scholarly journals On the determination of molecular fields. —II. From the equation of state of a gas

1924 ◽  
Vol 106 (738) ◽  
pp. 463-477 ◽  
Keyword(s):  
Equation Of State ◽  
Molecular Model ◽  
Virial Coefficients ◽  
Preceding Paper ◽  
Molecular Field ◽  
Inverse Power ◽  
Theoretical Expression ◽  
Inverse Power Law ◽  
Special Cases ◽  
Molecular Fields

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 formulæ for B are easily deduced from the general one given here.

scholarly journals On the atomic fields of helium and neon

1925 ◽  
Vol 107 (741) ◽  
pp. 157-170 ◽  
Keyword(s):  
Temperature Variation ◽  
Power Law ◽  
Molecular Model ◽  
Second Virial Coefficient ◽  
Inverse Power ◽  
Rigid Sphere ◽  
Inverse Power Law ◽  
Molecular Fields ◽  
First Time ◽  
Distinct Maximum

The methods of determining molecular fields, described in previous papers, are here applied with certain modifications to the cases of helium and neon. There is considerable observational material available in the case of helium, both as regards its isotherms and as regards its viscosity over a large range of temperature, but it has not as yet been used with any success to yield satisfactory information about its outer field. Keesom, who obtained a number of theoretical formulæ for the second virial coefficient of the equation of state, was unable to find any which would satisfactorily explain the temperature variation of this coefficient in the case of helium, as determined experimentally by Kamerlingh Onnes; in fact, the experimental coefficient showed at the higher temperatures a distinct maximum, and this property none of his theoretical formulæ possessed. The maximum property has, moreover, been confirmed by the later experimental work of Holborn and Otto. The formula, given in a recent paper, is, however, more successful in this direction, and the method of using it, there described, leads to the conclusion that the field of helium can well be represented by the superposition of repulsive and attractive fields, each according to an inverse power law. Furthermore, the values of the force constants are here determined. It has long been recognised that the temperature variation of the viscosity of helium cannot adequately be represented by the theoretical Sutherland formula, with the obvious inference that helium cannot be regarded (even roughly) as a rigid sphere with a superimposed attractive field. This is perhaps not surprising in view of the comparatively simple electronic structure of helium. Kamerlingh Onnes has shown that the variation can best be represented by the simple law, in which the viscosity varies as a power of the temperature. This formula, although first given as an empirical result, corresponds theoretically to a molecular model, in which the molecules are regarded as point centres of force repelling according to an inverse power law. The information available concerning the viscosity of helium is here used, for the first time, to determine the actual value of the repulsive force constant. It is further shown that the result is consistent with that found by the other (entirely independent) method, above described. The field thus determined is one of repulsion according to an inverse 14th power of the distance and a very weak attraction according to an inverse 5th power.

scholarly journals The molecular fields of hydrogen, nitrogen and neon

1926 ◽  
Vol 112 (760) ◽  
pp. 214-229 ◽  
Keyword(s):  
Special Interest ◽  
Power Laws ◽  
Experimental Information ◽  
Repulsive Force ◽  
Independent Method ◽  
Molecular Field ◽  
Cohesive Force ◽  
Inverse Power ◽  
Molecular Fields

Since the publication of some recent papers on molecular fields, some new experimental information has become available, which permits of further determinations of the forces between molecules. Hydrogen, nitrogen and neon are now added to the list of gases whose isotherms have been obtained by the precise methods of Holborn and Otto. The publication of these results for neon is of special interest, because one determination of the molecular field of neon has already been made, and it is valuable to have another independent method of attacking the same problem. A method of determining molecular fields from measurements of the isotherms of a gas has been described in an earlier paper. It proceeds on the assumption that the molecular field is spherically symmetrical and that it can be expressed in terms of inverse power laws, one to represent the repulsive force and one to represent the cohesive force. The method shows whether any particular model is a suitable one or not, and when it is, leads to a determination of the force constants.

