Van-der-Waals diagram

Superposition diagrams can also be constructed using depictions of space-filling models. A van der Waals surface depiction for each molecule is constructed, the diagrams so obtained are superimposed on each other. Then one van der Waals diagram is subtracted from the other.The... 

Fig. 18.1. Van der Waals diagrams, a A-DNA b B-DNA. Both duplexes are dodecamer fragments with the sequence [dfCGCGAATTCGCG j and idealized conformations. The drawings illustrate the differences between the geometries of the major grooves (base atoms dashed) and the minor grooves of the two conformation families. Whereas the base pairs are tilted with respect to the helix axis in A-DNA, they are practically normal to the helix axis in B-DNA. Carbon atoms are black, nitrogen atoms are blue, oxygen atoms are red, phosphorus atoms are yellow and hydrogen atoms are white...

Plotting the volume-pressure relation similar to a van-der-Waals diagram for TmSei- -Te we obtain fig. 97 (Boppart and Wachter 1984a). We have been able to produce a TmSeo.5oTeo.50 sample which has practically the critical composition, and for which... 

Fig. 97. The pressure versus volume change of TmSe, Te plotted in the form of a van-der-Waals diagram. The critical point is indicated by the solid circle. (After Boppart and Wachter 1984a.)...

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

Figure A2.5.13. Global phase diagram for a van der Waals binary mixture for whieh The...

In reeent years global phase diagrams have been ealeulated for other equations of state, not only van der Waals-like ones, but others with eomplex temperature dependenees. Some of these have managed to find type VI regions in the overall diagram. Some of the reeent work was brought together at a 1999 eonferenee [4]. 

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

One can effectively reduce the tliree components to two with quasibinary mixtures in which the second component is a mixture of very similar higher hydrocarbons. Figure A2.5.31 shows a phase diagram [40] calculated from a generalized van der Waals equation for mixtures of ethane n = 2) with nomial hydrocarbons of different carbon number n.2 (treated as continuous). It is evident that, for some values of the parameter n, those to the left of the tricritical point at = 16.48, all that will be observed with increasing... 

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

Liquid crystals stabilize in several ways. The lamellar stmcture leads to a strong reduction of the van der Waals forces during the coalescence step. The mathematical treatment of this problem is fairly complex (28). A diagram of the van der Waals potential (Fig. 15) illustrates the phenomenon (29). Without the Hquid crystalline phase, coalescence takes place over a thin Hquid film in a distance range, where the slope of the van der Waals potential is steep, ie, there is a large van der Waals force. With the Hquid crystal present, coalescence takes place over a thick film and the slope of the van der Waals potential is small. In addition, the Hquid crystal is highly viscous, and two droplets separated by a viscous film of Hquid crystal with only a small compressive force exhibit stabiHty against coalescence. Finally, the network of Hquid crystalline leaflets (30) hinders the free mobiHty of the emulsion droplets. 

Figure 4.17 Schematic diagram of bound tyrosine to tyrosyl-tRNA synthetase. Colored regions correspond to van der Waals radii of atoms within a layer of the structure through the tyrosine ring. Red is bound tyrosine green is the end of P strand 2 and the beginning of the following loop region yellow is the loop region 189-192 and brown is part of the a helix in loop region 173-177.

Raveau now calculated the values of p, v from van der Waals equation, plotted the logarithms, and compared the diagram with a similar one drawn from the experimental results. The results showed that the diagrams could not be made to fit in the ease of carbon-dioxide and acetylene, the divergencies being very marked near the critical point. 

Figure 6.2. Potential energy diagram showing the attractive Van der Waals interaction and the repulsive interaction due to Pauli repulsion,...

Fig. 2. Components of Li enthalpies of complexation with methylamines. Successive steps indicate the effect on energy of interaction between Li and the amine of inclusion of additional components of the binding energy. The diagram shows that the permanent dipoles on amines (the charge on the nitrogen of the isolated amine) favor ammonia over trimethylamine complexation, but that polarizability and inductive effects (shift of negative charge onto the nitrogen in the complex) cause a massive turnaround in favor of complexation with trimethylamine rather than ammonia. Of particular importance is the near inversion of order caused by the addition of repulsive van der Waals terms. Modified after Ref. (9).

Fig. 6. Comparative projections along the c axis of the diol molecules and the canals they enclose in 1, 2,3,8 and 9. The bond thickening signifies depth in individual molecules only, because the helical characteristic is absent from these projections of the lattice. The canal boundaries are marked as the intersecting projected van der Waals spheres of the hydrogen atoms which line the canals. All five diagrams are presented on the same scale. Significant hydrogen atoms are marked as filled circles, and the spines are circled...

Chapter 6. The outer contour in this map is for a density of 0.001 au, which has been found to represent fairly well the outer surface of a free molecule in the gas phase, giving a value of 190 pm for the radius in the direction opposite the bond and 215 pm in the perpendicular direction. In the solid state molecules are squashed together by intermolecular forces giving smaller van der Waals radii. Figure 5.2b shows a diagram of the packing of the Cl2 molecules in one layer of the solid state structure of chlorine. From the intermolecular distances in the direction opposite the bond direction and perpendicular to this direction we can derive values of 157 pm and 171 pm for the two radii of a chlorine atom in the CI2 molecule in the solid state. These values are much smaller than the values for the free molecule in the gas phase. Clearly the Cl2 molecule is substantially compressed in the solid state. This example show clearly that the van der Waals of an atom radius is not a well defined concept because, as we have stated, atoms in molecules are not spherical and are also compressible. 

Figure 5 P — V diagram of the Van der Waals equation of state. The solutions to these simultaneous equations are...

The Maxwell construction produces a Van der Waals phase diagram that resembles the experimental results of argon, shown in figure 1, in all respects. 

Figure 5.11 The p-T(a) and the T-p (b) phase diagrams of H2O calculated using the van der Waals equation of state.

It has been proposed to define a reduced temperature Tr for a solution of a single electrolyte as the ratio of kgT to the work required to separate a contact +- ion pair, and the reduced density pr as the fraction of the space occupied by the ions. (M+ ) The principal feature on the Tr,pr corresponding states diagram is a coexistence curve for two phases, with an upper critical point as for the liquid-vapor equilibrium of a simple fluid, but with a markedly lower reduced temperature at the critical point than for a simple fluid (with the corresponding definition of the reduced temperature, i.e. the ratio of kjjT to the work required to separate a van der Waals pair.) In the case of a plasma, an ionic fluid without a solvent, the coexistence curve is for the liquid-vapor equilibrium, while for solutions it corresponds to two solution phases of different concentrations in equilibrium. Some non-aqueous solutions are known which do unmix to form two liquid phases of slightly different concentrations. While no examples in aqueous solution are known, the corresponding... 

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