Fourth Virial Coefficient Research Articles

On the internal consistency of the hyper-chain approximation in the theory of classical fluids

The hyper-chain approximation in the theory of classical fluids, and a second approximation in which some account is taken of the first elementary diagram, are examined for internal consistency by calculating virial expansions of the pressure both directly and from the compressibility equation. Both theories are internally inconsistent at the fourth virial coefficient. Numerical results are presented for both fourth and fifth virial coefficients for a gas of hard spheres, and compared with those given by earlier treatments. A brief comment is made on the consistency of the equations for fluid mixtures.

Fluid Superposition Approximation and the Fourth Virial Coefficient

A new superposition approximation is given for relating the triplet distribution for a classical system of identical particles in interaction (fluid) to the pair distribution. The new approximation is based upon the well-known expression due to Kirkwood, but explicitly accounts for the particle density through a power series expansion. It is proposed that the coefficients of this expansion should be chosen in order to ensure internal consistency in the calculation of the virial coefficients of the pressure (when expressed in powers of the density) from either the pressure equation or the relative compressibility equation, and the necessary formulas are given. The equations are checked numerically for the ideal case of a gas of rigid spheres, as far as the fourth virial coefficient. The modified form of the original Kirkwood superposition approximation, which is the new approximation, is found for this case to be g(3)(r, s, t) = g(2)(r)g(2)(s)g(2)(t) [1+0.1014bn] where g(3) is the triplet distribution, g(2) the pair distribution, n the particle number density, and b is equal to four times the volume of each particle. The theory then leads to a fourth virial coefficient of value 0.2885b3 which is to be compared with the exact value 0.2869b3. This value is found to be a marked improvement on the values calculated by previous authors. The higher powers of n in the new superposition approximation necessary for the correct calculation of higher virial coefficients than the fourth are not considered in this paper.

Fourth Virial Coefficient for the Square Well Potential

The fourth virial coefficient $D$ and its components ${D}_{1}$, ${D}_{2}$, ${D}_{3}$ for molecules interacting with a square-well potential have been obtained as functions of the temperature by the use of Fourier transformation and the addition theorem of Bessel functions. Each component and their sum are nearly constant above the Boyle temperature and tend to negative infinity as temperature tend to zero. The total $D$ is expressed by $\frac{D}{{b}^{3}}=0.28642+1.5397f\ensuremath{-}23.554{f}^{2}+53.645{f}^{3}+69.859{f}^{4}\ensuremath{-}170.01{f}^{5}\ensuremath{-}14.777{f}^{6},$ where $f={e}^{\frac{\ensuremath{\epsilon}}{\mathrm{kT}}}\ensuremath{-}1$, and well width is equal to the hard core diameter. Corrections of the critical data and some related problems are discussed.

Intermolecular forces and properties of fluid. I. The automatic calculation of higher virial coefficients and some values of the fourth coefficient for the Lennard-Jones potential

The prediction of the virial coefficients for particular intermolecular potentials is generally regarded as a difficult mathematical problem. Methods have only been available for the second and third coefficient and in fact only few calculations have been made for the latter. Here a new method of successive approximation is introduced which has enabled the fourth virial coefficient to be evaluated for the first time for the Lennard-Jones potential. It is particularly suitable for automatic computation and the values reported here have been obtained by the use of the EDSAC I. The method is applicable to other potentials and some values for these will be reported subsequently. The values obtained cannot yet be compared with any experimental results since these have not been measured, but they can be used in the meantime to obtain more accurate experimental values of the lower coefficients.

Intermolecular forces and properties of fluids II. A general functional representation of inter molecular potentials, and some values of the second, third and fourth virial coefficients for systematically differing potentials

A new functional representation of angle-independent intermolecular potentials is described, having an unlimited number of parameters in the form of linear coefficients in an expansion depending on a complete system of functions. The basic single-term form of th is function is practically equivalent to the Lennard-Jones (6-12) potential. Three particular examples of this potential, having different widths of the potential well, have been examined, and the second, third, and fourth virial coefficients for them have been calculated for a number of temperatures. It is shown that the new functional representation should enable better and more systematic progress in the estimation of intermolecular potentials to fit experimental data.

Fourth Virial Coefficient for the Square Well Potential

The fourth virial coefficient $D$ and its components ${D}_{1}$, ${D}_{2}$, ${D}_{3}$ for molecules interacting with a square-well potential have been obtained as functions of the temperature by the use of Fourier transformation and the addition theorem of Bessel functions. Each component and their sum are nearly constant above the Boyle temperature and tend to negative infinity as temperature tend to zero. The total $D$ is expressed by $\frac{D}{{b}^{3}}=0.28642+1.5397f\ensuremath{-}23.554{f}^{2}+53.645{f}^{3}+69.859{f}^{4}\ensuremath{-}170.01{f}^{5}\ensuremath{-}14.777{f}^{6},$ where $f={e}^{\frac{\ensuremath{\epsilon}}{\mathrm{kT}}}\ensuremath{-}1$, and well width is equal to the hard core diameter. Corrections of the critical data and some related problems are discussed.

