Entropic principle

The most straightforward application of the entropic principle is the driving force that induces crystallization of polymers with chains in s M/N) helical conformation in crystalline lattices containing the screw M/N operator, so that the local helical symmetry of the chains becomes a crystallographic symmetry. According to this... 

In a recent work [46], experiments and simulations have been used to directly test the form of the volume distribution. Equation 5.5. The results show that the distribution can indeed be written in this very special form, which is a nontrivial result that supports the entropic principles behind the Edwards hypothesis. As the authors point out, however, testing Equation 5.5 does not test the equiprobable hypothesis. The analog of density of states, G V) involves the microscopic Wy,... 

The highest ES descriptors reflect some known differences between synthetic and natural molecules, including, for example, the degree of saturation or aromatic character. It is also interesting to note that the descriptor with the highest ES value, a ICM, is itself calculated using entropic principles. It accounts for the entropy of the distribution of the elemental composition of the compound. 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accmacy within a finite simulation time. It is, however, possible to calculate differences in such quantities. Of special importance is the Gibbs free energy, as it is the natoal thermodynamical quantity under normal experimental conditions (constant temperature and pressme. Table 16.1), but we will illustrate the principle with the Helmholtz free energy instead. As indicated in eq. (16.1) the fundamental problem is the same. There are two commonly used methods for calculating differences in free energy Thermodynamic Perturbation and Thermodynamic Integration. 

Once we have determined the entropy and enthalpy of polymerization, we can calculate the free energy of the process at a variety of temperatures. The only time this is problematic is when we are working near the temperatures of transition as there are additional entropic and enthalpic effects due to crystallization. From the free energy of polymerization, we can predict the equilibrium constant of the reaction and then use this and Le Chatelier s principle to design our polymerization vessels to maximize the percent yield of our process. 

Besides the energy factors, defined by the close-packing principle, entropic factors are also involved in determining the mode of packing of molecules. A molecule in a crystal tends to maintain part of its symmetry elements, provided that this does not cause a serious loss of density. In a more symmetric position a molecule has a greater freedom of vibration, that is, the structure corresponds to a wider energy minimum.126... 

It is important for the theoretical understanding of the formation of various topologies that these aggregates have entropic contributions on the scale of the objects, i.e. on a much larger scale than set by the molecules. These cooperative entropic effects should be included in the overall Helmholtz energy, and they are essential to describe the full phase behaviour. It is believed that the mechanical parameters discussed above kc,k and J0, control the phase behaviour, where it is understood that these quantities may, in principle, depend on the overall surfactant (lipid) concentration, i.e. when the membranes are packed to such a density that they strongly interact. 

When thiophene dioxide (106) was used as the diene component, true catalysis was observed with 107, affording the capsule bound adduct 108 (equation 32)93. The displacement of a single molecule of adduct by two molecules of starting material is, in principle, disfavored on entropic grounds, but turnover took place in this case due to the poorer affinity of the Diels-Alder adduct for the capsule. The rate enhancement of this reaction, based on the ratio of the half-life for the reaction outside vs inside the capsule, was 10-fold. 

The basic structure of taxol is that of a tetracyclic compound -A, B, C and D rings-, in which the central B ring is an eight-membered carbocycle. In principle, the formation of this medium sized ring appears somewhat problematic because of both entropic and enthalpic factors. 

In principle, it is a simple matter to include solvent water molecules directly in MD simulations, since appropriate intermolecular potential energy functions for water are available (1Z 37,38) one would just surround the solute molecules with a sufficient number of water molecules to approximate a bulk solution. Unfortunately, a "sufficient number of water molecules might be enormous, since many of the effects of aqueous solvation are long range or are due to entropic contributions arising from "structuring of the solvent, which may be cooperative in nature. 

The constant a is proportional to the step formation energy, f and b and c account for step-step interactions. Entropic, dipole and elastic interactions between steps give rise to the pfterm [24,25] whereas the p term may be due to electronic effects [26], The experimental data of fig. 1 can be fitted by y(0) = jo cosq f(p) over the whole range of orientations when all terms of eq. (1) are allowed [27]. More about this point in section 6. In principle, it is a matter of great interest to test the validity of eq. (1) and to determine the step as well as step interaction energies. 

It is observed that this principle correctly describes the trends observed in Figures 4.34-4.36. When the approach to equilibrium is decreased, less product intermediates are present on the surface, and the optimum will move towards the more reactive surfaces to compensate. A temperature increase will remove adsorbates from the surface, due to the entropic push towards the gas phase, and the optimal catalyst must thus be a more reactive surface to compensate. An increase in the reactant pressure will increase the surface coverage. The optimal catalyst under the higher pressure must therefore be a nobler surface to conserve the optimal coverage. 

Another way to interpret the above observations would be in terms of the general principle that effective steric stabilization of polymer-coated droplets requires that the continuous phase be a good quality solvent for the polymeric stabilizer. Under poor quality solvent conditions (asi-casein at high ionic strength), the required entropic stabilizing repulsion of the adsorbed protein layer is converted into a destabilizing polymer-mediated attraction (Dickinson and Stainsby, 1982 Dickinson, 2006). 

The lack of significant ring strain and favorable entropic factors results in facile cyclofunctionalization to form five- or six-membered rings. Emphasis in this review is placed on examples which illustrate general principles of regiochemical and stereochemical control. 

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