Force virial, intermolecular

In a homogeneous fluid, the pressure is the normal force exerted by the fluid on one side of a unit area on the fluid on the other side expressed somewhat differently, it is the momentum transfer per unit area per unit time across an imaginary (flat) fixed surface. There are both kinetic and virial contributions to the pressure. The first arises from the momentum transported across the surface by particles that cross the surface in the unit time interval it yields the ideal-gas contribution, fld = Nk T/V, to the pressure. For classical particles interacting via pair-additive, central forces, the intermolecular potential contribution to the pressure can be determined using the method introduced by Irving and Kirkwood [43]. A clear discussion of this approach is given by Davis in [44], where it is shown to lead to the virial equation of state of a homogeneous fluid,... 

The model presented here develops these ideas and introduces features which make the calculation of mixture properties simple. For a polar fluid with approximately central dispersion forces together with a strong angle dependent electrostatic force we may separate the intermolecular potential into two parts so that the virial coefficients, B, C, D, etc. of the fluid can be written as the sum of two terms. The first terms B°, C°, D°, etc, arise from dispersion forces and may include a contribution arising from the permanent dipole of the molecule. The second terms contain equilibrium constants K2, K, K, etc. which describe the formation... 

In addition to the above effects, the intermolecular interaction may affect polymer dynamics through the thermodynamic force. This force makes chains align parallel with each other, and retards the chain rotational diffusion. This slowing down in the isotropic solution is referred to as the pretransition effect. The thermodynamic force also governs the unique rheological behavior of liquid-crystalline solutions as will be explained in Sect. 9. For rodlike polymer solutions, Doi [100] treated the thermodynamic force effects by adding a self-consistent mean field or a molecular field Vscf (a) to the external field potential h in Eq. (40b). Using the second virial approximation (cf. Sect. 2), he formulated Vscf(a), as follows [4] ... 

In principle, the expressions for pair potentials, osmotic pressure and second virial coefficients could be used as input parameters in computer simulations. The objective of performing such simulations is to clarify physical mechanisms and to provide a deeper insight into phenomena of interest, especially under those conditions where structural or thermodynamic parameters of the studied system cannot be accessed easily by experiment. The nature of the intermolecular forces responsible for protein self-assembly and phase behaviour under variation of solution conditions, including temperature, pH and ionic strength, has been explored using this kind of modelling approach (Dickinson and Krishna, 2001 Rosch and Errington, 2007 Blanch et al., 2002). 

Our expression for AS n— on which this term is based —was derived by assuming purely random placement of polymer segments and solvent molecules. This may not be fully justified because of intermolecular forces that we did not consider in arriving at Equation (66). We take advantage of this opportunity to allow for some bias in the placement of particles on the lattice and replace 1/2 R with AS as the contribution of entropy to the second virial coefficient. 

Real gases, on the other hand, consist of atoms or molecules that interact through intermolecular forces. Atoms/molecules attract at distant range and repel at near range they may be thought of as having a finite size. The theory of real gases accounts for these facts by means of a virial expansion,... 

One source of information on intermolecular potentials is gas phase virial coefficient and viscosity data. The usual procedure is to postulate some two-body potential involving 2 or 3 parameters and then to determine these parameters by fitting the experimental data. Unfortunately, this data for carbon monoxide and nitrogen can be adequately represented by spherically symmetric potentials such as the Lennard-Jones (6-12) potential.48 That is, this data is not very sensitive to the orientational-dependent forces between two carbon monoxide or nitrogen molecules. These forces actually exist, however, and are responsible for the behavior of the correlation functions and - In the gas phase, where orientational forces are relatively unimportant, these functions resemble those in Figure 6. On the other hand, in the liquid these functions behave quite differently and resemble those in Figures 7 and 8. 

At normal temperature levels, intermolecular attractions prevail, and the second virial coefficients are negative. (See Sec. 16.2 for a discussion of the connection between intermolecular forces and the second virial coefficient.) If interactions between unlike molecular pairs are weaker than interactions between pairs of molecules of the same kind,... 

In dilute solution, all macromolecular chains undergo interactions with each other resulting in the so-called intermolecular excluded volume effect, corresponding to the intermolecular potential. This effect is also observed if one does not assume particular cohesive forces to occur between the macromolecular chains. Under these conditions, the second virial coefficient is calculated from the equation1,2) ... 

