This is the basic formula that will be applied to various properties Q sections. (5) in subsequent 3. DIELECTRIC CONSTANTS AND REFRACTIVE INDICES If Q is the total dielectric polarization E- V , where E is the static dielectric E + 2 constant, then 4n 3 4 = -(a + p2/3kT) where a is the polarizability and p the dipole moment.'Thus, from (5), the '' second dielectric virial coefficient " BD is given by where a12 is the polarizabdity of the interacting pair of molecules (regarded as a single system) and (p1 + p2) the corresponding resultant dipole moment. At densities high enough for triple and higher interactions to be important, long- range dipolar effects resulting from the so-called " boundary field " have to be considered,l but these do not affect the coefficient BD. In a similar way, the " second refractivity Virial coefficient " BR in the expansion n2- 1 n2 + 2 of - V, where n is the refractive index, is given by the first part of (7), (8) &N2 352 BR = -A .D. BUCKINGHAM AND J. A. POPLE 19 In a precise treatment, the a's would be frequency-dependent? but provided the refractive index is measured at a frequency which is not close to a natural frequency of the molecule, the static polarizability should be a good approximation to the true a. For monatomic substances, there can be no dipole moments in the cases of monomers and dimers, and BD reduces to its first term. Thus, if dispersion effects are neglected, BD = BR. The difference between *a12 and a~ has been the subject of several investigations, both classical and quantum-mechanical.2-5 It appears that for atoms BD is usually positive but its magnitude is relatively small. For gases of non-polar molecules with higher multipole moments, a temperature- dependent contribution to E will arise from the second term in (7).Carbon dioxide, for example, has a large quadrupole moment @ the field of which will induce a dipole in a neighbouring molecule. If a is the molecular polarizability, this dipole will be proportional to a@. The mean square dipole moment of the pair can be evaluated by averaging over orientations, and a simple expression is obtained for the second term in (7).6 Using a value of @ = 5.29 x 10-26 e.s.u. (which fits data on the C02 equation of state and crystal7), it is found that cal- culated values of the two parts of BD at 50" C are 8 and 38 cm6 mole-2. The experimental values of BD are 36 f 3 cm6 mole-2 (calculated by Brown * from the data of Michels and Kleerekoper 9) and 34 cm6 mole-2 (Keyes and Oncley 10).From the data of Michels and Hamers 11 for the refractive index of compressed CO2 for a wavelength of 5876 A it appears that BR at 500 C is approximately - 0.6 cm6 mole-2, but this result is almost certainly too small, and its negative sign seems unlikely.lz Turning to polar substances, BD will be larger and normally dominated by the second term. In a simple model of a polar substance, one assumes that the only important contributors are the permanent dipoles, so that The value of BD would then be proportional to the average value of cos 8, where 6 is the angle between the dipoles. According to the simple Stockmayer potential (dipole-dipole interaction together with a central-force potential) the mean value of cos 6 should be positive, but the experimental value of BD for methyl fluoride at 50" C is - 600 cm6 mole-2 (calculated from the measurements of David, Hamam and Pearse 13).The inclusion of a increases the mean square dipole moments of the interacting molecules and so makes the calculated BD'S even larger.14 The observed negative BD can be explained by supposing that shape effects encourage the dipoles to have opposed directions, thereby reducing the mean square dipole moment of the dimer. As illustrated in the figure, two rod-like polar molecules (4 (b) FIG. 1 .-Favourable configurations for rod and plate-like polar molecules. will tend to cluster in an anti-parallel manner, whereas plate-like ones might be expected to favour the arrangement shown in fig. l(b). If this explanation of an observed negative BD is correct, then we should expect plate-like molecules (such as paraldehyde and arsenic trifluoride) to have large positive BD's. The pressure dependence of the refractivity of polar gases should be consider- ably larger than that for non-polar onesp but unfortunately at present there are20 ELECTROMAGNETIC PROPERTIES no data from which accurate values of BR could be computed.Anisotropy in the polarizability tensor and the dependence of polarizability on electric field strength (" hyperpolarizability ") are two factors which might be expected to contribute to B~.12 4. ELECTRIC AND MAGNETIC DOUBLE REFRACTION The molecular Kerr constant ,K of a non-polar substance is given by where y measures the field dependence of the induced dipole moment of a mole- cule and where K is a number describing the anisotropy of the polarizability tensor; for an isolated axially-symmetric molecule (11) p = a E + + y E 3 + .. (10) K = (all - aL)/3a0; all and cc1 are the polarizabilities along and at right-angles to the axis. From (9), one finds,l5 on omitting y, where 6 is the angle between the axes of the pair. The density dependence of the molecular Kerr constant is therefore dependent upon the average value of (3 COS~ 8 - 1). This may be appreciated by examining the values of the product of the polarizability and the anisotropy of the pair for various configurations.15 Since the mean of (3 cos2 6 - 1) is zero if the forces between the molecules do not depend on relative orientation, the density dependence of ,K for non-polar substances should be small-this is found to be true for some simple gases such as C02 and C2H4.15 The molecular Cotton-Mouton constant, describing birefringence in a strong magnetic field, also has a BC proportional to the mean of (3 cos2 6 - l), and Benoit and Stockmayer16 have shown that the density dependence of the de- polarization of scattered light is proportional to the same quantity.5. MAGNETIC SUSCEPTIBILITY The effect of pressure on the magnetic susceptibility of a paramagnetic gas might also yield information about molecular interactions. The theory of this phenomenon is somewhat different, however. We shall discuss the theory for a molecule such as oxygen, where the paramagnetism arises from the spin and not the orbital moment of the electron.The ground state of an 0 2 molecule is 3 c . Two interacting ones will therefore give rise to three states instead of one, as for two diamagnetic molecules. These three states will be a singlet, triplet and quintuplet and with each there will be associated a different interaction potential energy $(I), #3) and 4(5). The a priori weights of the three states for any given configuration are 1 : 3 : 5. The second virial coefficient in the equation of state is therefore given by where Bi is defined byA . D . BUCKINGHAM A N D J . A . POPLE 21 If the molar magnetic susceptibility is expanded in the form then the leading term is given by 17 8 Np2 A , = - 4Np2 S(S + 1) = =, 3kT where ,8 = eh/4.nmc is the Bohr magneton and S the spin quantum number (equal to unity for an 0 2 molecule). To find B, we must consider corrections because, for some of the time, each molecule has to be considered as half of one of the three types of dimer. Now the probability of a given molecule having a neighbour in a region of relative configuration space d7 such that the pair are in the singlet state is N - exp [- #l)/kT]d.r.9 VQ Also, for the singlet dimer, S(S 3. 1) = O(0 + 1) = 0. Had we continued to treat the molecules independently, however, we should have had 2 x l(1 + 1) == 4 instead. Thus the contribution to B,/ V from singlet dimers is 4*2 LJ' exp (- @)/kT)[O - 4]d~. 3kT 2VQ 9 There are similar expressions for the other two states. Collecting terms, we find that B, can be written in terms of B1, B3 and Bg defined by (14), '*g2[2 3kT 9 1+-B3--B5 9 .3 5 1 B =-- -B (19) Eqn. (19) shows that measurements of B, would give an independent linear combination of B1, B3 and Bg. This would in fact give information about the magnitude of the separation of the intermolecular potentials. It is probable that the singlet state is the lowest, for the susceptibility is partly quenched in the liquid; 18 measurements on a compressed gas might enable this to be made more quantitative. The susceptibilities of solutions of oxygen in liquid nitrogen were measured over a considerable concentration range by Perrier and Onnes,lg and Kanzler20 has published data on the pressure and temperature dependence of xm for oxygen. Both these sets of data indicate that B, is negative at the tempera- tures studied. 1 Kirkwood, J. Chem. Physics, 1939, 7, 911. 2 Kirkwood, J. Chem. Physics, 1936,4, 592. 3 de Boer, van der Maesen and ten Seldam, PhySica, 1953, 19, 265. 4 Jansen and Mazur, Physica, 1955,21, 193,208. 5 Buckingham, Trans. Faraday SOC., 1956,52, 1035. 6 Buckingham and Pople, Trans. Faraday SOC., 1955,51, 1029. 7 Buckingham, J. Chem. Physics, 1955,23,412. 8 Brown, J. Chem. Physics, 1950, 18, 1200. 9 Michels and Kleerekoper, Physica, 1939, 6, 586. 10 Keyes and Oncley, Chem. Rev., 1936,19, 195. 11 Michels and Hamers, Physica, 1937, 4, 995. 12 Buckingham, Trans. Faraday SOC., 1956,52,747. 13 David, Hamann and Pearse, J. Chem. Physics, 1952, 20, 969. 14 Buckingham and Pople, Trans. Faraday Soc., 1955,51, 1179. 15 Buckingham, Proc. Physic. SOC. A , 1955, 68,910. 16 Benoit and Stockmayer, J. Phys. Radium, 1956,17,21. 17 Van Vleck, The Theory of Electric and Magnetic Susceptibilities (Oxford University 18 Kamerlingh Onnes and Perrier, Leiden Comm., 1910, 116. 19 Perrier and Kamerlingh Onnes, Leiden Comm., 1914, 139d. 20 Kanzler, Ann. Physik, 1939, 36, 38. Press, 1932), p. 266.

