Models van der Waals

Utilization of equations of state derived from the Van der Waals model has led to spectacular progress in the accuracy of calculations at medium and high pressure. 

R. Evans. The role of capillary wave fluctuations in determining the liquid-vapor interface. Analysis of the van der Waals model. Mol Phys 42 1169-1196, 1981. 

Here we present and discuss an example calculation to make some of the concepts discussed above more definite. We treat a model for methane (CH4) solute at infinite dilution in liquid under conventional conditions. This model would be of interest to conceptual issues of hydrophobic effects, and general hydration effects in molecular biosciences [1,9], but the specific calculation here serves only as an illustration of these methods. An important element of this method is that nothing depends restric-tively on the representation of the mechanical potential energy function. In contrast, the problem of methane dissolved in liquid water would typically be treated from the perspective of the van der Waals model of liquids, adopting a reference system characterized by the pairwise-additive repulsive forces between the methane and water molecules, and then correcting for methane-water molecule attractive interactions. In the present circumstance this should be satisfactory in fact. Nevertheless, the question frequently arises whether the attractive interactions substantially affect the statistical problems [60-62], and the present methods avoid such a limitation. 

Space filling van der Waals models (A3) are useful for illustrating the actual shape and size of molecules. These models represent atoms as truncated balls. Their effective extent is determined by what is known as the van der Waals radius. This is calculated from the energetically most favorable distance between atoms that are not chemically bonded to one another. 

The van der Waals model of monomeric insulin (1) once again shows the wedge-shaped tertiary structure formed by the two chains together. In the second model (3, bottom), the side chains of polar amino acids are shown in blue, while apolar residues are yellow or pink. This model emphasizes the importance of the hydrophobic effect for protein folding (see p. 74). In insulin as well, most hydrophobic side chains are located on the inside of the molecule, while the hydrophilic residues are located on the surface. Apparently in contradiction to this rule, several apolar side chains (pink) are found on the surface. However, all of these residues are involved in hydrophobic interactions that stabilize the dimeric and hexameric forms of insulin. 

The conformation of DNA that predominates within the cell (known as B-DNA) is shown schematically in Fig. A2 and as a van der Waals model in Fig. B1. In the schematic diagram (A2), the deoxyribose-phosphate backbone is shown as a ribbon. The bases (indicated by lines) are located on the inside of the double helix. This area of DNA is therefore apolar. By contrast, the molecule s surface is polar and negatively charged, due to the... 

The illustration opposite shows selected nucleic acid molecules. Fig. A shows various conformations of DNA, and Fig. B shows the spatial structures of two small RNA molecules. In both, the van der Waals models (see p. 6) are accompanied by ribbon diagrams that make the course of the chains clear. In all of the models, the polynucleotide backbone of the molecule is shown in a darker color, while the bases are lighter. 

There are several ways that dispersion (and the corresponding length scale) can be brought at the phenomenological level into the conservative part of the model. The two most well known examples of such theories are gradient (or van der Waals) models with energy... 

In conclusion we have presented a model for an electronic complex system with coexistence of different electronic phases at critical densities and coexistence of different liquids described by the modified van der Waals model as proposed for supercooled water. We discuss the critical values of the anisotropic interactions for the spinodal lines. We find that this model is able to describe the evolution of the pseudo-gap temperature versus doping in different cuprate families. 

The Tait equation predicts that the isothermal compressibility of a liquid approaches zero as the pressure becomes infinitely large. Is this reasonable in terms of the van der Waals model of real fluids ... 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

All of the models developed for predicting and correlating the properties of polymer solutions can be classified into two categories lattice models or van der Waals models. These two approaches can be used to derive activity coefficient models or equations of state. Activity coefficient models are not functions of volume and therefore are not dependent on... 

In both the lattice models and the van der Waals models, the behavior of the molecules is described as the sum of two contributions. The first contribution assumes that there are no energetic interactions between the molecules only the size and shape of the molecules need to be considered for this part. This is the contribution that would be predominant at very high temperatures where the kinetic energy of the molecules would be large compared to any interaction energies between the molecules. This interaction-free contribution is generally called the combinatorial or the athermal term. In the case of the van der Waals model, it is frequently referred to as the free volume term. 

In the van der Waals model the volume in which the molecules can translate is determined by the total volume of the system less the volume occupied by the molecules. Thus, the term "free volume." In this part of the treatment of the system intermolecular attractions are not taken into account, so this free volume term is the combinatorial (athermal) contribution. 

The second contribution in either the lattice or the van der Waals model is that originating from intermolecular attractions. This contribution is commonly referred to as the attractive energy term, the residual term, or the potential energy term. It is also known as the enthalpic contribution since the differences in interaction energies are directly responsible for the heats of mixing. This contribution is calculated by a product of a characteristic energy of interaction per contact and the number of contacts in the system. Van der Waals models use a similar expression for the interaction energy. 

Flory et al. (1964) developed an equation of state based on a van der Waals model given in reduced variables by ... 

Once you have the chemical potential for the van der Waals model of the previous exercise, find the equation of state by integrating the Gibbs-Duhem relation. Compare your result with the equation of state obtained from the approximate partition function using... 

Here we identify a natural extension of the van der Waals theory above this also serves to elaborate some notation that will be helpful subsequently. The van der Waals model was based upon the estimate (e) 6 (e — (( ))r). With the... 

We focus first on the outer-shell contribution of Eq. (7.8), p. 145. That contribution is the hydration free energy in liquid water for a distinguished water molecule under the constraint that no inner-shell neighbors are permitted. We will adopt a van der Waals model for that quantity, as in Section 4.1. Thus, we treat first the packing issue implied by the constraint Oy [1 i>a (7)] of Eq. (7.8) then we append a contribution due to dispersion interactions, Eq. (4.6), p. 62. Einally, we include a contribution due to classic electrostatic interactions on the basis of a dielectric continuum model. Section 4.2, p. 67. 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

---