The statistical mechanical verification of the adsorption Equation 11 proceeds most conveniently with use of the expression for y given by Equation 5. An identical starting formula is obtained via the virial theorem or by differentiation of the grand partition function (3). We simplify the presentation, without loss of generality, by restricting ourselves to multicomponent classical systems possessing a potential of intermolecular forces of the form... 

For higher ionic strength, e g. highly saline waters the PITZER equation can be used (Pitzer 1973). This semi-empirical model is based also on the DEBYE-HUCKEL equation, but additionally integrates virial equations (vires = Latin for forces), that describe ion interactions (intermolecular forces). Compared with the ion dissociation theory the calculation is much more complicated and requires a... 

The virial expansion has enjoyed greater appeal, especially as applied to lyotropic systems. Onsager s classic theory rests on analysis of the second virial coefficient for very long rodlike particles. It was the first to show that a solution of hard, asymmetric particles such as long rods should separate into two phases above a threshold concentration that depends on the axial ratio of the particles. One of these phases should be anisotropic (nematic), the other completely isotropic. The former is predicted to be somewhat more concentrated than the latter, but it is the alignment (albeit imperfect) of the solute molecules that is the predominent distinction. The phase separation is a consequence of shape asymmetry alone intervention of intermolecular attractive forces is not required. 

A standard work of reference on intermolecular forces is due to appear very shortly.] The different methods of approach to the study of intermolecular forces between like and unhke molecules are carefully discussed in this book. These methods include studies of both thermodynamic properties (e.g. virial coefficients) and also of non-equilibrium measurements (e.g. thermal conductivity, diffusion and thermal diffusion). 

The results of virial coefficient measurements are also discussed by E. A. Guggenheim.il For an interesting recent discussion of the data concerning intermolecular forces between unlike molecules see A. Michels and A. J. M. Boerboom. The relationship between virial coefficients and intermolecular forces has also been discussed in some detail by Guggenheim, and by J. S. Rowlinson. ... 

The fundamental assumption of the vdW eos is that the intermolecular force is separated into repulsive and attractive pressures. The separate representation of the repulsive and attractive forces is maintained in the vdW-type eos and the perturbation equations, but is dropped in the virial equations and the extended virial equations. 

After the pioneering quantum mechanical work not much new ground was broken until computers and software had matured enough to try fresh attacks. In the meantime the study of intermolecular forces was mainly pursued by thermodynamicists who fitted model potentials, often of the Lennard-Jones form [10] 4e[(cr/R) — (cr// ) ], to quantities like second virial coefficients, viscosity and diffusion coefficients, etc. Much of this work is described in the authoritative monograph of Hirschfelder et al. [11] who, incidentally, also gave a good account of the relationship of Drude s classical work to that of London. 

Ajit Thakkar, bom in Poona, India in 1950, left home at 17 to explore the West. A circuitous route led him to Queen s University in Kingston, Ontario. A summer job programming calculations of virial coefficients and transport cross-sections using FORTRAN IV, dreadful JCL, and punched cards on an IBM 360/50 drew him to computational chemistry. In 1976, he completed a Ph.D. in theoretical chemistry guided by Vedene Smith and influenced by Robert Parr. His faculty career began at the University of Waterloo and, since 1984, continued at the idyllic Fredericton campus of the University of New Bmnswick. He is now a University Research Professor, and author of more than 200 articles on molecular properties, electron densities and intermolecular forces. 

Polyelectrolyte molecules in highly dilute aqueous solutions exert strong electrical repulsions on each other. These repulsive forces are long range (proportional to l/r ) by comparison with normal dispersion forces (proportional to 1/r ), and as a consequence the intermolecular interactions persist down to the lowest measured concentrations. In osmotic-pressure measurements on polyelectrolytes, the Donnan membrane equilibrium must be satisfied and experimental results indicate that the second virial coefficient in the osmotic-pressure equation (p. 915) becomes very large. 

If the gas particles interact through a pairwise potential, then the contribution to the virial from the intermolecular forces can be derived as follows. Consider two atoms i and separated by a distance ry. 

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