ISSN:0366-9033    

DOI:10.1039/DF9562200017

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

INFRA-RED SPECTRA OF GASEOUS MIXTURES UNDER HIGH PRESSURE BY R. COULON, L. GALATRY, J. ROBIN AND B. VODAR Laboratoire des Hautes Pressions, Bellevue (S. and 0), France Received 2nd July, 1956 The absorption spectra of compressed gases have been studied in the infra-red. The regular pressure range is up to 1200 atm, but an apparatus for 6000 atm had been de- veloped. The measurements are concerned particularly with mixtures of a polar gas (hydrogen halide) with a non-polar gas (H2, N2, 0 2 , A). The following effects have been observed : (i) appearance of a Q-branch in the vibration-rotation spectrum of the hydrogen halide ; (ii) rapid increase of the forbidden vibrational band of the non-polar molecule under the influence of the polar molecules ; (iii) simultaneous transitions in a mixture of HCl and H2, giving rise to an absorption band at a frequency equal to the sum of the vibrational frequencies of the two separate molecules. In addition to the role played by the atomic distortion moments, the particularly strong effects of the dipole (and also at higher pressures of the quadrupole) moments have been described in terms of a simple theoretical concept taking into account the induced moments.GENERAL FEATURES Infra-red absorption spectra of a pure substance or of a mixture in the gaseous state and at a low enough density are considered to be produced by all the mole- cules, without any interaction between them. Indeed, in normal conditions,the frequency of the collisions and the mean intermolecular distances are such that each molecule spends most of its time far from its neighbours, and so the inter- molecular actions are negligible.The spectrum then characterizes a property of an isolated molecule of each component of the mixture. But if by a suitable physical means (liquefaction, compression, dissolution) the average distance between the molecules is made appreciably shorter, they are then submitted to the perturbing action of their neighbours during an important part of the time. This generally modifies their intrinsic properties with regard to incident electromagnetic waves. These modifications perturb significantly the infra-red spectrum of the substance or mixture under investigation and so provide information on both the intermolecular fields and dynamical state of the molecules in dense media, these two aspects of the problem being intimately connected.Experiment has shown that when two molecules come nearer to each other an interaction energy develops and modifies their mutual motion and vibrational or rotational state. Convenient approximations to this interaction energy allow the separation of it into different well-defined types. For molecules in which we are interested here, there are besides the London energy that is responsible for perturbations of atomic spectra by foreign gases, other forces due to the '' classical " structural properties of electronic shells. The order of magnitude of these forces is largely dependent on the type of molecules (whether polar or non-polar). It will be seen that we can classify the available experimental data in the same way as these different kinds of forces.22R . COULON, L. GALATRY, J . ROBIN AND B . VODAR 23 Let us emphasize first the importance of the theoretical problem. How can we from a priori knowledge of the force between two given molecules, deduce an explanation of the optical properties of the media in which these forces are acting? When the interacting systems are atoms, we assume, with Margenau, that the variation of potential energy of the pair results in equal variations of the atomic energy levels. This fact, together with the property of additivity of London forces provides a basis for the so-called statistical theory.1.2 In this theory, the effect of all the perturbations is replaced by afield, independent of the optically-active atom.But when the perturbated system is a molecule, its change of state, due to the perturbing systems acts on the other molecules in turn. These can therefore play an actual role in the absorption process, even though they do not absorb in normal conditions. In practice we do not take into account the action of the perturbation by an external field and we consider the whole system as the sum of the perturbed molecule + perturbator. With this system we associate an interaction Haniiltonian depending upon the configuration variables of each molecule. When the density is not too large (corresponding to pressures below 1000 atm, for instance), most spectral changes result from systems consisting of two molecules. Though this model is imperfect, it has the advantage of avoiding the difficult problem of the non-additivity of the forces under consideration.Strictly speaking, all the variables describing the state of the mixture should be treated quantially, but we prefer to treat classically the motion of the mass centres of the molecules (“ classical path ” method).3 Thus we may, especially in the previous pair approximation, introduce the radial distribution function as a weighting factor, in order to account for the relative motions of the molecules. The expression thus obtained3 for the absorption by a system of molecules involves, as for an isolated molecule, the matrix element of the electric moment of the whole system with respect to the wave function describing the internal states of this system, As a result, the intermolecular forces are involved in the spectral perturbations in two ways : (i) they modify the electric moment of each molecule of the system, and (ii) they modify the wave function of the system which is no longer a product of the wave function of the isolated molecules. We have then to consider the intermolecular forces according as to whether they are more or less able to produce changes (i) and (ii).So, we shall distinguish, in increasing order, of shortest distances of appreciable action : (1) the repulsive potential due to the overlapping of electron shells (a very (2) the potential between quadripoles and induced dipoles ; (3) the potential between dipoles and induced dipoles (an important directional (4) the potential between two permanent dipoles.We should, finally, mention forces of the “charge transfer” type.4 In contrast to the preceding ones, they are much more chemical in character. They have a very important directional effect, which so far has been detected only in solutions.5 Nevertheless, they may provide a better explanation of certain spectra of gas mixtures under high pressure. In general, corresponding to the forces of types (1) and (2) are the spectra induced by pressure in non-polar gases. These will be studied below and later we shall consider the spectral changes of polar media where forces of type 3 or 4 are predominant. In the last section, we shall tentatively compare these spectral modifications with those observed in liquids and solutions. weak effect on the mutual orientation of the molecules 3) ; effect) ;INFRA-RED SPECTRA 24 INDUCED ABSORPTION Homonuclear diatomic molecules have no dipolar infra-red absorption spectrum under normal conditions, their dipolar moment being always zero owing to sym- metry.But, in 1948, Crawford, Welsh and Locke observed an absorption spectrum of compressed oxygen at Raman vibrations frequencies of this gas (fundamental 1556 cm-1; 6 and first overtone, 2900-3300 cm-9.7 They had observed an analogous phenomenon in N2 and H2 and in mixtures of these gases alone and together with rare gasesg-ll), and we have studied the fundamental spectra of N2 and 0 2 up to 800 atm.12 In the same way, Ketelaar and Coll13 observed some pure rotational lines in compressed hydrogen up to 100atm. In polyatomic molecules, forbidden transitions appear too when the density increases (CH4,15 CO2,8.16 17 C2H2 34).There is another noteworthy spectral effect as the density increases, viz., ‘‘ simultaneous transitions ” which give rise to absorption bands corresponding to the sum (or difference for liquids) of the eigen frequencies of two neighbouring molecules (combination frequencies). Thus, in the first overtone regions of H2, Welsh and Coll7 had distinguished the induced first overtone (8080 em-1) and the simultaneous transitions vH2 + vH2 (8320 cm-1). Combination frequencies were observed too in mixtures of COz with H2, N2 and 02.25 and in liquid mixtures (cp. later). The main properties of induced spectra are : (i) up to about IOOOatm the integrated absorption: I = J e(v)dv, (+) s extinction coefficient) is proportional to the square of the density in a pure gas and to the product of the densities of components in a mixture. These properties suggest 6 that induced absorption is due to bimolecular collisions. More exactly, as Kranendonk has ~hown.3~18~19 the pair of molecu!es in collision shows an induced dipole moment, responsible for the absorption.For 0 2 , I increases more slowly than the square of density,l2 this fact being connected with the possible formation of Q4 molecules. band (ii) Since the first experiments,a the structure of induced vibrational spectrum of diatomic molecule has been found not to obey the infra-red (AJ = 1) but Raman (AJ = 0, f 2) selection rules. These selection rules were theoretically deduced by Kranendonk and Bird 1% 20 who successfully calculated the area I of induced bands for H2.The dipolar electric moment of the pair in collision is due to (i) the asymmetric distortion of electronic wave functions by overlapping of atomic orbitals and (ii) the polarization of one molecule by the quadrupole moment of the other? Mizushima23 proposed a theory taking into account only the quadrupole field. Therefore he is unable to explain entirely induced spectra of symmetric diatomic molecules by the high pressure of a rare gas. These can be wholly explained only by considering the distortion of electronic clouds.18 The rota- tional selection rules are simply related to the deformation properties of sym- metric diatomic molecules ; these properties have the symmetry of a polarizability tensor instead of that of an electric moment.The numerical calculatioii of effects (i) and (ii) 18 shows that distortion forces have only a very weak directional power. They are therefore principally responsible for the Q branch (AJ = 0) ; quadru- polar potentials, which are more angle-dependent, take an important part in the induction of the lines of S and 0 branches. It must be noticed that here, the effects of rotational hindrance during collision are neglected. This is permissibleR . COULON, L. GALATRY, J . ROBIN A N D B. VODAR 25 in view of the relatively weak orientation power of the acting forces (no permanent dipole). It will be seen in the next section that it is not likely to be true for polar molecules. For light H2 molecule, the Q branches are separated into two maxima due to the rotation of H2-H2 pairs; 11 we can observe a splitting of the same type for the S(1) line (fig.1) at 100 atm and 140" C and have noted further details in the Q-branch structure-three peaks at 4124, 4138.5 and 4149.5 cm-1 lying in the Qp sub-branch. These details are visible, in our experimental conditions, only between 80 and 300 atm and for t > 80" C. The length of the cell was 42 cm. An elementary computation shows that these lines cannot be assigned to rotational structure of the double molecule since they are too far apart. - I - ' E x : U- INDUCED SPECTRA o f H, 2 2 Induction by electric field J - 0 1 2 3 I N - ' I 6 m x N . . I I I Q R v QP Induction by pressure FIG. 1. It is therefore interesting to compare this structure with that obtained by Crawford and Dagg21 for the induced absorption spectrum of H2 by an electro- static field (fig.1) previously predicted by Condon.22 Using this parallelism we think our structure would be due to the removal of the forbidding of the Q branch by the intermolecular electric field. Our larger width could be explained by the fluctuating character of this intermolecular electric field. On the other hand, the differences between our frequencies and those of Crawford and Dagg, which decrease with increasing J, could be explained by