Two dimensional analogs

Extensions to Higher Dimensional Lattices Higher dimensional versions of each of the above measures may be obtained in a straightforward fashion. The two dimensional analogs of the set and measure dimensions, for example, are given by considering the sets of possible configurations on x — y patches of size B x By ... 

It is advantageous with a new drug substance to be able to estimate what its solubility will be prior to carrying out dissolution experiments. There are several systems of solubility prediction, most notably those published by Amidon and Yalkowsky [14-16] in the 1970s. Their equation for solubility of p-aminobenzo-ates in polar and mixed solvents is a simplified two-dimensional analog of the Scatchard-Hildebrand equation and is based on the product of the interfacial tension and the molecular surface area of the hydrocarbon portion of a molecule. 

Since surface pressure is a free energy term, the energies and entropies of first-order phase transitions in the monolayer state may be calculated from the temperature dependence of the ir-A curve using the two-dimensional analog of the Clausius-Clapeyron equation (59), where AH is the molar enthalpy change at temperature T and AA is the net change in molar area ... 

These arc the two-dimensional analogs of the infamous (see Introduction) 2,3,4-trihydroxy-glutaric acid. 

The problem of calculating the degree of orientation and size from a scattering pattern has been widely addressed in the scientific literature [86, 88]. In the study of regenerated tissues a two-dimensional analog of the Hermans orientation index [89, 90], previously used to specify the orientation of a polymer from X-ray scattering data, was adapted [81, 91]. 

Many surface states are two-dimensional analogs of three-dimensional states. As with any analogy, however, there are points of similarity and points of difference between the surface and bulk states. For most of the states we discuss, we consider the phenomenological behavior as represented by Figure 7.6 and Figure 7.7. 

For osmotic equilibrium, the chemical potential of the solvent must be the same on both sides of the membrane. In the two-dimensional analog also the chemical potential must be the same for the water on both sides of the float. The presence of the solute lowers the chemical potential of the solvent, but the excess pressure compensates for this. Therefore, by analogy with Equation (3.19), we write... 

The coefficients a and b are two-dimensional analogs of the three-dimensional van der Waals coefficients. 

An isotherm that is not too difficult to derive by the methods of statistical mechanics assumes an adsorbed layer that obeys the two-dimensional analog of the van der Waals equation. The result of such a derivation is the equation (see Table 9.1)... 

A two-dimensional analog of PV work can be recognized in fluid films that exhibit surface tension (tendency of the film surface to contract against an opposing spreading force). The surface tension work wsurf (of, for example, a soap film) can be measured by a rectangular wire-frame device with moveable edge, as shown in Fig. 3.5. 

This is clearly the two-dimensional analog of PV work. Note that the sign differs in (3.10) and (3.14) because work is performed on the system by reducing the volume in (3.10), but by expanding the area in (3.14). 

As the flow exponent becomes smaller, the polymer melt becomes increasingly shear thinning, and the pressure required to achieve a specific flow rate becomes even smaller (Fig. 8.10). We also note that the intersection of the curve with the x-axis is not dependent on the flow exponent. There is no axial pressure gradient at the intersection. A simple two-dimensional analogy model is the planar channel. If only the walls are moved and no pressure gradient is overlaid, the flow rate, according to an analytical solution, is not dependent on the flow exponents. 

Figure 1.11 Portion of a Penrose tiling based on two rhombuses. Penrose tilings are nonperiodic tilings of the plane and are two-dimensional analogs of quasicrystals. (Diagram created by the free Windows application Bob s Rhombus Walker, v. 3.0.19, JKS Software, Stamford, CT.)...

---