Inverse Power Law Behavior in a Memory Scanning Experiment

2006 ◽  
Author(s):  
Gerardo Ramirez ◽  
Sonia Perez ◽  
John G. Holden
Keyword(s):  
Power Law ◽  
Inverse Power ◽  
Memory Scanning ◽  
Inverse Power Law

Critical consolute point in hard-sphere binary mixtures: Effect of the value of the eighth and higher virial coefficients on its location

2010 ◽  
Vol 75 (3) ◽  
pp. 359-369 ◽  
Author(s):  
Mariano López De Haro ◽  
Anatol Malijevský ◽  
Stanislav Labík
Keyword(s):  
Equation Of State ◽  
Binary Mixtures ◽  
Hard Sphere ◽  
Hard Spheres ◽  
Virial Coefficients ◽  
Fluid Mixture ◽  
Binary Fluid ◽  
Virial Series ◽  
Consolute Point ◽  
Critical Constants

Various truncations for the virial series of a binary fluid mixture of additive hard spheres are used to analyze the location of the critical consolute point of this system for different size asymmetries. The effect of uncertainties in the values of the eighth virial coefficients on the resulting critical constants is assessed. It is also shown that a replacement of the exact virial coefficients in lieu of the corresponding coefficients in the virial expansion of the analytical Boublík–Mansoori–Carnahan–Starling–Leland equation of state, which still leads to an analytical equation of state, may lead to a critical consolute point in the system.

The Bender Equation of State and Virial Coefficients

2000 ◽  
Vol 65 (9) ◽  
pp. 1464-1470 ◽  
Author(s):  
Anatol Malijevský ◽  
Tomáš Hujo
Keyword(s):  
Equation Of State ◽  
Virial Coefficient ◽  
Virial Coefficients ◽  
Low Density ◽  
Second Virial Coefficients ◽  
The Third ◽  
Temperature Dependences ◽  
Experimental Values

The second and third virial coefficients calculated from the Bender equation of state (BEOS) are tested against experimental virial coefficient data. It is shown that the temperature dependences of the second and third virial coefficients as predicted by the BEOS are sufficiently accurate. We conclude that experimental second virial coefficients should be used to determine independently five of twenty constants of the Bender equation. This would improve the performance of the equation in a region of low-density gas, and also suppress correlations among the BEOS constants, which is even more important. The third virial coefficients cannot be used for the same purpose because of large uncertainties in their experimental values.

scholarly journals On Using the BMCSL Equation of State to Renormalize the Onsager Theory Approach to Modeling Hard Prolate Spheroidal Liquid Crystal Mixtures

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 846
Author(s):  
Donya Ohadi ◽  
David S. Corti ◽  
Mark J. Uline
Keyword(s):  
Equation Of State ◽  
Second Virial Coefficient ◽  
Virial Coefficients ◽  
Pure Component ◽  
Excluded Volume ◽  
Theory Approach ◽  
Prolate Spheroidal ◽  
Empirical Correction ◽  
Prolate Spheroids ◽  
Onsager Theory

Modifications to the traditional Onsager theory for modeling isotropic–nematic phase transitions in hard prolate spheroidal systems are presented. Pure component systems are used to identify the need to update the Lee–Parsons resummation term. The Lee–Parsons resummation term uses the Carnahan–Starling equation of state to approximate higher-order virial coefficients beyond the second virial coefficient employed in Onsager’s original theoretical approach. As more exact ways of calculating the excluded volume of two hard prolate spheroids of a given orientation are used, the division of the excluded volume by eight, which is an empirical correction used in the original Lee–Parsons resummation term, must be replaced by six to yield a better match between the theoretical and simulation results. These modifications are also extended to binary mixtures of hard prolate spheroids using the Boublík–Mansoori–Carnahan–Starling–Leland (BMCSL) equation of state.

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