the fact that the higher levels are less and less perturbed by neighbouring molecules. AS a matter of fact, the localization region of the rotator is narrower with respect to angular variables. SPECTRAL EFFECTS IN DENSE POLAR MEDIA In pure HC1,26 compressions below the critical temperature allows a gradual transition from the absorption spectrum of the gas ( P and R branches resolved in rotational lines) to that of the liquid (a broadened structureless band, with a maximum shifted towards smaller frequencies).In order to specify the intermolecular actions in polar media, it seemed of interest to investigate the spectra of mixtures of polar and non-polar gases at various densities of each component. We studied the three following effects which are very different in character.26 INFRA-RED SPECTRA PERTURBATION OF ALLOWED 'FREQUENCIES OF A POLAR GAS These were studied in mixtures containing a small amount of polar gas (for instance 1 Amagat * HCI with N2 up to 500 Amagats) so that each molecule of the polar gas is interacting only with non-polar gas molecules.The fundamental vibration-rotation bands of HCl 27* 28 and HF 29 compressed by H2, 0 2 , He, A, shows, in addition to a progressive disappearance of rotational structure with increased non-polar gas density, an increase of the R branch intensity and an im- portant deformation of the whole band. On the other hand, N2 has a very specific effect on HCI and HF : an intense maximum of absorption appears at the place of the Q-branch forbidden under normal conditions (more exactly, this maximum is slightly shifted towards smaller frequencies (5 to 30 cm-1) particularly with HF. In these cases, the increase of the area I of the band is proportional to the non-polar gas density.With CO as the active gas 28 the rotation-vibration band (non-resolved) is very slightly deformed, i.e. there is a filling of the central minimum and a lowering of P and R branches, the total area of the band remaining nearly constant. An analogous experiment was made on the v3 frequency of CHq.14 The authors attempted an explanation by taking account only of the " backward induction effect ", but they have themselves stressed the approximate character of this model.24 In the same way, in the pure rotation spectra of HBr and HCI, the absorption increases proportionally to the square of pressure, or proportionally to the foreign gas pressure (up to 20 atm).33 ADDITIVE COMBINATIONS FREQUENCIES We have observed the appearance of an absorption band in the region of the sum of the proper frequencies of the two molecules in HCI + €32 mixtures.30 The centre of this band lies near 7050 cm-1 and the sum VHCl + vHZ is 2886 4- 4155 cm-1 =1 7041 cm-1.Fig. 2 shows this combination band (A) and the mutual perturbation of the HC1 second overtone (B) and of the H2 first overtone (C). In analogous conditions this combination band does not appear in HCl + N2 and HCl + 0 2 mixtures (see below). PERTURBATION OF INDUCED FREQUENCIES OF NON-POLAR GAS In the same mixtures HCl + Hz, the dipole moment of the polar molecule is able to induce an absorption of the normally forbidden frequency of a non-polar gas; this is much more important than the one induced by another non-polar gas (N2) by a Kranendonk-type effect. This result 20 is evident for the two follow- ing mixtures of widely different compositions.In H2 at low density (10 Amagats) the induced absorption is not evident; when a relatively large proportion of HC1 (up to 100 Amagats) is added a large absorption at V H ~ (4155 cm-1) is induced In the same way when we add a small quantity of HC1 (6 Amagats) to H2 at 100 Amagats this is sufficient to double the intensity of the H2 induced band although the same quantity of N2 (6 Amagats) gives only small increase (fig. 3 ~ ) . Nevertheless, it has been proved 16 that a gas with a large quadrupole moment (C02) is able to produce an analogous effect. The three effects mentioned above can be justified, qualitatively at least, by a simple molecular rnodeI3Lj2 under the condition that the vibrations of each molecule is not greatly perturbed by the interaction potential.When the density of gases in the mixture is small enough, spectral modifications can be considered * The density in Amagats is the ratio of the gas density under experimental conditions to those under normal conditions. (fig. 3A).R. COULON, L. GALATRY, J. ROBIN AND B . VODAR 27 as due only to pairs of different molecules. In such a pair, the inactive molecule is polarized by the polar one. The moment so induced varies with the field due to the polar molecule and with the polarizability of the non-polar one. It can, therefore, absorb the incident radiation according to the classical theory. The A 1 I E l l FIG. 2. V cm" Y c m-I FIG. 3.-Absorption of H2 in H2 + HC1 mixtures. I pure H2 10 Am.I pure H2 92 Am. 11 99 + HCl 50 Am. 11 99 +N26Am. 111 99 + HCl 110 Am. 111 t s + HC13 Am. IV 9s + HC16 Am. total dipole moment is as a first approximation the expansion limited to the r - 3 term (r = intermolecular distance) of the general expression of the dipole moment of such a pair9 The matrix element according to the vibration- independence assumption, can be written as : + 3 -+ + + fLZF == < #:2t+$ I Ax1 f Bx2 f CxlX;! + D + 1 $q $ r ~ ~ $ " l ), (fiV vibration wave function, (f) rotation wave function).28 INFRA-RED SPECTRA The first three terms of this polynomial show the influence of decreasing inter- molecular distance and therefore of increasing density, on the spectral regions of: (i) the fundamental frequency of the polar molecule, (ii) the fundamental frequency of the non-polar molecule (absorption of a (iii) the sum of the fundamental frequencies of the two molecules.diflerent type from that described by Kranendonk) ; x1 and x2 are the vibration parameters of the two molecules; this shows that the intensities depend on the reduced masses of the molecules. In particular, the ratio of the intensities of combination bands in HCl + H2 and HCl + N2 mixtures is ~ N ~ V N ~ / ~ H , V H , - 8, which explains the missing combinations frequency for a HC1+ N2 mixture in our experimental conditions. More detailed analysis of the D term allows one to predict moreover some possible features in the pure rotation frequencies range 49 (when rotations can also be considered as independent). An analogous model, but including the quadrupole moment of the non-polar molecule, has been used by Fahrenfort,l6 who has given some numerical calculations for C02 + foreign gas mixtures.It is worth remembering that conclusions following from this model may be wrong if we have sufficiently strong directive electrostatic potential, similar to a patential barrier during collisions. The possibility of a specific electronic inter- action (quasi-chemical) is also possible for vigorous collision. It is perhaps what happens in HCl + N2 or HF 4- N2 mixtures, in which the central maximum of the perturbed polar gas band would be due to an electronic rearrangement, the polar molecule taking a d i k e structure in the collision complex. However, this may be the case if the angular potential barrier is high enough to stop the molecular rotation during collision, so that the system can absorb near the pure vibrational frequency of the polar molecule.3 CONCLUSION The purpose of spectroscopic reasearch on compressed gases is to establish a consistent model of molecular interaction ; this model should be able to connect spectral data of compressed gases with those of liquids and solutions. Unfor- tunately the experimental conditions are rarely propitious, the gases under investigation having a low critical temperature. Nevertheless, symmetric diatomic substances (N2, 0 2 , H2) in the liquid state show the same fundamental band as in the gaseous compressed state.W37,38 In liquids, too, Ketelaar aiid Hooge 3 9 ~ 4 8 have observed combination frequencies for various mixtures (e.g.Br2 + CS2). These authors think that it is a general type of absorption.40 However. the greatest interest lies in studies of the modi- fications of vibrational frequencies of various dissolved substances with the nature of the solvents. Melle Josien and co-workers systematically studied the shift and the shape of the fundamental band of NH,41 C = O,42 HC1,43 HBr,M HI 44 vibrators in various solvents. They stress the change in shape of the HCl funda- mental band in solvents like C6H6 on the one hand, and CCl4 on the other. It is evident that there is also, in liquid mixtures, a specific effect of the perturbing molecules. It seems evident, however, that the study of infra-red or Ranian spectra of one substance dissolved in a compressed gas or in a liquid solvent cannot lead to exactly similar absorption profiles.The difference between the dynamic states of the active molecules must indeed be reflected in the shape of the spectrum. In fact, the collision being stronger in compressed gases than in liquids, theR . COULON, L. GALATRY, J . ROBIN AND B . VODAR 29 minimum mutual distances are smaller and it is possible that, even for non-polar gaseous solvents, the validity of the Kirkwood-Bauer-Magat relation 35 (neighbour- hood similar to a continuous polarizable dielectric) is greatly reduced. The comparison of these different spectra should be able to give some informa- tion about the state of motion of dissolved molecules 45,46 47 both in compressed gases and in liquids. 1 Margenau, Physic. Rev., 1951,82, 156.2 see review by Robin, Robin, J. Phys. Radium, 1956, 17, 143. 3 Van Kranendonk, 7hesis (Amsterdam, 1952). 4 Mulliken, J. Amer. Chem. Soc., 1952,74,811. 5 Collin and D’Or, J. Chem. Physics, 1955, 23, 397. 6 Crawford, Welsh and Locke, Physic. Rev., 1949,75,1607. 7 Welsh, Crawford, McDonald and Chisholm, Physic. Rew., 1951,83, 1264. 8 Crawford, Welsh and Locke, Physic. Rev., 1949,76,580. 9 Crawford, Welsh, McDonald and Locke, Physic. Rev., 1950, 80,469. 10 Chisholm, McDonald, Crawford and Welsh, Physic. Rev., 1952,88,957. 11 Chisholm and Welsh, Can. J. Physics, 1954,32, 291. 12 Coulon, Oksengorn, Robin and Vodar, J. Phys. Radium, 1953,14, 63. 13 Ketelaar, Colpa and Hooge, J. Chem. Physics, 1955,23,413. 14 Welsh, Pashler and Dunn, J. Chem. Physics, 1951, 19, 340.15 Coulon, Oksengorn and Robin, J. Phys. Radium, 1953, 14, 347. 16 Fahrenfort, Thesis (Amsterdam, 1955). 17 Birnbaum, Maryott and Wacker, J. Chem. Physics, 1954,22,1782. 18 Van Kranendonk and Bird, Physica, 1951,17,953. 19 Van Kranendonk and Bird, Physica, 1951,17,968. 20 Coulon, Galatry, Robin and Vodar, J. Phys. Radium, 1955, 16,728. 21 Crawford and Dagg, Physic. Rew., 1953,91,1569. 22 Condon, Physic. Rev., 1932, 41, 759. 23 Mizushima, Physic. Rm., 1949, 76, 1268 ; 77, 150. 24 Welsh and Sandiford, J. Chem. Physics, 1952, 20, 1646. 25 Fahrenfort and Ketelaar, J. Chem. Physics, 1954, 22, 1631. 26 West, J. Chem. Physics, 1939, 7, 795. 27 Coulon, Oksengorn, Robin and Vodar, Compt. rend., 1953,236,1481. 28 Coulon, Galatry, Oksengorn, Robin and Vodar, J. Phys. Radium, 1954,15, 58,641. 29 Coulon, Oksengorn and Vodar, Compt. rend., 1954,239,964. 30 Coulon, Robin and Vodar, Compt. rend., 1955,240,956. 31 Galatry and Vodar, Compt. rend., 1955,240, 1072. 32 Galatry, Dipldme #Etudes Supbrieures (Paris, 1955). 33 Birnbaum and Maryott, Physic. Rev., 1954,95,622. 34 Kiyama, Minomura and Ozawa, Proc. Japan Acad., 1954,30,758. 35 Bauer and Magat, J. Phys. Radium, 1938,7, 319. 36 Lee Smith, Keller and Johnston, Physic. Rev., 1950,79,728. 37 Allin, Hare and McDonald, Physic. Rev., 1955, 98, 554. 38 Hare, Allin and Welsh, Physic. Rev., 1955, 99, 1887. 39 Ketelaar and Hooge, J. Chem. Physics, 1955, 23, 749. 40 Ketelaar and Hooge, J. Chem. Physics, 1955,23, 1549. 41 Josien and Fuson, J. Chem. Physics, 1954,22, 1169,1264. 42 Josien and Lascombe, J. Chem. Physics, 1955,52, 162. 43 Josien and Sourisseau, Bull. SOC. Chim., 1955, 178. 44 Josien, Sourisseau and Castinel, Bull. SOC. Chim., 1955, 1539. 45 Crawford, Welsh and Harrold, Can. J. Physics, 1952,30, 81. 46 Brandmuller, 2. Physik, 1955,40, 75. 47 Breneman and Williams, Physic. Rev., 1953, 91,465. 48 Hooge, Thesis (Amsterdam, 1956). 49 Galatry and Vodar, Compr. rend., 1956, 242, 1871. 50 Buckingham and Pople, Trans. Faraaby SOC., 1955,51, 1173.

