Van der Waals expression

Fig. 23. Graph of equation of state, such as Van der Waals expression, showing metastable conditions.

The mutual dependence of the pressure, volume, and temperature of a substance is described by its equation of state. Many such equations have been proposed for the description of the actual properties of substances (and mixtures) in the gaseous and liquid states. The van der Waals expression is just one of these and of limited applicability. The virial equation of state ... 

Does the precision of the measurements justify the van der Waals expression, Eq. (35), instead of the perfect-gas expression, Eq. (34) Justify your answer in terms of your... 

To compare a very accurate and very rough approximations for repulsive term the classical van der Waals expression... 

An improvement of a classical repulsive expression (2) for one dimensional system of hard sphere by the very accurate presentation of the Liu s EoS (Figure 7) doesn t change a topologic picture of phase diagram in comparison with classical van der Waals expression for repulsive term. It seems that an improvement of repulsive term makes more plausible of isotherm behavior near second critical point. To analyze a qualitative behavior of thermod5mamic surface anomalies in whole via simpler model is preferable due to a topological equivalence of models under consideration. 

We can check that a substitution of n, from Equations 5.13 and 5.23 into Equation 5.27 yields, respectively, the Frumkin and van der Waals expressions for y,j, specifically Equations 5.17 and... 

For a substance obeying van der Waals expression we would have—... 

In the classical van der Waals expression (Equation (64)) we considered the gas molecules to be hard spheres of diameter, <7. If we write this expression in terms of the molecular parameters of a gas then we have... 

In his thesis van der Waals expressed his desire to determine a quantity that plays a peculiar role in Laplace s theory of capillarity. He was referring to a molecular pressure, a measure for the cohesion of matter. He was—in the Newtonian tradition—looking for a way of grasping inter-molecular forces, the forces that would appear in his own equation of state. 

The usefulness of the van der Waals expression for attractive interactions has been demonstrated in many examples. After Frank [1945] and Flildebrand and Scott [ 1950, 1962], intermolecular interaction potentials such as... 

A similar analysis may be applied to the partially ordered nematic fluids composed of molecules comprising the mesogenic unit and flexible chain segments. In the LC state, one must consider the orientation-dependent interactions in addition to those of the isotropic nature. As mentioned earlier, the volume dependence (1/V ) incorporated in the Maier-Saupe expression may be replaced by MV. In its modified form, Maier-Saupe potential can easily be accommodated by introducing an additional term in the conventional van der Waals expression ... 

Show that the van der Waals equation of state gives = -ap for a pure fluid. Here p is the molar density while a is the parameter in the equation of state and is assmned to be constant. What is the significance of the sign of u 7 What is the ideal-gas limit for the van der Waals expression for u ... 

A quadratic force field is limited to terms like (/ — Iq), and higher terms such as (/ — /q) are excluded for bending and stretching, although the torsional and van der Waals expressions may be more complicated. 

Dispersion forces are universal because they attract all molecules together, regardless of their specific chemical nature. The potential energy of dispersion attraction between two isolated molecules decays with the sixth power of the separation distance. Based on the so-called Hamaker theory (i.e., the method of pair-wise summation of intermolecular forces) or the more modern Lifshitz macroscopic treatment of strictly additive London forces, it is possible to develop the so-called Lifshitz-Van der Waals expression for the macroscopic interactions between macroscopic-in-size objects (i.e., macrobodies) [19, 21], Such an expression strongly depends on the shapes of the interacting macrobodies as well as on the separation distance (non-retarded or retarded interaction). For two portions of the same phase of infinite extent bounded by parallel flat surfaces, at a distance h apart, the potential energy of macroscopic attraction is ... 

The macroscopic effect of cohesion due to dispersion forces is usually calculated from the Lifshitz-Van der Waals expression (6.6), providing that the separation distance h is known. Israelachvili [21, 22] has proposed a universal value of 0.165nm to describe the effective spacing h between molecular planes in all liquids with molecules interacting solely through dispersion forces. In this case, the Gibbs energy of cohesion may be evaluated as... 

To calculate the potential energy (/2) associated with each molecule we use Eq. 17.4.4 and replace the lower limit by a, since g(r) is zero for r < a. van der Waals, in effect, assumed that g r) is independent of temperature and volume for all values of r as a result, the integral is a constant, which van der Waals expressed as (-2alN ), where a is a positive constant (Vera and Prausnitz). Thus, on a molar basis ... 

The classic theory due to van der Waals provides an important phenomenological link between the structure of an interface and its interfacial tension [50-52]. The expression... 

The quantity zoi will depend very much on whether adsorption sites are close enough for neighboring adsorbate molecules to develop their normal van der Waals attraction if, for example, zu is taken to be about one-fourth of the energy of vaporization [16], would be 2.5 for a liquid obeying Trouton s rule and at its normal boiling point. The critical pressure P, that is, the pressure corresponding to 0 = 0.5 with 0 = 4, will depend on both Q and T. A way of expressing this follows, with the use of the definitions of Eqs. XVII-42 and XVII-43 [17] ... 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

Van der Waals and especially van Laar simplified these expressions by assuming a geometric mean for and an aritlmietic mean for... 

Besides the expressions for a surface derived from the van der Waals surface (see also the CPK model in Section 2.11.2.4), another model has been established to generate molecular surfaces. It is based on the molecular distribution of electronic density. The definition of a Limiting value of the electronic density, the so-called isovalue, results in a boundary layer (isoplane) [187]. Each point on this surface has an identical electronic density value. A typical standard value is about 0.002 au (atomic unit) to represent electronic density surfaces. 

Two potential energy expressions used for van der Waals interactions are the Fennard-Jones 6/12 potential function or some modification thereof. 

Stretching, bond bending, torsions, electrostatic interactions, van der Waals forces, and hydrogen bonding. Force fields differ in the number of terms in the energy expression, the complexity of those terms, and the way in which the constants were obtained. Since electrons are not explicitly included, electronic processes cannot be modeled. 

The virial equations are unsuitable forhquids and dense gases. The simplest expressions appropriate (in principle) for such fluids are equations cubic in molar volume. These equations, inspired by the van der Waals equation of state, may be represented by the following general formula, where parameters b, 9 5, S, and Tj each can depend on temperature and composition ... 

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