Virial coefficient

Abstract: The three-dimensional classical many-body system is approximated by the use of collective coordinates, through the assumed knowledge of two-body correlation functions. The resulting approximate statistical state is used to obtain the two-body correlation function. Thus, a self-consistent formulation is available for determining the correlation function. Then, the self-consistent integral equation is solved in virial expansion, and the thermodynamic quantities of the system thereby ascertained. The first three virial coefficients are exactly reproduced, while the fourth is nearly correct, as evidenced by numerical results for the case of hard spheres.

Abstract: The investigation of a preceding paper has shown that the temperature variation of viscosity, as determined experimentally, can be satisfactorily explained in many gases on the assumption that the repulsive and attractive parts of the molecular field are each according to an inverse power of the distance. In some cases, in argon, for example, it was further shown that the experimental facts can be explained by more than one molecular model, from which we inferred that viscosity results alone are insufficient to determine precisely the nature of molecular fields. The object of the present paper is to ascertain whether a molecular model of the same type will also explain available experimental data concerning the equation of state of a gas, and if so, whether the results so obtained, when taken in conjunction with those obtained from viscosity, will definitely fix the molecular field. Such an investigation is made possible by the elaborate analysis by Kamerlingh Onnes of the observational material. He has expressed the results in the form of an empirical equation of state of the type pv = A + B/ v + C/ v 2 + D/ v 4 + E/ v 6 + F/ v 8, where the coefficients A ... F, called by him virial coefficients , are determined as functions of the temperature to fit the observations. Now it is possible by various methods to obtain a theoretical expression for B as a function of the temperature and a strict comparison can then be made between theory and experiment. Unfortunately the solution for B, although applicable to any molecular model of spherical symmetry, is purely formal and contains an integral which can be evaluated only in special cases. This has been done up to now for only two simple models, viz., a van der Waals molecule, and a molecule repelling according to an inverse power law (without attraction), but it is shown in this paper that it can also be evaluated in the case of the model, which was successful in explaining viscosity results. As the two other models just mentioned are particular cases of this, the appropriate formulae for B are easily deduced from the general one given here.

Abstract: 1. Units and Fundamental Constants: 1.1 Units. 1.2 Fundamental physical constants. 2. General Physics: 2.1 Measurements of mass, pressure and other mechanical quantities. 2.2 Mechanical properties of materials. 2.3 Temperature and heat. 2.4 Acoustics. 2.5 Radiation and optics. 2.6 Electricity and magnetism. 2.7 Astronomy and geophysics. 3. Chemistry: 3.1 The elements. 3.2 Properties of inorganic compounds. 3.3 Properties of organic compounds. 3.4 Vapour pressures. 3.5 Critical constants and second virial coefficients of gases. 3.6 Properties of solutions. 3.7 Properties of chemical bonds. 3.8 Molecular spectroscopy. 3.9 Electrochemistry. 3.10 Chemical thermodynamics. 3.11 Miscellaneous data 4. Atomic and Nuclear Physics: 4.1 Electrons in atoms. 4.2 Absorption of photons. 4.3 Work function. 4.4 Free electron and ions in gases. 4.5 Absorption of particles and dosimetry. 4.6 Radioactive elements. 4.7 Nuclear fission and fusion and neutron interactions. 4.8 Nuclei and particles. 5. Miscellaneous engineering data. 6. Statistical methods for the treatment of experimental data. 7. Laboratory safety. 8. Introduction to quality assurance in measurement. Index.

Abstract: The quantum mechanics of proper open systems yields the physics that governs the local behavior of the electron density, ρ(r). The Ehrenfest force F(r) acting on an element of ρ(r) and the virial field ν(r) that determine its potential energy are obtained from equations of motion for the electronic momentum and virial operators, respectively. Each is represented by a “dressed” density, a distribution in real space that results from replacing the property in question for a single electron with a corresponding density that describes its average interaction with all of the remaining particles in the system. All bond paths, lines of maximum density linking neighboring nuclei in a system in stable electrostatic equilibrium, have a common physical origin in terms of F(r) and ν(r), regardless of the nature of the interaction. Each is homeomorphically mirrored by a virial path, a line of maximally negative potential energy density linking the same nuclei. The presence of a bond path and its associated virial path...

Abstract: An equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid sphere molecules in contact with a rigid sphere solute of arbitrary size. A number of exact relations which describe rather fully the functional form of G(r) are derived. These are based on both geometrical considerations and the virial theorem. A knowledge of G(a) where a is the diameter of a rigid sphere enables one to arrive at the equation of state. The resulting analytical expression which is exact up to the third virial coefficient gives the fourth virial coefficient within 3% and the fifth, insofar as it is known, within 5%. Furthermore over the entire range of fluid density, the equation of state derived from theory agrees with that computed using machine methods. Theory also gives an expression for the surface tension of a hard sphere fluid in contact with a perfectly repelling wall. The dependence of surface tension on curvature is also given. ...