ISSN:0366-9033    

DOI:10.1039/DF9562200022

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

PERTURBATION OF ELECTRONIC RESONANCE SPECTRA BY FOREIGN GASES UNDER HIGH PRESSURE BY J. ROBIN, R. BERGEON, L. GALATRY AND B. VODAR Laboratoire des Hautes Pressions, Bellevue (S. and O.), France Received, 2nd July, 1956 Absorption spectra of resonance lines of the alkali metals, Hg and some other metals have been observed in foreign gases at pressures up to 1200atm and higher, and at temperatures up to 1200°C. By introducing an interaction potential of the Lennard- Jones type in the Margenau statistical theory, calculations of lines shapes and shifts have been made using an electronic computer. Qualitatively, the role of the repulsive forces is shown directly in some cases by the observed reversal of the direction of the shift (from red to blue). Quantitatively the agreement between theoretical and experimental values is fairly good, with only one arbitrary parameter (the coefficient of the repulsive term) ; in addition, some information about the interaction with excited states may be obtained from these measurements.An accurate comparison of the line shapes gives some data on the radial distribution function in the compressed gases. In addition to the perturbed normal resonance frequencies, other satellite bands have been observed, which become very intense at the highest pressures ; a survey of the data concerning these bands is presented, and their origin is tentatively ascribed to “non- adiabatic transitions ”. The influence of pressure on the breadth and position of spectral lines has been known for a long time.1 Natural and Doppler breadths, independent of neighbouring atoms or molecules soon become negligible compared with pressure effects.When only short-range interaction forces are considered, resonance forces derived from a potential proportional to r-3 must be taken into account for a pure substance where each absorbing atom is perturbed by similar ones (self-broadening). But when the absorbing atom is surrounded by neutral molecules of a different species, the perturbing potential is proportional to r-6. At low pressure, observed broadenings and shifts are nearly proportional to the pressure of the perturbing gas, in agreement with “collision theories”. At higher pressures, Margenau’s “ statistical theory ” with a perturbing potential proportional to r-6 leads to broadening and shift both proportional to the square of the compressed gas density.But, at pressures above 100 atm, shifts are not quadratic functions of the density nor are they proportional to the broadening. We have therefore thought it useful to investigate the perturbation of the spectral lines of metals by foreign gases up to pressures much higher than those at which many other authors have been working, i.e. pressures of the order of 100 atm, and which are outside the range of validity of the available theories. BY extending our pressure range to 1500, and more recently to 6000 atm, we observed some new phenomena. For the interpretation of the new results, it is necessary to treat by more refined theories the change in the properties of atoms and mole- cules which are forced by high pressure to be very near together.Our experimental studies are concerned essentially with absorption spectra. Perturbations of the resonance line or second doublet of the principal series have been investigated for mercury, and for alkali metals and thallium, with compressed hydrogen, nitrogen, argon or helium as the perturbing gas.2 The metal was deposited in a small glass or quartz absorption cell, which was itself placed in a metal gas-tight vessel, having quartz or sapphire windows with faces perpendicular 30J . ROBIN, R . BERGEON, L. GALATRY AND B. VODAR 31 to the principal axis. Optical contact insured their being leak-proof at high pres- sures. The foreign gas had to be very pure; therefore it was compressed by heating at constant volume after previous cooling (H2 and He), or by liquefying the gas (A and N2) in a steel cylinder.In order to reach 6OOO atm A and N2 had to be solidified in the cooled cyl- inder so as to take advantage of the volume contraction on freezing.3 To get sufficiently high metallic vapour densities in the absorption cell, it was necessary to work at various temperatures, depending on the nature of the metal and the pressure, viz., 250" to 400" C for alkali metals, 90" to 160" for mercury and 900" to 1200°C for thallium. Below 400" C, the metal pressure vessel could be heated extern- ally, but for higher tempera- tures we devised an internally heated pressure bomb.4 The observed shifts are plotted in fig. 1, 2, 3 as a function of pressure and den- sity of the foreign gas.Except for rubidium the two com- ponents of the alkali doublets merge into a single non- resolved band, so that we could measure only the shift and broadening of the whole. This measurement is nevertheless meaningful due to the marked broadening observed at very high pressures. It is obvious that, above about 50 atm, the shift of the maxima is not a quadratic function of the den- sity. The tendency is rather to FIG. 1.-Shifts of the 1st doublet of Rb resonance lines perturbed by (a) argon, (b) nitrogen, and (c) helium. be-a linear function with, in some cases, a maximum, corresponding to a reversal of the shift (fig. 1). Observed wave-length shifts may be several hundreds kngstroms. The determination of broadening could be made over the whole pressure range only in a few cases, because of the appearance of diffuse bands at the edge of the broadened line (these bands will be dealt with later).Fig. 4 shows a plot of the half-width for Na + A and Hg + H2 and fig. 5 the broadening and shift induced by compressed argon on the second doublet of the rubidium principal series. At a pressure of 10oO atm, the shift is 275A and the half-width about 130 A or 650 cm-1. We have also observed that, as long as no new band appeared, high-pressure broadened lines are nearly symmetrical in shape, whereas they are generally less so at lower pressures. Fig. 6 illustrates this for the rubidium second doublet perturbed by argon and for the mercury line at 2537A perturbed by hydrogen. As mentioned above, it was not possible to measure broadenings at very high pressures in many cases, because of the appearance of diffuse bands in the neigh- bourhood of the line under investigation.32 ELECTRONIC RESON ANCE SPECTRA FIG.2.-Shifts of the 25365A mercury line perturbed by (a) argon, (6) nitrogen, (c) helium and (d) hydrogen. Curves (b) and (c) are limited because of the satellite band near the band under in- vestigation. FIG. 3.-Shift of the second doublet of Rb principal series (42155 and 4201.8 A) perturbed by argon. Above 50 Arnagats. the two lines merge into one band. 0 FIG. 4.Half-widths : (a) Na reson- ance lines (merged) perturbed by argon; (b) 2536.5 A Hg line per- turbed by hydrogen.FIG. 5.-Rb second doublet perturbed by argon ; spectra from 1 to 1000 atm illustrating the important broadenings and shifts (300 A).[To face puge 32.J . ROBIN, R . BERGEON, L. GALATRY A N D B . VODAR 33 Many authors have in fact observed that pressureinduced bands appear 195 near some nietal lines. In our absorption spectroscopic studies one band gener- ally appears on the short wave-length side of the main line at pressures below 1200 atm.2 In several cases, it was observed that the intensity of this band was enhanced with respect to the main line when the density of the perturbing gas increased; there is evidence also, in some cases, of a shift of these diffuse bands with varying pressure. For K + A mixtures, we have observed 2 bands near 7380 and 7670 A. I I b FIG. 6.-(a) Contour of 2536.5A Hg line perturbed by hydrogen at 120, 520 and 1400 atm; (b) contour of Rb second doublet perturbed by argon at 70 and lo00 atm.FIG. 7.Cuccessive aspects of the line R (25365A) and its two satellite bands S1 and S2 as pressure increases from 10 to 6,000 atm. For Hg perturbed by A and N2 (fig. 7) a second satellite band appears at about loo0 atm.5 The main line R becomes merged in the first band S1, then a second one S 2 appears and its intensity increases more quickly than that of S1. Both bands are shifted to the shorter wave-lengths as shown in fig. 8. In an attempt to explain theoretically the observed broadening and shifts we require a simple model of perturbation which is likely to give a first approximation B34 ELECTRONIC RESONANCE SPECTRA to the observed dependence of shift and broadening on density. A theory of the “statistical” type was therefore considered.The main assumption is that the field acting on the absorbing atom varies sufficiently slowly compared with the frequency vo of the absorbed radiation ; the field due to the surrounding molecules being then quasi-static, we can assume that the intensity distribution I(v)dv is proportional to the statistical weight of the configuration which gives rise to a frequency perturbation between v - vo and v - v0 + dv. It is further assumed that the energy levels of the absorbing atom are perturbed by an amount equal to the potential energy exerted by the surrounding molecules. The energy jumps in transitions are then perturbed by a function of the configuration only. Such a concept was the starting point of Margenau’s statistical theory.;’ P I<q/cm2 d A FIG.8.-Shifts of R, S1 and S2 as a function of pressure and density. The perturbing gases are argon (full m e ) and nitrogen (dashed curve). Margenau assumed that the lower and higher state of the absorbing atom interacted with the surrounding molecules according to a London-type law, viz., A v = v - v ~ = - ult-6. (1) In this case, a simple expression can be derived for the line shape, and the breadth and displacement of the maximum are found to be proportional to the square of the density. In order to introduce short-range forces between the absorbing atom and per- turbing molecules we write for the perturbing energy (2) E(r) = pr-12 - ur-4 The first term is intended to represent the difference between the repulsive potential to which lower and excited states of the atom are submitted.With the preceding assumption, the intensity of the line I(v)dv will be pro- portional to the probability of occurrence of configurations consistent with the inequalities N 1-1 v-*dv< z(Wj-12-prj-6) &v+*:dv. (3) The sum is extended to all the perturbing molecules surrounding the atom, i.e. the repulsive terms are treated as if they were additive, or, in other words, independent of the configuration and degree of packing of the molecules. ThisJ . ROBIN, R . BERGEON, L. GALATRY A N D B . VODAR 35 is not true for the very close-packing at very high pressures so that Pr-12 must be considered as giving only the general trend of the actual perturbation law. Finally, assuming that molecules are uniformly distributed around the atom, i.e.that the probability of their being at a distance between r and r + dr is p(r)dr = 4+dr/V, the intensity distribution of N perturbing molecules is the integral extending over that part of the N space for which inequality (3) is satisfied. Markoff’s analysis of random processes allows a determination of I(v), giving I N ( v ) = - s” exp (- pwN(p)dp ; 2n -a-J AN@), the Fourier transform of IN@), is then directly related to physical data (probability p(r) and perturbation) by Letting N go to infinity and performing some transformations we get finally (4) I(q) = Srn exp [- 8A’(x) cos [GB‘(x) + 7x21” dx, 0 where we have introduced for simplicity the following symbols : 6 =: (1~/3)(/3/a))q (a reduced density), q = (/l/a2)v (a reduced frequency). ( 5 ) (6) The variable x is related to p and A’ + iB’ to A N @ ) .This last quantity can be obtained in the form of a power series in x and tabulated. The integral (4) had to be evaluated on a I.B.M. punched card calculating machine. With the reduced variable 8 and 7 (cf. eqn. (5) and (6)) the calculated line shape is the same for any set of values of cc and p which are speciric to each pair consisting of an interacting atom and molecule. The maximum of the band has been determined for reduced gas densities up to 6 = 1.5 (350 Amagats for Na + A). At larger densities, the line shape is too asymmetrical and it is meaning- less to compare the location of the maximum with that of the more symmetric observed line. In fig. 9 we compare the calculated shift with that observed for Na atoms perturbed by argon.8 Here we adopted for a the value 1.9 x 10-42 cm-1 cm6 obtained by Lindholm from the collision shift at low pressures.9 With the value /3 = 3.6 x 10-82 a satisfactory agreement could be achieved for densities up to 350 Amagats.The line breadth, with the same values is given in fig. 9 ; for the higher pressures, the calculated breadth is somewhat too low. This discrepancy corresponds to the excessive asymmetry of calculated line shapes (fig. 10) which have a too steep “ red ” wing. Nevertheless these calculated lines have the same property as some observed lines : the shift is reversed at a certain (high) pressure and is afterwards directed toward short wave-lengths. Our simple model leads to fairly good agreement with experiment for the broadening and shift but fails at too high pressures.Its advantage is to allow a comparison of statistical theory with experimental results by introducing a n = molecular density (molecules/cm3), and36 ELECTRONIC RESONANCE SPECTRA repulsive term into the perturbing potential. This term, though it cannot account accurately for the true short-range forces of the excited state, gives a reasonab1e Picture of the trough of the perturbing potential ~ ( r ) (the ordinate difference between the two curves of fig. 11). 