Theories and computation of second virial coefficients of electromagnetic phenomena.
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Many bulk properties of gases depend linearly on the gas density at lower densities, but as the density increases departures from linearity are observed. The density dependence of a bulk property Q may often be discussed systematically by expanding Q as a power series in l/Vm, to yield: Q=Aq+Bq/Vm+Cq/V2m+..., where Bq is known as the second virial coefficient of the property Q. Aq is the ideal gas value of Q, and Bq describes the contribution of molecular pair interactions to Q. Theories of Q may be regarded as having two main components, one describing how the presence of a neighbour of a given molecule can enhance or detract from its contribution to Q, and the other the molecular interaction energy which determines the average geometry of a pair encounter. The latter component is common to all theories, and the former requires detailed derivations for each specific bulk property Q. In this work we consider the second virial coefficients of five effects, namely the second pressure virial coefficient B(T), and also the second dielectric, refractivity, Kerr-effect and light-scattering virial coefficients, Be, Br, Bk, and Bp, respectively. Using a powerful computer algebraic manipulation package we have extended the existing dipole-induced-dipole (DID) theories of the second dielectric, refractivity and Kerr-effect virial coefficients to sufficiently high order to establish convergence in the treatment of both linear and non-linear gases. Together with the established linear theory of the second pressure virial coefficient, the extended theory of the second light- scattering virial coefficient developed by Couling and Graham, and their new non-linear theory of the second pressure and light-scattering virial coefficients, our new theories provide a comprehensive base from which to calculate numerical values for the various effects for comparison with experiment. We have collected as much experimental data of the various second virial coefficients as possible, for a wide range of gases. The ten gases chosen for detailed study comprise a selection of polar and non-polar, linear and non-linear gases: the linear polar gases fluoromethane, trifluoromethane, chloromethane and hydrogen chloride; the non-polar linear gases nitrogen, carbon dioxide and ethane; the non-linear polar gases sulphur dioxide and dimethyl ether; and the non-linear non-polar gas ethene. Using the best available measured or calculated molecular parameter data for these gases, together with the complete theories for the second virial coefficients, we have attempted to find unique sets of molecular parameters for each gas which explain all the available experimental data. In general, reliable measured or calculated molecular properties are regarded as fixed, and only the Lennard-Jones and shape parameters in the molecular interaction energy are treated as best-fit parameters within the constraints of being physically reasonable. Many of the apparent failures of second virial coefficient theories have been due to the lack of convergence in the series of terms evaluated. It is essential to work to sufficiently high orders in the polarizabilities and various multipole moments to ensure convergence for meaningful comparison with experiment. This often requires the manipulation of extremely long and complicated expressions, not possible by the manual methods of our recent past. The advent of computer manipulation packages and fast processors for numerical integration have now enabled calculation to high orders, where the degree of convergence can be sensibly followed. Our efforts to describe all of the effects for which data is available met with mixed success. For four of the gases, fluoromethane, chloromethane, dimethyl ether and ethene, a unique parameter set was found for each which described all of the available effects reasonably well. For the three gases, trifluoromethane, nitrogen and sulphur dioxide, one interaction parameter set explained all but one of the effects for which data was available to within experimental uncertainty. For trifluoromethane the parameter set which yielded good agreement for B(T), Be, and Bk could not explain the observed values of Br, while for nitrogen one parameter set produced reasonable agreement for all of the effects except Bp and a different set, which yielded good agreement for Bp, did not explain the remaining four effects as well as the first set. The parameter set which explained B(T), Bk and Bp very well for sulphur dioxide, yielded a value for Be, which was much larger than the experimental value, although of the correct sign and order of magnitude. Hydrogen chloride posed a special problem as data was only available for two of the effects, B(T) and Be. It was possible to find a set of interaction parameters in good agreement with the measured values of B(T), but the experimental data for Be was an order of magnitude larger than the largest calculated values. Since the remaining effects have not been measured for this gas it was not possible to test the theory more rigorously. For the remaining gases carbon dioxide and ethane, it was impossible, based on the existing measured values, to select a unique parameter set which explained all of the effects. In many of the cases where definite conclusions could not be drawn, it was not possible to decide whether the disagreement between theory and experiment was due to the large scatter and uncertainty of the experimental data or failure of the theory. However, there were very few instances of complete failure of the theory to explain experiment, and no one effect showed consistent disagreement, so that in general it may be said that the mechanisms of the second virial coefficients under study are reasonably well understood. It would require more precise measurements of the various effects, as well as more measured or calculated molecular property tensor components, such as the hyperpolarizability and the A- and C-tensors , to test the DID molecular interaction model more stringently.