Macroscopic example

Symbol Symmetry Elements Molecular Examples Macroscopic Examples... 

Although, as we have already stressed, analogies between the macroscopic and the molecular world often lead to misunderstandings and must be used carefully, it is tempting to compare the relationship between chirality and motion in Feringa s systems with that seen in a macroscopic example the windmill. When the wind blows at the vanes of a windmill, its rotor immediately knows which direction to rotate in. When blown at by the same wind, two enantiomeric windmills will rotate in opposite directions, and if the vanes were symmetric with respect to the axis of rotation, the windmill just... 

Bose-Einstein condensates are unusual in numerous ways. With careful study physicists will gain basic knowledge about the material and quantum worlds. The atoms in a condensate are indistinguishable. All atoms move at the same speed in the same space. One can ask How can two objects occupy the same place at the same time A condensate is a macroscopic quantum wave packet and a macroscopic example of Heisenberg s uncertainty principle. Condensates hold the promise of bringing new insights to the strange world between the microscopic quantum and the macroscopic classical domains. 

The precious metal composition is typically uniform in the radial and axial directions of the monolith structure, although different designs have been described in the patent literature and have even been used in some selected applications. However, much more common is a nonuniform distribution of the precious metals within the washcoat layer. One - macroscopic - example of nonuniform distribution is that the amount of one precious metal component decreases from the part of the washcoat that is in contact with the gas phase towards the part of the washcoat that is in contact with the monolith wall and eventually vice-versa for the second precious metal component. Another - microscopic - example of nonuniform distribution within the washcoat is that each precious metal component is selectively deposited on different washcoat components. These nonuniformities are intentional and are desirable for kinetic reasons or because of specific beneficial interactions between the precious metals and the washcoat oxides. The type of nonuniformity that can be achieved depends strongly on the production procedure of the catalyst. 

We have also discussed the formation of spatio-temporal patterns in non-variational systems. A typical example of such systems at nano-meter scales is reaction-diffusion systems that are ubiquitous in biology, chemical catalysis, electrochemistry, etc. These systems are characterized by the energy supply from the outside and can exhibit complex nonlinear behavior like oscillations and waves. A macroscopic example of such a system is Rayleigh-Benard convection accompanied by mean flow that leads to strong distortion of periodic patterns and the formation of labyrinth patterns and spiral waves. Similar nano-meter scale patterns are observed during phase separation of diblock copolymer Aims in the presence of hydrodynamic effects. The pattern s nonlinear dynamics in both macro- and nano-systems can be described by a Swift-Hohenberg equation coupled to the non-local mean-flow equation. 

A perfect helical main chain conformation always leads to a rodlike or cylindrical external shape. But each monomeric unit in such a rod contributes a certain flexibility. So, the flexibility of the rod, as a whole, must increase with increasing degree of polymerization, even when the flexibility per monomeric unit remains constant. A macroscopic example of this would be the flexibility of steel wires of equal diameter but different lengths. Thus, even a perfect helix will adopt coil shape if the molecular mass is very high. Because of this, helically occurring macromolecules, and other stiff macromolecules, can often be well represented by what is known as the wormlike screw model for macromolecular chains at low molecular masses, the chains behave like a stiff rod, but for high molecular masses, the behavior is more coil-like. Examples are nucleic acids, many poly(a-amino acids), and highly tactic poly(a-olefins). 

We have developed the relativistic theory of molecular science from the first principles offundamental physics, namely from quantum mechanics and from the special theory of relativity. In principle, we are now able to study any molecular system using quantum chemical methods of controllable accuracy. Comparisons with purely mmrelativistic calculations highlight so-called relativistic effects. Prominent macroscopic examples are the yellowish color of elemental gold and the fluidity of mercury at ambient temperature. This final chapter comprises some important examples for which relativity is of paramount importance. 

This must be combined with the macroscopic sweep efficiency to determine the recovery factor (RF) for oil (in this example). 

The topic of capillarity concerns interfaces that are sufficiently mobile to assume an equilibrium shape. The most common examples are meniscuses, thin films, and drops formed by liquids in air or in another liquid. Since it deals with equilibrium configurations, capillarity occupies a place in the general framework of thermodynamics in the context of the macroscopic and statistical behavior of interfaces rather than the details of their molectdar structure. In this chapter we describe the measurement of surface tension and present some fundamental results. In Chapter III we discuss the thermodynamics of liquid surfaces. 

Since solids do not exist as truly infinite systems, there are issues related to their temiination (i.e. surfaces). However, in most cases, the existence of a surface does not strongly affect the properties of the crystal as a whole. The number of atoms in the interior of a cluster scale as the cube of the size of the specimen while the number of surface atoms scale as the square of the size of the specimen. For a sample of macroscopic size, the number of interior atoms vastly exceeds the number of atoms at the surface. On the other hand, there are interesting properties of the surface of condensed matter systems that have no analogue in atomic or molecular systems. For example, electronic states can exist that trap electrons at the interface between a solid and the vacuum [1]. 

The size of the exciton is approximately 50 A in a material like silicon, whereas for an insulator the size would be much smaller for example, using our numbers above for silicon dioxide, one would obtain a radius of only 3 A or less. For excitons of this size, it becomes problematic to incorporate a static dielectric constant based on macroscopic crystalline values. 

To define the thennodynamic state of a system one must specify fhe values of a minimum number of variables, enough to reproduce the system with all its macroscopic properties. If special forces (surface effecls, external fields—electric, magnetic, gravitational, etc) are absent, or if the bulk properties are insensitive to these forces, e.g. the weak terrestrial magnetic field, it ordinarily suffices—for a one-component system—to specify fliree variables, e.g. fhe femperature T, the pressure p and the number of moles n, or an equivalent set. For example, if the volume of a surface layer is negligible in comparison with the total volume, surface effects usually contribute negligibly to bulk thennodynamic properties. 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

Computer simulations act as a bridge between microscopic length and time scales and tlie macroscopic world of the laboratory (see figure B3.3.1. We provide a guess at the interactions between molecules, and obtain exact predictions of bulk properties. The predictions are exact in the sense that they can be made as accurate as we like, subject to the limitations imposed by our computer budget. At the same time, the hidden detail behind bulk measurements can be revealed. Examples are the link between the diffiision coefficient and... 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

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