0 , L I 1 100 2 0 0 FIG. 9.-Calculated and observed shifts and half-widths as functions of foreign gas density for Na lines perturbed by argon; shifts on the left, broadenins on the right. Circles are experimental values; 8 dashed curves represent the quadratic relationship;' full curves give results of calculations for a 6 : 12-perturbation model.-' P - 4 0 ' FIG. 10.-Calculated and observed contours : Na lines perturbed by argon. Full curves 1, 3 and 4 are calculated contours for reduced densities 6 = 0.8, 1 and 2.2 respectively. The dashed curve 2 gives for comparison the measured contour for 6 = 0-932. As regards satellite bands, a detailed explanation would require some know- ledge of the relation between the bands mentioned above and those observed at low foreign gas pressures (a few flltn of mercury) which have considerable intensities.11 Their temperature dependence should also be known. As suchJ . ROBIN, R . BERGEON, L. GALATRY AND B. VODAR 37 information is as yet lacking, we shall only draw attention to some plausible explanations.So far, the authors 1 2 ~ 1 3 have considered, especially for low pressures, that the Franck-Condon principle applied to transitions between the potential energy curves of the two states concerned in this transition. But it must be noticed that the rigorous application of this principle to a system of atoms is suspect because of our ignorance of the potential energy curve of the excited state. This leaves us little hope of verifying qualitatively this model for all observed cases. Moreover, this model neglects the short-range interaction of electronic clouds during a collision, which interaction could give rise, as shown by Kuhn and Oldenberg13 to a splitting of the upper potential energy curve.This splitting can account for the possible existence of two satellite bands. This could lead to a generalization of the correlation diagrams of diatomic molecules to the case of two atoms. I I I FIG. 11 .-Interaction potentials Na + A : e w e I : Na in the normal state (1s) ; curve II : On the other hand, one might think that the frequency of these bands would be equal to the frequency of the transition in the statistically perturbed atom, plus an amount that would correspond to a change in the relative kinetic energy of the two atoms. Moreover, this energy exchange would take place about a preferred value related to the mass of the interacting atoms. As a matter of fact the optical transition of an electron in an alkali metal is accompanied by an expansion of the localization region, which, from a naive comparison would correspond to a force tending to pull the atoms apart.The energy necessary for the Franck-Condon principle strictly to apply is then very high, and more so as higher terms of the principal series are considered. As for two non-bonded atoms the electron shells are more affected as they come nearer to each other than for a diatomic molecule, it is seen that J #*qdq is probably more dependent on the internuclear distance, which suggests a departure from the Frank-Condon principle in this case. By taking into account the translational wave functions in the formulation, one introduces a dynamical point of view which is perhaps related to the fact that the distance between the main line and the satellite band is roughly propor- tional to p-* ( p = reduced mass).l4 Finally, this model, if it proves to be justified could be partially retained for high pressures, since the transitions under consideration are of the s-p type and correspond to an increased localization of the electron in one direction only. The perturbations in other directions are of little interest here, so we are led once more to the simple uni-dimensional case dealt with above. excited state (2P). Zero energy is that of the free atom for each curve.38 ELECTRONIC RESONANCE SPECTRA 1 See review by Robin, S., and Robin, S., J. Phys. Radium, 1956,17,143. 2 Robin, J., Robin, S, and Vodar, Compf. rend., 1951, 232, 1754 ; 1951, 233, 928 ; 3 Robin, J. and Vodar, J. Phys. Radium, 1956,17,500. 4 Robin, J. and Robin, S., J. Phys. Radium, 1956,17,499. 5 Ch’En, Bennet and Jehenko, J. Opt. SOC. Amer., 1956,46, 182. 4 Robin, J. and Vodar, Compt. rend., 1956,242,2330. 7 Margenau, Physic. Rev., 1935,48,755 and 1951,82,156. 8 Robin, S., Thesis (Paris, 1951). 9 Lindholm, Diss. (Uppsala, 1942). 10 Bergeon, Robin, S. and Vodar, Compt. rend., 1952,235, 360 ; 1954,238,2507 and 11 Preston, Physic. Rev., 1937, 51, 298. 12 Oldenberg, 2. Physik, 1929, 55, 1. 13 Kuhn and Oldenberg, Physic. Rev., 1932, 41, 72. 14 Robin, S., private communication ; see also Krefft and Rompe, 2. Physik, 1932,73, 1951,233,1019 and 1954,238,1954. 1955 240, 172. 681.

ISSN:0366-9033    

DOI:10.1039/DF9562200030

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

THE EFFEXT OF PRESSURE ON THE VIBRATIONAL FREQUENCY OF BONDS CONTAINING HYDROGEN * BY A. M. BENSON AND H. G. DRICKAMER Dept. of Chemistry and Chemical Engineering, University of Illinois, Urbana, Illinois Received 30th April, 1956 Using a technique described earlier, measurements have been of the effect of pressure on the vibrational frequency of a series of chemical bonds mostly containing hydrogen. The systems were studied in CS2 and CFC13. The shifts vary from the “red shifts” resulting from van der Waals’ forces in OH to a definite “ blue shift ” for the antisym- metric CH2 vibration. In general the repulsive forces are effective at lower relative densi- ties in CFCI3 than in -2. While the data are not yet extensive, certain regularities are noted which should have theoretical significance.A possible application of the pressure coefficient of the frequency to the identification of molecular structure is noted. In previous papersl.2 a method for studying infra-red spectra in solutions under pressures to 12,000 atm, has been described and applied to the OH stretching vibration of n-butanol in a series of solvents. A “ red-shift ” (to lower frequencies) was obtained which varied approximately as p2/r6 and which was roughly propor- tional to the polarkability of the most polarizable bond in the solvent. This shift seems clearly due to the attractive interaction between the solvent, or at least the more polarizable bonds of the solvent, and the hydrogen in the OH bond. At some density the controlling interaction must become repulsion rather than attraction.Since this critical density should be different for different bonds, we have extended our investigation to include a series of other bonds mostly involving hydrogen. The apparatus used was essentially the same as described in the previous papers. The spectrometer was a Perkin-Elmer single-beam double-pass type, employing LiF optics. The systems studied are shown below. TABLE 1 bond molecule solvent concentration vibration NH SH CH CH CH2 CH2 OH OH CF C6H5NH2 n-C3H7SH CHC13 CHCl3 CH2CI2 CHzCl;! n-C4HgOH CfjH5OH CFCl3 CFC13 -2 cs2 CFCI3 cs2 CFCl3 CFCl3 a 2 CFC13 0 4 % vNH stretch (s and a) 5 % vSH stretch 3 %,5 % vCH stretch vCH stretch 6 % 5 % vCH stretch (s and a) 5 % vCH stretch (s and a) 0 7 % vOH stretch 0 6 mg/d vOH stretch 100 % 3vCF stretch The chemicals used were generally the best available commercially, in some cases purified further by distillation.The runs we= made in a room maintained at constant temperature and each point was re-run 3 or 4 times and the average shift taken. While the accuracy vaned with the dispersion and with other factors, the shifts were reproducible generally to a fraction of a wave number. * This work was supported in part by the U.S. Atomic Energy Commission. 3940 VIBRATIONAL FREQUENCY OF BONDS No density data were available for CFC13 as a function of pressure. We have made some crude p , w, t measurements at 24-25’ C and supplemented these 1 I I 3 9 CF s t r e t c h I I 1 1.3 -201 I *o 1 . 1 1.2 FIG. 1. from Bridgman’s data on CCl4 and CHBr3. The relative den- sities used in the plots were: TABLE 2.-RELATIVE DENSITY OF CFC13 P (atm) P I P 0 1 l*OOo 500 1.053 1 ,OOo 1.092 ~ , O O o 1.151 4,000 1.215 6YOOo 1.255 8YOOo 1.285 10,OOo (extrapolated) 1.305 The results are shown in fig.1-6. A really satisfactory theoretical analysis was not possible before the deadline for papers, but the salient features will be indicated. From fig. 1 and 2 it can be seen that the CF and OH stretch- ing frequencies shift to lower frequencies (“ red shift ”) with a roughly linear density dependence. A somewhat better correlation is obtained by plotting against p2. If only inter- action with nearest neighbours is considered a p2/r6 dependence would be pre- dicted. If the surroundings are treated as a continuum, a pJr3 dependence is predicted.The experimental results lie between, but nearer the former case. The magnitude of the shift for n-C4HgOH in CFC13 is consistent with the polar- izability of the CCl bond (see Fishman and Drickamer 1). The red shift for phenol in CS2 is very close to that obtained previously for n-C4HgOH and CH30H in CS2 for all except the two highest pressures. This is apparently due to phenol freezing out at these pressures. In general for these dilute solutions there seems to be very little effect of the size of the group attached to the OH, although there was some indication in our earlier data for t-C4HgOH of shielding of the OH by the adjacent CH3 groups. It is relatively easy to show that the magnitude of the shift should be propor- tional to the number of quantum jumps between the initial and the final state, i.e.to the number of the overtone. Since we measured the second overtone (34 for CF, the fundamental would show a maximum shift of about five wave numbers at the highest density. This is consistent with the relatively weak attractive forces in fluorocarbons, as evidenced by low cohesive energy and large molar volume. In fig. 3-5 we show the shifts with pressure of the CH and CH2 stretching vibrations for CHCl3 and CH2Cl2 in CS2 and in CFC13. Here, for the first time, one obtains “blue shifts”, i.e. to higher frequency, with increasing density at pressures well below 10,OOO atm. The CH stretching vibration in CHCl3 and the symmetrical CH2 stretching vibration in CH2C12 behave in a qualitatively similar manner.Both are of the same symmetry type (A1). In CS2 they both give a red shift throughout the entire pressure range. In CFCl3 both show a small red shift at low pressures followed by a small but distinct blue shift at high density. Apparently theA . M . BENSON AND H. G . DRXCKAMER 41 repulsive forces become significant at lower relative densities in CFC13 than in CS2. Of course, since the molar volumes and other parameters are not identical for the two solvents at 1 atm, we are not measuring our relative densities from the same fiducial conditions for the two solvents. I C 5 A 3 (cm": C - 5 -- - - 0 . 7 5 % n BuOH i n CSZ 0 7 5 x M e O H in CS, 1.0 1.1 1.2 1.3 (f 1 Po FIG. 2. C H s t t t t c h ( a n t . ) CH, CI,42 VIBRATIONAL FREQUENCY OF BONDS The antisymmetric (type Bj) CH2 vibration in CH2Cl2 shows a greater im- portance for repulsive forces at any given density than does the symmetric vibra- tion.This is true in both solvents. It is interesting to note that the difference between the shifts for the symmetric and antisymmetric vibrations are nearly the same at the same relative density in each solvent. For this vibration also repulsive effects are more important in CFC13 than in CS2. FIG. 5. It should be noted that the CF stretching vibration shows no apparent effect of repulsive forces even at the highest pressures, whereas the CH stretch in CHC13 shows a distinct blue shift. One might expect that the electrons clustered on the fluorine would lead to repulsion at relatively low densities. Apparently the attractive potential for CF is rather shallow, but the repulsive potential rises steeply only at quite small intermolecular distances.A. M.BENSON AND H. G . DRICKAMER 43 Fig. 6 shows the results for the NH2 vibrations of aniline in CFC13 and for the SH vibration of n-C3H7SH in CS2. The N H 2 vibrations show somewhat less effect of repulsion in the pressure range than do the CH2 vibrations. As might be expected, the effect lies between that for OH and for CH2. It may be noted that there is no significant difference between the symmetric and antisymmetric N H 2 vibrations such as was found for CH2. This point requires further in- vestigation. 3 SH stretch 5X.n-PrSH i n CS2 --I O - 1 - ' / - - 2 - V N H stretch(ant.) - 0-4%@-NH inCFC13 I ! I I - 2 ~ N H stretch (sym.) 0.4% @-NH i n C F C l j The SH stretching vibration shows the effect of repulsion at somewhat lower pressures than does the OH, as might possibly be expected. The results obtained to date raise several interesting theoretical questions which we plan to treat, along with more extensive results, in a future paper, One interesting possible application of the work 3 lies in the field of identification of molecular structure. If each vibration has not only a characteristic frequency but also a characteristic pressure coefficient, by running spectra at several pressures, it might be possible to resolve many ambiguities which arise in the usual 1-atm spectrum. This work was supported in part by the U.S. Atomic Energy Commission. A. M. Benson would like to acknowledge a fellowship from the National Science Foundation. 1 Fishman and Drickamer, J. Chem. Physics, in press. 2 Fishman and Drickamer, Anal. Chem., May, 1956. 3 Douglas Applequist, private communication.

