Van der Waals theories

The van der Waals theory assumes a molecular structure of matter, where matter means gases or liquids. The interaction between molecules requires modification of 

Molecules, or atoms in the case of noble gases, at close proximity tend to repel. The attending volume reduction is —nb, where is a parameter accounting for the exclusion of (molar) volume that a particle imposes on the other particles in V. At large distances particles attract, which in turn reduces the pressure. The particular form of this pressure correction, i.e. ainjvV, may be motivated as follows. The number ofparticlepairsinasystem consisting of Aparticles is A(A — l)/2 N /2. 

Expressed in moles this leads to the factor n. The attraction is limited to not too large particle-to-particle separation. We assume that two particles feel attracted if they are in the same volume element AV. The probability that two particular particles are found within A V simultaneously (A V/ V). Assuming this to be true for all possible pairs leads to an overall number of attracted molecules proportional to (n/ V). The resulting Eq. (4.1) is the van der Waals equation of state for gases and 

Hentschke, Thermodynamics, Undergraduate Lecture Notes in Physics, 

The parameters a and b may be estimated by measuring the pressure as function of temperature at low densities. The result may then be approximated using the following low density expansion of Eq. (4.1)  

---

The gradient model has been combined with two equations of state to successfully model the temperature dependence of the surface tension of polar and nonpolar fluids [54]. Widom and Tavan have modeled the surface tension of liquid He near the X transition with a modified van der Waals theory [55]. 

Kao M, Uhlenbeck G E and Hemmer P 1963 Gn van der Waals theory of vapor-liquid equilibrium. I. Discussion of a one-dimensional model J. Math. Phys. 4 216... 

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

To describe the simple phenomena mentioned above, we would hke to have only transparent approximations as in the Poisson-Boltzmann theory for ionic systems or in the van der Waals theory for non-coulombic systems [14]. Certainly there are many ways to reach this goal. Here we show that a field-theoretic approach is well suited for that. Its advantage is to focus on some aspects of charged interfaces traditionally paid little attention for instance, the role of symmetry in the effective interaction between ions and the analysis of the profiles in terms of a transformation group, as is done in quantum field theory. 

Similarly, examples of barriers arising largely from simple steric hindrance can be found, as for instance in the hindered diphenyls.35 On the other hand there are many arguments suggesting that this is not the important force in ethane and similar molecules. It would be difficult to understand the relatively slow fall in barrier from ethane to methyl silane to methyl germane on a van der Waals repulsion basis. Furthermore, the small effect of substituting F, Cl, or Br on one end would also seem mysterious. The equilibrium orientation in propylene is opposite to the predictions of one of the quantitative van der Waals theories. Finally, the apparently small effect of bending back the C—H bonds is not in accord with either the electrostatic or van der Waals pictures. 

Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite constants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 

Van der Waals further finds a relation between the temperature coefficient of surface tension and the molecular surface energy which is in substantial agreement with the Eotvos-Ramsay-Shields formula (see Chapter V.). He also arrives at a value for the thickness of the transition layer which is of the order of magnitude of the molecular radius, as deduced from the kinetic theory, and accounts qualitatively for the optical effects described on p. 33. Finally, it should be mentioned that Van der Waals theory leads directly to the conclusion that the existence of a transition layer at the boundary of two media reduces the surface tension, i.e., makes it smaller than it would be if the transition were abrupt—a result obtained independently by Lord Rayleigh. 

The most controversial and contradicting problem is, perhaps, the natural and collectorless floatability of sulphide minerals. Gaudin (1957) classified the natural hydrophobicity of different minerals according to their crystal structure and showed that most sulphide minerals were naturally hydrophobic to some extent, which had been fiirther proved based on van der Waals theory by Chander (1988, 1999). Lepetic (1974) revealed the natural floatability of chalcopyrite in dry grinding. Finklestein (1975, 1977) demonstrated that orpiment, realgar and molybdenite were naturally floatable, and that pyrite and chalcopyrite had natural floatability at certain conditions due to the formation of surface elemental sulphur. Buckley and Woods (1990,1996) attributed the natural floatability of chalcopyrite... 

The viewpoint parallels that of many other theories of condensed state behavior. The van der Waals theory develops an equation of state for dense gases from the assumption that each molecule moves in an average field provided by its neighbors and that the molecules contribute additively to the pressure. The Flory-Huggins thermodynamic theory of concentrated polymer solutions proceeds similarly. Chains select configurations on a lattice partially occupied by... 

