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. 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