ISSN:0366-9033    

DOI:10.1039/DF9562200039

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

PRESSURE-INDUCED METALLIC TRANSITIONS IN INSULATORS BY B. J. ALDER AND R. H. CHRISTIAN University of California, Radiation Laboratory, Livermore, California Received 17th October, 1956 Experimental evidence is presented that several ionic and molecular crystals change their conductivity into the range of metallic conductivities when subjected to a pressure of about 250,000 atm. For some of these substances the pressures at which this transition occurs is roughly defined. It is known from qualitative theoretical arguments that all materials will exhibit metallic behaviour when subjected to a sufficiently high pressure. For systems under extreme pressures, for example, as found in astrophysical situations, the Thomas-Fermi model is applied. This model, in which all the electrons are assumed to be statistically placed about the nucleus, would exhibit metallic con- ductivity.For hydrogen a rough quantum-mechanical calculation has been carried out 1 which predicts that a pressure in the neighbourhood of 5 x 105 atm would be needed to produce the metallic state. Such pressures would be very diffcult to reach by static means ; however, the recent development of techniques utilizing high explosive systems permit measurements to be made at these pres- sures.2s3 Although the shock wave generated by these explosive systems keeps the substance at these high pressures for only microseconds, this time is quite long compared to times required for the establishment of equilibrium for electrons. Static experiments would be possible for those materials that conduct below approximately 100,OOO atm.Indeed such measurements would be preferable since they can be carried out under more controllable conditions. In fact, some such static experiments are in progress by Dr. D. T. Griggs, W. C. McMillan and C. P. Nash at the University of California at Los Angeles having been started independent and previous to this work. Their results should soon be published. THEORETICAL CONSIDERATIONS A treatment of an electron in a box, or a hydrogen atom in a box 4 has shown that compression has the effect of raising the energy levels of the electron. In fact, when the size of the cage is reduced to 1.835 Bohr radii, the kinetic energy of the electron of the hydrogen atom is greater than the attractive energy of the central positive charge. The pressure required for this is about 6.3 x 106 atm and the system would then certainly be metallic.However, in this model the walls of the box are infinite and prevent the escape of the electron. In solid materials the walls represent the repulsive interaction of neighbouring atoms and the system has to be considered in its entirety, the energy levels being replaced by energy bands. Compression in the region where repulsive forces are dominant again has the effect of raising the bands and the further effect of broadening them. Metallic conduction will then be achieved when the highest filled band of an in- sulator overlaps with an unoccupied higher band (the conduction band). This must occur before pressure ionization takes place, that is, at a pressure lower than that necessary to overcome the central coulomb attraction for an outer electron of an atom.The choice of insulators that might conduct at the lowest possible pressures 44B. f . ALDER AND R. H. CHRISTIAN 45 hence is governed by the criteria that they must have an empty band not far removed from the highest filled band. This certainly occurs when an atom has a loosely bound electron. The hydrogen atom would be a poor choice since the 2p energy level is about lOeV removed from the 1s level in the free atom. The materials that have been found in this work to give indication of metallic con- duction are believed to have an empty band less than 5 eV above the occupied band. Other substances whose bands were separated by a larger energy were also tested and they did not conduct at the highest pressures used so far.These substances were used to establish the validity of the experimental procedure. EXPERIMENTAL METHOD The pressure was created by a shock wave generated by a high explosive system similar to one previously used.2 This assembly induced a strong plane shock into a series of plates in contact with the high explosive. This series of plates consists of three aluminium plates, individually grounded, and separated by two Teflon plates. This is believed to produce an uncharged shock at the samples although it might not be necessary to use such precautions. Each sample was pressed against the last plate by spring-loaded pins. Several ar- rangements of these pins have been used, all yielding the same results.In one case two pins were merely pressed against the surface of the crystal.3 One of these pins was charged to 300V, while the other was grounded. These pins constituted part of an appropriate r.c. circuit. A " raster " type oscilloscope was attached across a resistor of this circuit, so that any discharge could be observed and photographed. The shape of the signal indicates roughly the resistance of the sample. In the second type of arrangement the pins were buried half-way into the crystal. In the third case only the charged pin was inserted half-way into the crystal. The aluminium plate on which the crystal rested was substituted for the grounded pin, so that the resistance was measured in the perpendicular direction to the previous cases. EXPERIMENTAL CONSIDERATIONS The pressure was determined from the known equation of state of aluminium2 by means of an accurate measurement of the velocity of the aluminium surface.This was accomplished by a group of seven pins, spaced at 1-5 mm intervals away from the aluminium plate. From a knowledge of the pressure in the aluminium, the pressure in the crystal itself can be estimated from its initial density and measured shock velocity. A shock velocity was obtained by measuring the time difference between the signal from the pins in the crystal and the closure of an additional pin located beside the base of the sample and 0.2mm away from the surface of the aluminium plate. The shock velocity obtained from this transit time must be consistent with one predicted from com- pressibility data in order to interpret the data as implying metallic conduction.It some- times happened that the pins in non-conducting material discharged also, which might be due to the arrival of the aluminium plate itself at the position of the pins. However, since this occurs considerably later the timing of this phenomenon allows a clear dis- tinction to be made between metallic behaviour and spurious discharge. At present no method is available for measuring the temperature of the sample during the time the shock passes through, however, an estimate 2 of it can be made. For most substances considered here the temperatures will range between 1OOO" and 2000" C. As the shock reflects off the free surface the sample will be adiabatically cooled by the rare- faction wave to somewhere near 600" C and the pressure resulting from the propagation of the shock in the gas phase surrounding the sample would maintain a pressure of about 500 atm in the crystal.Although these ionic crystals have melting points in the region of 600 to 700" C and quite high conductivities 5 in the molten state under atmospheric conditions, the experiments with buried pins rule out the possibility that ionic conduc- tivity discharged the pins. This is because the conductivity with buried pins is certainly measured while the crystal is under high pressure, where melting occurs at much higher temperatures and ionic mobilities are quite low. An approximate calculation with the aid of the Clapeyron equation shows that at 2000°C the ionic crystals are solid above 50,000 atm.46 METALLIC TRANSITIONS RESULTS InitialIy all samples had a resistance greater than 108 ohms, except red phosphorous whose resistance was about 5 x 106 ohms.The materials that showed a resistance of Iess than 100 ohms, the minimum observable, at a pressure in the aluminium plate of 280,000 atm, were a single crystal of CsI, and the following powders which were com- pressed into pellets to densities near that of the crystal; commercial-grade 12, red phos- phorous, LiAlfi and RbI. Furthermore, a powdered pellet of CsI seemed to behave simifarly to a single crystal. The materials which definitely did not conduct at this pres- sure were Teflon and pressed NaCI. Several substances had resistances of the order of 100 to lo00 ohms which might mean that they are in the transition region from an in- sulator to a conductor at this pressure ; that is, the bands were approaching each other within thermal energies.RbT already gave some indication of having a resistance of the order of 100 ohms while KI definitely did. CsBr and particularly CsCl had resistances of the order of 1000 ohms. When the pressure in the aluminium plate was about 100,000 atm only phosphorous of the above materials conducted. It seems likely that the metallic transition observed is the same one, namely the one to black phosphorous, that was observed by Bridgman 6 at about 40,000 atm. At 150,OOO atm in the aluminium plate, which corresponds to 50,000 atm in the crystal, LiAIH4 conducted but appeared to have a resistance of the order of 1000 ohms.It should be mentioned that the L m might have contained a fair amount of metallic aluminium as an impurity, as evidenced by its grayish appearance. However, if these results are reliable, the transition for LiAlI& would be in this pressure region. It might be expected that LWH4 would have the lowest transition pressure of all the ionic materials studied, since the Alfi- ion would hold the electron least tightiy. At the same pressure in the aluminium plate as the above experiment, the iodine sample conducted also. This corresponds to a pressure in the iodine of roughly 130,OOO atm. Hence, the transition occurs below this pressure and above 80,OOO atm where it did not conduct according to the experiment previously mentioned. Iodine absorbs in the visible and has an energy level roughly 1.3 eV above the filled band. In another experiment in which a pressure of 200,000 atm was obtained in the aluminium plate, CsI showed a resistance in the neighbourhood of 1OOO ohm. Thus a pressure in the CsI of 200,000 atm must be near the transition point. RbI and KI seem to be in a similar position. CsBr, on the other hand, still had a resistance above 106 ohms under these experimental conditions and hence its transition lies at higher pressures. Considerably more effort is planned in order to improve the precision of the resistance measurements, and to determine more accurately the transition pressure, the temperature dependence of this phenomenon and the nature of any volume change associated with the transition for these as well as other substances. A detailed comparison with theoretical predictions could then be made. 1 Wigner and Huntington, J. Chem. Physics, 1935,3, 764. 2 Walsh and Christian, Physic. Rev., 1955,97, 1944. 3 Goranson et al., J. Appl. Physics, 1955,26, 1472. 1 DeGroot and Seldam, Physica, 1946, 12,669. 5 Yaffe and Van Artsdalen, J. Physic. Chem., 1956,60, 1125. 6 Bridgman, Proc. Amer. Acud., 1937,71,424 ; 1945,76,64.