An alternative, already implicit in van der Waals theory of the surface tension [237], exploits an interconnection between the coefficient Co and the pair correlation function g(ra, rp) of the inhomogeneous system. As the local density distribution determining g(ra, rp) is unknown, one has to resort to a... 

It is desirable to have something more sophisticated than the van der Waals theory. Such sophisticated theories inform us of the relation between the thermodynamic properties of the fluid and the interactions between the molecules or particles and also of the structure of the fluid. Even though there is no static structure, we can still speak of a structure. We can measure experimentally the radial distribution function (RDF), g(i 12), of a fluid. The RDF gives the probability of finding a pair of particles, whose centers are located at ri and r2, separated by a distance Rn — ri —r2. ... 

Types ionic, covalent, metallic, hydrogen bonding, van der Waals theory (including London dispersion forces)... 

John Prausnitz I d like to make one quick addendum. I want to defend the use of molecular simulations because we have gained insight from them as well. Let me mention one outstanding example. Until about 20 years ago, we believed that you could only condense a phase with attractive forces, and no one ever questioned that myth. Then computer simulations were done in the 1960s by Bemi Alder and his associates. The results showed that even for hard spheres, without any attractive forces, you can get a phase transition. This was never present in the van der Waals theory. I want to emphasize that simulations also add to our conceptual knowledge. 

The other cause, the density effect, is especially important at high densities, where molecules are more or less confined to cells formed by their neighbors. In analogy to the well-known quantum mechanical problem of a particle in a box, the translational energies of such molecules are quantized, and this has an effect on the thermodynamic properties. In 1960 Levelt Sengers and Hurst [3] tried to describe the density quantum effect in term of the Lennard-Jones-Devonshire cell model, and in 1980 Hooper and Nordholm proposed a generalized van der Waals theory [4]. The disadvantage of both approaches is that, in the classical limit, they reduce to rather unsatisfactory equations of state. 

In the context of van der Waals theory, a and b are positive parameters characterizing, respectively, the magnitude of the attractive and repulsive (excluded volume) intermolecular interactions. Use this partition function to derive an expression for the excess chemical potential of a distinguished molecule (the solute) in its pure fluid. Note that specific terms in this expression can be related to contributions from either the attractive or excluded-volume interactions. Use the Tpp data given in Table 3.3 for liquid n-heptane along its saturation curve to evaluate the influence of these separate contributions on test-particle insertions of a single n-heptane molecule in liquid n-heptane as a function of density. In light of your results, comment on the statement made in the discussion above that the use of the potential distribution theorem to evaluate pff depends on primarily local interactions between the solute and the solvent. 

The reappreciation of the van der Waals theory refines these two arguments. A decisive feature is a sharp distinction between the differing roles of attractive and repulsive interactions the fact that these interactions of differing physical... 

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... 

The increase of h depends on the deviations from the simple gas laws. According to Van der Waals theory, h must be a maximum when the product fv is a minimum. This relationship is confirmed quahtatively by Koch s experiments. 

This result can be deduced from Van der Waals theory. Since there is no interchange of heat with the surroundings, the change in the total energy of the gas is equal to the work... 

Real solid bodies, therefore, differ considerably from the perfect solid at higher temperatures, but appear to approach asymptotically to the perfect condition as the temperature is lowered. The conception of a perfect solid body hke that of a perfect gas is only true in the limiting case. It will, perhaps, be possible to build up a complete theory of the solid state on the basis of Einstein s hypothesis, as van der Waals theory was evolved from the conceptions of the classical theory of... 

The free ions of weak electrolytes, even in relatively concentrated solutions, are present in such small quantities that they still conform to the simple laws. It is possible that the free electric charges on the ions exert appreciable forces on one another in concentrated solutions. This would cause deviations from the simple laws analogous to the deviations from the simple gas laws which are accounted for by van der Waals theory. 

Van der Waals theory is a typical example of a mean field theory in the sense that z is the only position variable. The interfacial layer has the same averaged properties everywhere parallel to the surface. This model works well as long as the temperature is far below its critical value. When T - T fluctuations in the density profile start to become important so that taking average values becomes less accurate. This is the main reason why [2.5.29] is not a good approximation. 

---