Statistical Mechanics of Rigid Spheres

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. The expressions obtained correlate nicely with those adduced by other thermodynamic and statistical mechanical theories. They also suggest that macroscopic consideration on surface tension can sometimes be successfully extrapolated to molecular dimensions.

An approximation for the radial distribution function

The present paper is concerned with a new approximation for the distribution function of third order. This approximation may be regarded as an improved form of the well-known superposition assumption of Kirkwood. The idea is to add to Kirkwood’s expression a linear combination of distribution functions of the same type, the coefficients of which can be easily determined. The new approximation is introduced in the Bom—Green integral equation for the radial distribution function, for which an expansion into powers of the density is used. From this the terms proportional to the first and second power of the density are calculated. The first four virial coefficients can be expressed as functions of these terms, whether the pressure equation or the compressibility equation is used. Numerical evaluation is performed for the ideal case of a gas of rigid spheres. The value obtained for the fourth virial coefficient is compared with the exact one and those given by naing Kirkwood’s assumption by Rushbrooke & Scoins, and Nijboer & van Hove. It is seen to be more nearly exact and internally consistent. The term proportional to the square of the density, in the expansion of the radial distribution function, appears to be very similar to the exact one as calculated by Nijboer & van Hove. It can be seen to be better than the corresponding term when Kirkwood’s assumption or one proposed by Nijboer & van Hove is used. Finally, an alternative assumption is suggested, and applied to the case of hard spheres.

Velocity of Ultrasonics in Compressed Gases near the Critical Point and up to 1200 Atmospheres

The velocity of ultrasonics in compressed gases was measured by the optical method using the diffraction of monochromatic light. Our investigations may be summarized as follows: (1) Measurements on various compressed gases (A, N2, CH4) at pressures up to 1200 atmos, temperatures from 20°C to 200°C, and at frequencies ranging from 0.5 to 3.5 Mc. No definite dispersion was found in that range. The velocity increases as a function of pressure; at high densities its temperature coefficient is negative. The velocities calculated from the experimental PVT data of other authors are in good agreement with our measured velocities. The ratio of specific heats and the adiabatic compressibilities have been calculated. Calculations have been made for A, N2, and CH4 using a series expansion of the velocity of sound with respect to densities; the coefficients of this expansion are related to the first four virial coefficients and their first two derivatives. The usual (6–12) spherically symmetric intermolecular potential has been assumed. It is found that a satisfactory agreement between calculated and observed velocities can be achieved at densities up to 300 amagat units; this agreement is obtained for both PVT and sound velocity data, by taking into account only a constant value of the fourth virial coefficient in our temperature range. It is concluded that a purely thermodynamical interpretation is sufficient to explain the observed velocities in our range and that sound velocity data seem a convenient means to test, jointly with the equation of state, the accuracy of a given intermolecular potential law. (2) Measurements of ultrasonic velocity near the critical point: The decrease of the velocity, previously observed for CO2 and N2O, has been studied in C2H6 and C3H8. Systematic measurements on C3H8 show that no dispersion is present in our frequency range (0.585 to 3.2 Mc). Values of the velocity have been calculated on a purely thermodynamical basis from some semiempirical equations of state. The measured values are a little lower than the calculated values in the immediate vicinity of the critical point. Such a possible disagreement has been tentatively ascribed to the effect on the bulk viscosity of a structural relaxation process. (3) Measurements by a pulse method of the ultrasonic attenuation in nitrogen at pressures from 80 to 1400 atmos are presently in progress. Preliminary results indicate that the attenuation is strongly decreased under pressure.

Virial Coefficients of ``Parallel Square'' and ``Parallel Cube'' Gases

The second, third, fourth, and fifth virial coefficients are evaluated exactly for gases whose molecules interact in two dimensions as rigid elastic squares with their sides parallel or in three dimensions as rigid elastic cubes with their edges parallel. Upper and lower bounds on the virial coefficients of gases composed of rigid circles or rigid spheres are obtained in terms of the virial coefficients for the parallel squares and parallel cubes.

Virial Expansion for Mixtures of Rigid Spheres

The equation of state for a binary mixture of rigid spheres of arbitrary size ratio is developed up to the fourth virial coefficient. The superposition approximation is employed and thus the fourth virial coefficient is inexact; however, the second and third are evaluated precisely. The important thermodynamic functions are tabulated as well as the changes in these functions upon mixing at constant free volume. The fourth virial coefficient is evaluated exactly for the case where one of the components is a point sphere and the result is compared with the approximate one.