ISSN:0366-9033    

DOI:10.1039/DF9562200044

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

TRANSPORT OF ENERGY AND MOlMENTUM IN A DENSE FLUID OF HARD SPHERES BY H. C. LONGWET-HIGOINS AND J. P. VALLEAU Dept. of Theoretical Chemistry, University of Cambridge Received 2nd July, 1956 A simplified kinetic theory of the hard-sphere fluid is described. From certain simpli- fying assumptions about the pair distribution function, expressions are derived for the shear viscosity, bulk viscosity and thermal conductivity at high densities as functions of the temperature, pressure, particle radius and particle mass. These expressions, modified to take account of attractive forces, are compared with published experimental data on monatomic and diatomic fluids. Agreement is only moderate, but improves with increasing density. The theory leads to the relation thermal conductivity 5 Boltzmann’s constant’ particle mass _ - 2 x shearviscosity - which applies with reasonable accuracy for liquid argon and liquid nitrogen.1. INTRODUCTION In dilute gases and liquids, the fluxes of momentum and energy are primarily convective (i.e. due to the motions of the individual molecules in space) ; in dense gases and liquids, on the other hand, the chief contribution to the fluxes is from an additional mechanism, the collisional, in which momentum or energy is trans- ferred from one mass-centre to another through the action of intermolecular forces. To calculate this effect in a fluid where the intermolecular potential varies continuously with distance, the distortion of the radial distribution function must be detennined.1 No completely satisfactory way of doing so has yet been found.The work of Enskog and Chapman2 has provided a satisfactory theory of transport processes in non-uniform moderately dilute gases. Enskog extended this theory to dense fluids of rigid spheres by including collisional transport, and modifying the Boltzmann equation to take account of the finite size of the spheres and of the resulting increase in the frequency of collisions. He solved this equation approximately to get a modified single-particle distribution function. from which the transport coefficients may be calculated.2 So far, however, this theory bas been applied effectively only in a semi-empirical manner. The statistical mechanical calculations of higher order non-equilibrium distribution functions, as proposed by Kirkwood,lp 3 and Born and Green,4 are mathematically complicated even for pair distributions, and can as yet be applied only by making somewhat doubtful assumptions.Neither of these approaches, unfortunately, makes the physical nature of transport at high densities as clear as could be desired. It would seem that insight into the problem might be gained by applying explicit and highly physical assumptions to a very simple model. It is the purpose of this paper to describe such a theory for a fluid of hard spheres with no internal degrees of freedom. Transport of momentum and energy is assumed to occur as the result of the i.nstantaneous transfer of these quantities from masscentre to mass-centre during colIisions. With such a model, as we shall see, fluxes will occur even when the radial distribution function is isotropic.The coefficients of shear and bulk viscosity, and the thermal conductivity, are obtained in terms 4748 TRANSPORT OF ENERGY A N D MOMENTUM of the mass and size of the spheres and the density, pressure, and temperature of the system. The results are compared with experimental data on monatomic and diatomic fluids at relatively high densities. 2. THE HARD SPHERE FLUID With a hard sphere fluid, it is necessary to consider only binary collisions, so we are interested especially in pair distribution functions. Consider a pair of spheres 1 and 2 which are nearly in contact, and let 1 be the unit vector along the line from the centre of 1 to the centre of 2. Let w1, w2 be the components of velocity of the two spheres along 1.Then we shall require the value of the function which represents the number of pairs of spheres per unit volume which are within unit small distance of contact, which have velocity components in dvl at v1 and dw2 at w2 and whose relative orientation lies in solid angle do around the direction I, This function can be evaluated making just two assumptions : (a) The spatial pair distribution function depends only on the temperature and density, as in equilibrium, not on the gradients of temperature or hydro- dynamic velocity. Then a quantity go, the total number of pairs of molecules, per unit volume, within unit small distance of mutual contact, can be evaluated using equilibrium statistical methods, and is found to be 5 h12 0, W l , wddwd'U1dw2, ;(;- n), in which a is the radius of each sphere and It is the number density of the fluid.This assumption is only an approximation to the truth, in that the pair distribution function will in general be distorted under a gradient, but it should serve to predict the order of magnitude of the transport coefficients in hard sphere fluids. (b) The velocity distribution function of each particle is Maxwellian with a mean determined by the local hydrodynamic velocity, and a spread determined by the local temperature. The velocities of different molecules are assumed to be uncorrelated. Now the flux of momentum or energy, under a gradient of hydrodynamic velocity or temperature, may be expressed by an integral of the type. where $ ( l , q , w 2 ) is the transport of the quantity, in the direction in which the flux is being calculated, during the collision of two particles 1, 2 ; this depends on the components of velocity of the particles along the line of centres, and, of course, on the orientation of this line relative to the direction considered.We will now consider the separate cases. The details of these calculations may be found in a paper by Longuet-Hi& and Pople.5 (i) SHEAR VISCOSITY Consider the fluid under a gradient of x-velocity in the y-direction, (3e1,/3y) = - of. We wish to calculate the flow of x-momentum in the y-direction. In a collision between two molecules 1 and 2, each of mass m, an amount of momentum m(q - v2)l is transferred from one to the other. Only the x-component of this, m(w1- v 2)&, need be considered, and the y-component of its change of position is hi', so that (2.3) 40, Y, vd = 2m4b(wl - v2)*H.C . LONGUET-HIGGINS AND J . P. VALLEAU 49 We can, with no loss of generality, assume that the hydrodynamic velocity is zero at the point of collision of the spheres. Then the average velocity of the first sphere is av’l,, in the x-direction, and that of the second --m‘h. Appealing to assumption (b), and integrating the velocity distributions over components of the velocity perpendicular to the line of centres, we find Using (3) and (4), (2) can be evaluated to any desired approximation. The first approximation is linear in the velocity gradient v’, and when divided by this gradient gives the shear viscosity as 4a inkT 4 P rl = 5( 7) (fi- .)* (ii) BULK VISCOSITY Consider the fluid contracting uniformly at a rate given by - & - 3u, - 3 -_ v’* ax 3y 32 (2.5) We are interested in, say, the flux of z-momentum in the z-direction, for which (2.7) M Y 0 1 , v 2 ) = 2 m a M V l - v 2 ) .Employing the same procedure as before we can write Again we can evaluate (2), now the flux of z-momentum in the z-direction. The first terni, independent of the velocity gradient, represents the collisional contribu- tion to the pressure, namely (P - A T ) . The next term is proportional to the velocity gradient, and is interpreted as giving rise to the bulk Viscosity. When this is divided by the divergence of the rate of strain, - 3v’, it gives the bulk viscosity as 4a mkT4 P x=+) (RT-.). (2.9) (iii) THERMAL CONDUCTIVITY Consider the fluid under a temperature gradient (3T/3x) = - T’. In a collision an amount of energy m(v12 - v 2 2 ) / 2 is transferred from 1 to 2, so that the resulting transport of energy in the x-direction is (2.10) 40, V l , v 2 ) = mQlX(Vl2 - v22).Using assumption (b) once more, and again integrating over the components of velocity perpendicular to the line of centres, we find m22 ] . (2.11) [ - 2k(T - al,T’) When (2) is evaluated, the first approximation is proportional to T‘; when this quantity is divided by T’, it gives the thermal conductivity, (2.12)50 TRANSPORT OF ENERGY AND MOMENTUM It will be noted that for a fluid of hard spheres, these coefficients have a square- root dependence on temperature, at constant density. This could have been anticipated on dimensional grounds.We have considered only the collisional contributions to the fluxes. In addition there will be a convective contribution, the importance of which we would expect to diminish as the density increases. To assess its relative importance, we may recall that the collisional contribution to the pressure is (P - &T). Clearly then, we may expect the coefficients calculated by the theory to apply only for densities at which P]&T > 1. (2.13) 3. COMPARISON WITH EXPERIMENT Although the above theory applies to rigid spheres only, it is of interest to compare its results with the experimental transport coefficients of liquids and of gases at high pressure. In particular, the inert gases, where the attractive forces are not great and vary slowly with distance, may be described approximately by this model.Enskog has shown that even small diatomic molecules may be fairly well described using the hard-sphere model. Molten metals constitute another class of monatomic fluids, though in these, of course, heat conduction proceeds primarily by an electronic mechanism. In these real fluids there are, of course, attractive forces, which will upset the relation of go to the pressure and temperature. Now the pressure may be written, according to thermodynamics, as P = T(3P/JT)V - (3U/3V),. (3.1) In a hard-sphere fluid the second term of this expression vanishes, so that it seems reasonable to follow Enskog's suggestion of replacing the pressure, in our expres- sions, by T(3P/3aY the " kinetic pressure ',. To do this requires thermal ex- pansion and compressibility data which rarely accompanies transport data for dense fluids.In tables l(a), 2,3 and 4 are given some calculations on the transport coefficients of gases under fairly high pressures, along with experimental data obtained by Michels and his collaborators, who have also published the necessary density data for calculating the kinetic pressure of these systems. It is difiicult to choose a suitable value of the radius a, which is necessary for absolute calculations of the coefficients ; instead of making such calculations, we have quoted the effective radius a' necessary to give agreement with experiment. In the case of argon a' may be compared with the van der Waals radius of 1.43 A and the crystal radius of 1.91 A. It may be mentioned that the Viscosity of hydrogen is, as predicted, very closely 1/2/Fthat of deuterium, so a calculation would yield the same values of the effective radius.TABLE THERMAL CONDUCIlVITY AND VISCOSlTY OF AROON 900 -141 2 2 1100 0151 3.1 1300 *I61 4 9 1500 0168 4-5 (a) at 25" C 697 19.9 9.0 1-8 7-14 40 21.4 6 9 2 5 8-13 3.2 22.6 5-6 3.3 9.08 2 3 WO 5 3 3.6 1002 2 8 (b) at 87.3" K 11912* 13 1.2 *21 7.5 29 3.9 60 25 4.2H. C . LONGUET-HIGGINS AND J . P. VALLEAU 51 pressure (atm) 541.7 630-4 742.1 854.1 965.8 pressure (atm) 898 1080 1269 1450 1647 1931 pressure (atm) 901.2 1002.5 1 100.4 1292.4 1559.1 1661.9 1762.3 1853.1 TABLE ~.-V~SCOSITY OF NITROGEN * AT 50" C so87 1 0.59 3.51 5.9 -0947 0.80 3.79 4.7 -103 0.98 417 4 3 -1 10 1.2 4-55 3.8 -1 16 1.6 4.9 1 3.1 TABLE THERMAL C0M)UCIzWTY OF NITROGEN 9 AT 50" C -1 13 22 203 9.2 .122 26 222 8.5 -1 30 29 3 2 8.3 -136 35 263 7.5 *142 42 280 6 7 -150 52 307 5.9 TABLE 4.-viSCOSI"Y OF DEUTERIUM 10 AT soo c 10-23 n (em-3) 0134 0144 -152 0168 -1 86 -193 *199 .204 1-71 1.77 1 -82 1 -94 2.08 2.14 2.20 225 9.0 8 4 7.6 6 6 5 4 5.1 4.7 4.5 We have also made some similar calculations on liquid systems. Of special interest is liquid argon, for which kinetic pressure and density data are quoted by Rice.11 In table l(b), the effective radius a' is obtaimed from A and from 17 independently at 87-3" K.Both values are too large, and so are the values of a' obtained from the viscosities of liquid metals.14 For mercury, for example, the experimental shear viscosity of 1.55 centipoise calls for an effective ionic radius a' of 3.2 A ; this may be compared with the Pauling radius15 for Hg2+ of 1-10 A.For most liquid metals compressibility data are not available. If, as a rough approximation, the Compressibility of the solid metal is used, it is found that, as expected, the theoretical predictions improve as PIAT, or rather (aPlaT),lnk, becomes greater. The ratio of a' to the Pauling ionic radius ranges from 1.6 for silver at lo00" C, where (3P/X!")v/nk is 16, up to 10 for sodium16 at 103" C, where it is only 5-4. In View of the scarcity of available data and of the uncertainty about the ex- pression for go, it is of interest to examine the ratio In table 5 values of this ratio for four liquids are compared with the experimental values.As expected, the agreement is considerably better. The coefficients D, of self-diffusion may be obtained from our model by the use of an additional postulate.5 Its value is (qlA)tlmx. = 2m/5k* (3.2) (3.3)52 TRANSPORT OF ENERGY AND MOMENTUM TABLE RATIO q/h FOR VARIOUS LIQUIDS temp. 10’ (‘1IA)t)theor. 10’ (q/&xpt. 87.3 1-92 2.1 (OK) (deg. sec2 cm-2) (deg. sec2 cm-2) 2.5 0.192 0194 4.0 0.192 0 106 80.5 1 -49 1.17 64.8 1.35 1.37 To eliminate the uncertainty as to the radius to assign to metallic ions it is inter- esting to test the theoretical relation which does not involve the radius a. This is done for two liquid metals in table 6. Again tin, with a value of 12 for (3P/3T),lnk, gives a much better result than mercury, where this ratio is only 8.4. CONCLUSION Although in general the predicted coefficients are rather low, the results are of the right order of magnitude for a wide range of real systems. It will be noted that in every case there is improvement as the density increases. It has been explained that this is to be expected on account of the convective contribution to the transport, and that on these grounds the theory can be expected to apply only as (3P/3T),/nk becomes considerably greater than 1. In fact this ratio is not great (between 2 and 4) for the pressurized gases examined above. It would therefofe be interesting to have data for still higher pressures. It should, however, be pointed out that even at high pressures the use of the undistorted pair distribution function is somewhat inadequate.Indeed, it is only the discontinuity in the intermolecular potential between hard spheres which allows an instantaneous flux of energy or momentum to result from this distribution function-for “ soft spheres ” the resulting transport coefficients would vanish. It would not be surprising, then, if the theoretical results were still somewhat lower than the experimental, even at quite high densities. In addition to these two difficulties inherent in the assumptions, there are the limitations of applying to real fluids calculations based on a model of hard spheres. It has already been mentioned that attractive forces will upset the relation between go and the pressure and temperature. In addition the dynamics of “ collisions ” of real particles are only approximated by the hard sphere model.Even for the inert gases the continuous nature of the intermolecular potential will mean that our expressions are slightly inaccurate, while for metals, where the repulsive forces between the ions vary rather slowly in space, the model is less adequate. For diatomic molecules, the rotations of the molecules, as well, should be con- sidered though the rotational degrees of freedom will be less important at low temperatures. The scholarship awarded to one of us (J. P. V.) by the National Research Council of C mda is gratefully acknowledged.H . C. LONGUET-HIGGINS AND J . P . VALLEAU 53 1 Irving and Kirkwood, J. Chem. Physics, 1950,18,817. 2 Chapman and Cowling, The Mathematical Theory of Non-uniform Cases (Cambridge 3 Kirkwood, J. Chem. Physics, 1946,14, 180. Kirkwood, Buff and Green, J. Chem. 4 Born and Green, A General Kinetic Theory of Fluids (Cambridge, 1949). 5 Longuet-Higgins, and Pople, J. Chem. Physics, in press. 6 Michels, Botzen and Schuurman, Physica, 1954,20,1141. 7 Michels, Botzen, Friedman and Sengers, Physica, 1956,22, 121. 8 Michels and Gibson, Proc. Roy. SOC. A , 1932,134,288. 9 Michels and Botzen, Physica, 1954, 19, 585. 10 Michels, Schipper and Rintoul, Physica, 1954,19, 1011. 11 Rice, J. Chem. Physics, 1946, 14, 324. 12 Rudenko and Schubikov, Physik. 2. Sowjetunion, 1934,6,470. 13 Uhlir, J. Chem. Physics, 1952, 20,463. 14Panchenkov, Compt. rend. U.R.S.S., 1951,79,985. 15 Pauling, T%e Nature of the ChemicuZ Bond (Oxford University Press, 1940). 16Ewing, Grand and Miller, J. Amer. Chem. Soc., 1951,73, 1168. 17 Bowers and Mendelssohn, Proc. Roy. SOC. A, 1950,204,366. 18 DeTroyer, Van Itterbeek, and Van den Berg, Physica, 1951, 17, 50. 19 Bowers, Proc. Physic. SOC. A, 1952, 65, 511. 20 Hammann, Ann. Physik, 1938, 32, 593. 21 Careri and Paoletti, Numu Cimento, 1955, X , 2, 574. 22 Nachtrieb and Petit, J. Chem. Physics, 1956,24,746. University Press, London, 1939). Physics, 1949, 17, 988. Eisenschitz, Proc. Roy. SOC. A , 1952, 215, 29.

