Intermolecular virial

Buckingham R A and Corner J 1947 Tables of second virial and low-pressure Joule-Thompson coefficients for intermolecular potentials with exponential repulsion Proc. R. Soc. A 189 118... 

It is interesting to note that for a van der Waals gas, the second virial coefficient equals b - a/RT, and this equals zero at the Boyle temperature. This shows that the excluded volume (the van der Waals b term) and the intermolecular attractions (the a term) cancel out at the Boyle temperature. This kind of compensation is also typical of 0 conditions. 

Intermolecular potential functions have been fitted to various experimental data, such as second virial coefficients, viscosities, and sublimation energy. The use of data from dense systems involves the additional assumption of the additivity of pair interactions. The viscosity seems to be more sensitive to the shape of the potential than the second virial coefficient hence data from that source are particularly valuable. These questions are discussed in full by Hirschfelder, Curtiss, and Bird17 whose recommended potentials based primarily on viscosity data are given in the tables of this section. 

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 Fig. 5.1 we see that the intermolecular interactions accounting for VCIE (upper curve) and VPIE (lower curve) differ not in kind, only in degree. The well depth for gas-gas interaction is available from analysis of the virial coefficient of the parent isotopomer, that for the condensed phase can be obtained by combining the energy of vaporization and the zero point energies of the condensed and ideal vapor phases. 

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] ... 

A knowledge of the magnitude of these quantities and their quantitative contributions to /uE can give insight into the detailed character of the intermolecular interactions in a biopolymer solution, including the means by which their properties may be manipulated. The sign of the second virial coefficient provides a simple indicator of the type of interactions... 

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,... 

Collision-induced absorption takes place by /c-body complexes of atoms, with k = 2,3,... Each of the resulting spectral components may perhaps be expected to show a characteristic variation ( Qk) with gas density q. It is, therefore, of interest to consider virial expansions of spectral moments of binary mixtures of monatomic gases, i.e., an expansion of the observed absorption in terms of powers of gas density [314], Van Kranendonk and associates [401, 403, 314] have argued that the virial expansion of the spectral moments is possible, because the induced dipole moments are short-ranged functions of the intermolecular separations, R, which decrease faster than R 3. We label the two components of a monatomic mixture a and b, and the atoms of species a and b are labeled 1, 2, N and 1, 2, N, respectively. A set of fc-body, irreducible dipole functions U 2, Us,..., Uk, is introduced (as in Eqs. 4.46), according to... 

The virial coefficients B(T), C(T), D(T),... are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B T) to be explicitly evaluated by statistical mechanical methods as... 

The viscosity behavior of poly[(a-carboxymethyl)ethyl isocyanide] may be studied in neutral organic solvents. The concentration dependence of its reduced specific viscosity in 1,2-dichloroethane is shown in Fig. 11. A linear dependence indicates that the coefficients of higher concentration terms of the usual virial equations are negligibly small—a case which should be found with molecules, such as stiff rods, that give few intermolecular entanglements in dilute solution. 

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. 

These tables give "covolumes of propellants with a systematic error of less than 5%. The basis of Corner s theory is the expression of the 2nd virial coefficient of a gas as a simple function of the parameters of the intermolecular field... 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

Shielding Surfaces for Two Interacting Molecules. In the Maryland meeting Jameson and de Dios reported the first ab initio calculations of the rare gas pair intermolecular shielding surfaces for Ar2, ArNe, Ne2, and NeHe (55). With these shielding surfaces it was possible to calculate the second virial coefficient for nuclear shielding as a function of temperature, using the well established intermolecular potential functions V(R) for the pair. 

For gases, n = e 1 is an excellent approximation. The easiest approach to condensed phases maintains this approximation, where calculations of the molecular first-order and response properties are performed for the isolated molecule, while accounting for the effect of intermolecular interactions through the number density N = Aa/Vm, and therefore by taking appropriate values of Vm. This rough, often at best qualitative, approach is somewhat relaxed by employing expansions of the birefringence constant with the density, that is in inverse powers of Vm. This introduces the appropriate virial coefficients [15,16]... 

---