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.