ISSN:0366-9033    

DOI:10.1039/DF9562200047

出版商:RSC    

年代:1956

数据来源: RSC

摘要:

THE THERMAL CONDUCTIVITY OF ASSOCIATING GASES* BY E. WHALLEY Division of Applied Chemistry, National Research Council, Ottawa 2, Canada Receiwed 2nd July, 1956 A gas which associates to h e r s and trimers has critical properties. The thermal conductivity of such a gas has been calculated assuming that the diameter of the molecules is smatl compared witb the mean free path and that inelastic collisions have negligible direct effect on the transport properties. The equation for the transport of energy contains four terms, relating to the transport of kinetic and internal energy by both internal cir- culation and conduction. The CaIculations have been compared with the thermal conductivity of real gases to try to obtain some insight into the mechanism of transport in dense gases.1. INTRODUCTION The transport properties of a gas at relatively low pressures depend on pressure in a rather simple manner. The coefficients of viscosity and thermal conduc- tivity are independent of the pressure, and the various diffusion coefficients (con- centration, thermal, and pressure) are inversely proportional to the pressure. Kinetic theory predicts these pressure dependencies for a gas consisting of mole- cules which have random velocities and spatial distribution, and which are small compared with the mean distances between them; collisions between three mole- cules are of no importance. At moderately high pressures the gas no longer has these properties and its transport coefficients do not have a simple pressure dependence. The deviations can be ascribed, to a first approximation, to the following effects : (i) at a collision there is a rapid transport of molecular properties over the distance between the centres of the colliding molecules.This is known as collisional transfer, and it becomes important when the diameter of the molecules is not negligible compared with the mean free path. It is due mainly to the repulsive forces between the molecules; (ii) because of the attractive forces multiple collisions occur, and there is a local correlation of molecular velocities. The collisional transfer has been discussed rigorously by Enskog for a pure gas consisting of elastic spherical molecules, and extended by Thorne to binary mixtures.30 It is generally agreed that at temperatures well above the critical temperature, and up to densities of the order of the critical density the Enskog theory agrees moderately well with experimental measurements.From this we conclude that the pressure effect in this region is largely due to collisional transfer, i.e. to the repulsive forces, and that the attractive forces contribute a much smaller effect. This cannot be true in the critical region. Indeed the very existence of a critical region demands the existence of attractive forces, and it is reasonable to suppose that the attractive forces will play an important role in transport phenomena. This is fully confirmed by experimental measurements. The most striking anomalies are obtained with thermal diffusion.192,3 As the density is increased at constant temperature the thermal diffusion ratio changes sign as many as three * issued as N.R.C.no. 4021. 54E. WHALLEY 55 times. The viscosity and thermal conductivity are also anomalous,4-* and it is significant that the thermal conductivity departs qualitatively from the Enskog theory at a higher temperature than the viscosity.9 Sound absorption in the critical region is very high 1 0 s l l and this has been attributed to a configurational relaxation which must depend very largely on the attractive forces. The statistical-mechanical theory of transport phenomena 12q13 is in principle capable of taking account of the attractive forces, but because of the complexity no numerical calculations have been reported. This paper has been written mainly from the point of view of the experi- mentalist who is interested in qualitative, or semi-quantitative, explanation of these phenomena.A theory is required because of the insight obtained from even an approximate theory, and as a guide for further experiments. The theory, therefore, should be relatively simple so that numerical results can be obtained, and such complicated phenomena as thermal diffusion can be examined without undue effort. We have chosen a dilute gas which associates to form binary and ternary clusters in equilibrium. Tn such a model the attractive forces are taken account of at least to the extent that critical properties are exhibited. We have calculated the thermal conductivity of this model, and compared it with experi- mental measurements on real gases. The effect of the association on the transport properties is as follows.First, the presence of dimers and trimers alter the transport properties because the molecular composition of the gas is altered and the mean free path is no longer proportional to pressure. Secondly, internal energy can be carried by the dimers and trimers. Thirdly, if a temperature gradient is imposed on the gas, there is formed a concentration gradient of all species, the monomers increasing in con- centration with increasing temperature, and the dimers and trimers decreasing. This leads to an internal circulation, the polymers moving up the temperature gradient and monomers down; this will also affect the transport properties. It is immediately evident on this picture that, because of the internal circulation which occurs when a temperature gradient is imposed and the transport of internal energy, the thermal conductivity will depart from the Enskog theory at a higher temperature than the viscosity. This is observed9 The transport properties of gases which associate to h e r s have been dis- cussed several times.l4,15 SchSer and Foz Gazulla 16 showed that the effect of pressure on the thermal conductivity of polar gases at about 1 or 2 atm could be accounted for by assuming a small amount of association to dimers.The effect of the internal circulation was neglected. It was shown some years ago 17 that the internal circulation of NO2 + NzO4 in a mixture of H2 + N2 + NO2 + N204 could increase the relative separation of hydrogen and nitrogen by a factor of about two.Recently a number of papers has appeared in which the thermodynda of irreversible processes has been used to examine the tra1~3port properties Of associating fluids 18. 19 and this has been applied to the thermal ConductiVitY Of dense gases2os21 by assuming that dimerization can OCCUT. It was concluded, as is concluded here from kinetic theory, that transport of internal energy plays an important role in transport under pressure. 2. PRESENT MODEL 2.1 MOLECULAR INTERACTION Real molecules interact somewhat as shown in fig. la, in which the potential energy u is plotted as a function of the internuclear distance r. It has been shown above that the attractive part of this potential probably plays a rather important role in determining the transport properties in moderately dense gas- at tern- peratures not too far above their critical temperature.No calculations have56 THERMAL CONDUCTIVITY been done using a model of this kind because of the complexity of the problem. We have attempted here to develop a simple theory in the following way. Two molecules A and B (which may be identical) are assumed to have two possible kinds of interaction. The first is represented by the line DEF in fig. lb and corresponds to an elastic collision between rigid spherical molecules. The second kind is a simple harmonic vibration GHJ which corresponds to the existence of a double molecule AB. A crossing from DE to GHJ corresponds to the forma- tion of a molecule AB, and a crossing from GHJ to DE corresponds to the destruc- tion of a molecule AB.The probability of such a crossing is assumed to be (i) low" enough that the inelastic collisions do not contribute appreciably to (ii) high enough to maintain closely the local thermodynamic equilibrium between monomers and dimers which tends to be destroyed by diffusion. the transport properties, and FIG. 1.-Intermolecular potentials of molecules. la represents real molecules ; represents the present model. l b A double molecule AB is assumed to interact with a monomer C in a similar way, so that triple molecules ABC are formed in equilibrium with the monomers. Similar assumptions can be made leading to the formation of stable clusters of 4, 5, etc. atoms. In the numerical calculations we restrict the gas to forming binary and ternary clusters. The binary and ternary clusters are assumed to interact with one another only as rigid elastic spheres.2.2 TRANSPORT PROPERTIES There are several possible models of varying complexity for calculating the transport properties of our molecular model. Rigorous calculations can be done by applying the Enskog theory of dense gases and taking account of the chemical equilibria existing. Unfortunately the Enskog theory has been worked out for only pure substances and binary mixtures. The presence of trimers cannot be taken account of until the general theory for multiconiponent mixtures has been worked out, nor canjtransport properties in a gas mixture so that diffusion could not be studied. Since we have set out to describe a theory which can be used to calculate all the transport properties we have not used this approach.So far as the author is aware, there is no approximate theory of transport in a dense gas of elastic spheres, though it should be possible to take approximate account of the collisional transfer. Since the dense gas theory appears to be impracticable, we must assume that the molecules are small compared with their mean free path. The exact kineticE. WHALLEY 57 theory of multicomponent gas mixtures is available,22. 23 but numerical cal- culations are very complicated. We shall, therefore, use the mean free path theory which yields numerical results without too much trouble 24 even for such compIicated phenomena as thermal diffusion in multicomponent mixtures. 3. THE EQUATION OF STATE The discussion is here restricted to a pure one-component gas which contains the molecules A1, A2, and A3 in thermodynamicequilibrium.A1 is assumed to be monatomic. The subscript 1, 2, or 3 to any symbol means that the property represented is that of Al, A2, or A3 respectively. It is assumed that mixtures of A1, A2, and A3 behave like ideal mixtures of perfect gases, thus, p = nkT, (3.1) where p is the pressure, T the temperature, n the number density of clusters, and k is Boltmann’s constant. This is equivalent to assuming that the molecules are small compared with their mean free paths. The concentrations of the various clusters are determined by the equilibrium constants K2 and K3 for the formation of A2 and A3, thus, K2 = P21Pr2 = C2IC12P9 K3 = P31Pr3 = C31C13P2, (3.2) (3.3) where p1, p2 and p3 are the partial pressures, and c1, c2, and c3 are the mole fractions ; by definition ~1 + ~2 + ~3 = 1.The equation of state of such a mixture has been discussed by Woolley.25 He shows that the virial coefficients B and C of the equation of state (3.4) B C pv- 1 + - + - + . . . ., RT - v v2 (3.5) where V is the molar volume, and R the gas constant per mole, are given by B = - K2RT, C = (4K22 - 2Kj)R2T2. (3.6) Eqn. (3.5) is the simplest equation of state which predicts critical properties and it predicts that PcVc/RTc = +, B, = - V, C, = + Vc2, (3 07) where the subscript c refers to the value at the critical point. From eqn. (3.6) it seems that B can never be positive. Although it has been claimed that negative equilibrium constants are not meaningless 26.27 they are nevertheless unsatis- factory.Ginell28 has shown how to overcome this difficulty by taking account of the excluded volume of the molecules. B then depends on both this volume and K2, and it can be positive though K2 is positive. In this paper we avoid this difficulty by restricting ourselves to temperatures well below the Boyle temperature and assume that the excluded volume makes only a small contribution to the virial coefficients. For real non-polar gases the Boyle temperature is about 2-6 T,, where T, is the reduced temperature TITc ; we restrict the theory to reduced teniperatures of less than 1.6. We have now to evaluate K2 and K3. The thermodynamic equations (3.8) where H is the enthalpy per mole, are used. The total energy per molecule u dlnK2 H2-2Hl dlnK3 H3- 3Hl dT RT2 ’ dT RT2 ’ -= -=58 THERMAL CONDUCTIVITY is made up of translational, rotational, vibrational, and potential energies.It is conveniently divided into the translational energy -kT and the internal energy w, thus, (3.9) The potential energy is measured relative to two widely separated A atoms, hence, 3 2 3 2 u=-kT+ W. 5 2 2 w1= 0, u1 = 2kT, HI = -RT. (3.10) Species 2 is diatomic. The vibration frequency of A2 will be of the same order as the vibration frequency of the atoms in solid A1 ; it is sufficiently low to treat the vibrations classically and to neglect the zero-point energy. Hence wz = 2kT - E , (3.1 1) where E is the depth of the potential well H (fig. lb) below OD. For a Lennard- Jones 12 : 6 potential we have approximately 4 E = -kTc, 5 Hence (3.12) 9 4 2 5 2 5 w2 = 2kT - $Tm u2 = ZkT - tkTc, H2 = -RT - -RT,.(3.13) The ternary clusters A3 will consist of both linear and triangular molecules. The linear clusters will have a higher energy (Iower negative potential energy) than the triangular clusters and will have a lower concentration. For simplicity, we have neglected their presence and assume that A3 is triangular. If the potential energy of a triangular cluster is assumed to be three times that of a binary cluster, then 9 12 w3 = -kT - EkT,, u3 = 6kT - p T c , H3 = 7RT - -RT,. 2 5 5 (3.14) From eqn. (3.8), (3.10), (3.13) and (3.14) we find that -- dlnK2 d l n T --{!+A} 2 5Tr ’ dInK3=_ d l n T {’+”) 2 5T, a (3.15) It is convenient to develop the theory entirely in terms of reduced quantities so that we obtain reduced transport properties in terms of reduced temperature and pressure.Convenient reduced equilibrium constants K2, and K3r are K i r = K2~c, K3r = K3pc2. (3.16) By integrating (3.15) and using (3.7) to determine the integration constants we find that Kzr = 01498 T,-lI2 exp (4/ST,), (3.17) K3r = 0.01848 T,-’/2 exp (12/5T,). (3.18) From (3.6), (3.13, and (3.18) we find that the reduced second and third virial (3.19) (3.20) coefficients are given by B/ Vc = - 0*4492T;1” exp (4/5Tr)9 C/Yc2 = 0*8071T, exp (8/5T,)(l - 0.2060Tr1/2 exp (4/5 T,). From eqn. (3.2)-(3.4) and (3.16) we have CI’P?& + c12prK2r + ~1 -.T 1 = 0, (3.21)E. WHALLEY 59 thus giving cl as a function of T, and pr. c2 and c3 are then obtained from eqn.(3.2) and (3.3). The Pr against Vr isotherms obtained from eqn. (3.5), (3.19), and (3.20) are shown in fig. 2. 4. THE TRANSPORT PROPERTIES For the reasons outfined above we will use the mean free path theory for cal- calculating the transport properties. We consider a mixture of molecules l, 2, . . . i, without at present specifying whether any of these are associated. Tn a gas containing gradients of pressure, temperature, and composition there is, in general, a hydrodynamic flow superposed on the thermal velocities in order to ensure the correct transport across a plane fixed in the apparatus. Let this velocity be v, and we consider gradients in the x-direction only. The transport F(#j) of any property $ associated with species i is the sum of that due to the hydrodynamic flow v, and to diffusion.The transport due to hydrodynamic flow is ItjV&i. That due to diffusion is obtained by usual methods of the mean free path theory, i.e. taking the mean of that for all the molecules crossing a plane and originating from a distance I from the plane, where I is the mean free path. This is If required, the theory of Fiirth29 could be used. Then, due to the persistenCe of velocities, the properties ni, vi, and Jli do not have the same mean free paths.60 THERMAL CONDUCTIVITY If Zi, Z:, and Zi' are the mean free paths for the transport of ni, vi, and $i respectively. it is easily shown that (4.2) The effect of persistence of velocities could be taken account of, but it would affect the results by only a numerical factor near unity.In order to simplify the theory we have neglected it. We will therefore use eqn. (4.1). In order to eliminate v, we introduce the diffusion fluxes *Ti. These are obtained by putting #i = 1, thus By eliminating v, from (4.1) using (4.3) we obtain The total flux F($) of property $ is obtained by siimming Ft$i) over all species, thus The first term of eqn. (4.5) represents the transport of a property because the mole- cules carrying it are transported. The second term represents the transport because the property is passed from molecule to molecule at collision. 5. THERMAL CONDUCTMTY The thermal conductivity is obtained by putting *i = Ui- (5.1) It is more informative to consider the translational energy and the internal energy separately because only translational energy is carried at low pressures.The transport of internal energy is induced by pressure. Hence (5.2) 3 2 $I= -kT+ wi. The total flux of heat q is obtained by putting eqn. (5.2) into (4.5) to give where V' = Ji/ni is the mean velocity of molecules i. The four terms on the right hand side of eqn. (5.3) are respectively the flux of kinetic energy by internal circulation, the flux of internal energy by internal circulation, the flux of kinetic energy by conduction, and the flux of internal energy by conduction. By con- duction we mean the usual process of passing energy from molecule to molecule at collision. In a non-associating gas at steady state vt = 0, and the first two terms of eqn. (5.3) disappear, and there is no transport by internal circulation. In a gas at pressures low enough that dimers and trimers are not formed in appreciable amounts, both V, and wi are zero, and the only contribution to the thermal conductivity is the third term of eqn.(5.3).61 In a gas which can associate to dimers and trimers, the condition for no net E. WHALLEY transport of matter across a plane is c1 v1 + 2c2v2 + 3c3v3 = 0. (5.4) In the absence of a pressure gradient and neglecting the effect of the persistence of velocities after collision the relative velocities are given by 24 where ini is the nass of a molecule i. The mean free paths are given by li = (nTZcioli(l + m1/rni)1/2}-1 = (2/2nna2f1>-1, etc., (5.6) where qi is the mean of the collision diameters of molecules A1 and Ai, and 0 is the collision diameter of molecules Al.The quantitiesfl, etc., are of the form fi = XCl + YC2 + zc3, and x, y , and z are calculated by assuming that the volumes of A2 and A3 mole- cules are twice and thrice respectively that of an A1 atom. Thus 022 = 1.2600, 033 = 1.4420. Also the coefficient of thermal conductivity at low pressure is (5.7) It is convenient to consider that the coefficient of thermal conductivity is (5.8) the sum of four parts, corresponding to the four terms of eqn. (5.3), thus, A = hl + A2 + A3 + 4. From the above equations we find that - - 2 ~ 1 + c2 (- dc3 - SC3)}, (5.9) 4 3 f 3 d h T r (5.1 1) (5.12) where co = ct -t 2c2 + 3c3 = 1 + c2 + 2 ~ 3 , W; = WZikT, W$ = ~ 3 i k T . (5.13) (5.14)62 THERMAL CONDUCTIVITY DISCUSSION Numerical values of A&, etc., have been calculated over a range of reduced temperatures and pressures.The four contributions to the thermal conductivity and the total thermal conductivity are plotted in fig. 3 as a function of the reduced temperature at various reduced pressures. The co-existence line is sketched in fine lines. The greatest contribution is from A3, i.e. the transport of kinetic energy by conduction, and A& is about 0.8 in the critical region. The transport of I . . . . < . . . . ' 9 I011121314151617 91011 1 2 1 3 1 4 1 5 1 6 i 7 T, Tr FIG. 3.-The contributions to, and the total thermal conductivity. internal energy by conduction & is about 0.3 of the zero pressure thermal con- ductivity. The transport by internal circulation is rather small. The transport of kinetic energy by this mode is about one-tenth of the total, and that of internal energy is negative, i.e.internal energy is carried up the temperature gradient. The reason for this is straightforward. For every A2 molecule which moves up the temperature gradient, two A atoms move down. The transport of internal energy down the temperature gradient is thus 2wl - wz per A2 molecule. Since w 1 is zero, this is equal to - (BT- $Tc) =- kT(2 - g), which is negative for 7'' > 0.4. The transport of total energy (internal + kinetic) is and this is positive for all values of Tr less than 1-6. Similar remarks apply to the transport of energy by the internal circulation of A3 molecules. A glance at fig. 3 shows that the thermal conductivity of the present model is not very dependent on pressure, and A/&) departs from unity by only about 15 z.E.WHALLEY 63 Experimental values of A/& are about 2.5-3 in the critical region.7 The PVT properties are qualitatively predicted as fig. 2 shows. This is another example of the well-known fact that transport properties are very much more sensitive to the details of intermolecular interaction than equilibrium properties. The main contribution to the transport properties in a real gas which is not present in our model is the collisional transfer. Undoubtedly, most of thefactor of about 23 in the ratio of experimental thermal conductivities of dense gases to those calculated here is due to the neglect of this effect. To include it would increase the transport by conduction, but would decrease the transport by internal circulation because there is no collisional transfer in diffusion and the rate of diffusion would be decreased.We conclude, therefore, that in dense gases near the critical region most of the thermal conductivity is due to the Enskog effect on the transport of kinetic energy by conduction, and the transport of the internal energy of clusters plays a smaller, though still im- portant, role. The transport of energy by internal circulation is relatively un- important, and contributes no more than a few percent to the total conductivity. The author's thanks are due to Miss J. Taylor for help with the calculations. 1 Giller, Duffield, and Drickamer, 3. Chem. Physics, 1950,18, 1027. 2 Robb and Drickamer, J. Chem. Physics, 1951, 19, 818.3 Caskey and Drickamer, J. Chem. Physics, 1952,21, 153. 4 Mason and Maass, Can. J. Res. B, 1940,18,128. 5 Naldrett and Maass, Can. J. Res. B, 1940,18,322. 6 Michels, Botzen, Friedman, and Sengers, Physica, 1956,22,121. 7 Uhlir, J, Chem. Physics, 1952,20,463. 8 see Franck, Chm.-Ing. Techn., 1953, 25, 238, for a review of thermal conductivity 9 Michels, Cox, Botzen, and Friedman, J. Appl. Physics, 1955,26, 843. 10 Schneider, Can. J. Chem., 1951, 29, 243. 11 Schneider and Chynoweth, J. Chem. Physics, 1952,20, 1777. 12 Irving and Kirkwood, J. Chem. Physics, 1950,18,817. 13 Born and Green, A General Kinetic Theory of Liquids (Cambridge University Press, 14 Nernst, Boltzmann Festschrift, 1904, 504 : Sci. Abstr., 1904,7, 885. 15 Wildt, Astrophys. J., 1936, 83, 202; Mon.-Not. Roy. Astron. SOC., 1937, 97, 225. 16 SchXer and Foz Gazulla, 2. physik. Chem. B, 1942,52,299. 17 Whalley, Tram. Faraday SOC., 1951,47,1249. 18 Prigogine and Buess, Bull. Classe Sci. Acad. Roy. Belge, 1952,38,711, 8 5 2 19 Meixner, 2. Naturforsch. A, 1952,7,553. 20 Franck, 2. physik. Chem., 1952,201, 16. 21 Waelbroeck, Lafleur, and Prigogine, Physica, 1955,21,667. 22 Hellund, Physic. Rev., 1940, 57, 319, 328, 737, 743. 23 Curtiss and Hirschfelder, J. Chem. Physics, 1949, 17, 550. 24 Whalley and Winter, Trans. Faruday SOC., 1950, 46, 517 ; Whalley, Trans. Faraduy 25 Woolley, J. Chem. Physics, 1953,21,236. 26 Kitpatrick, J. Chem. Physics, 1953,21,1366. 27 Winston, J. Chem. Physics, 1953,21,2245. 28 Ginell, 3, Chem. Physics, 1955,23,2395. 29 Fiirth, Proc. Roy. SOC. A, 1942,179,461. 30 Chapman and Cowling, Mathematical Theory of Non-Uniform Gases (Cambridge measurements under pressure. 1949). SOC., 1951,47, 1249. University Press, 1952).

ISSN:0366-9033    

DOI:10.1039/DF9562200054

出版商:RSC    

年代:1956

数据来源: RSC