Randomness, molecular

Rate of Diffusion. Diffusion is the process by which molecules are transported from one part of a system to another as a result of random molecular motion. This eventually leads to an equalization of chemical potential and concentration throughout the system, and in the case of dyeing an equihbrium between dye in the fiber and dye in the dyebath. In dyeing there are three stages to diffusion diffusion of dye through the bulk solution of the dyebath to the fiber surface, diffusion through this surface, and diffusion of dye from the surface into the body of the fiber to allow for more dye to diffuse through the surface layer. These processes have been summarized elsewhere (9). 

A phenomenological description of the differential cross-section for emission of photoelectrons into solid angle O in the lab frame can be written, assuming random molecular orientation and an axis of cylindrical symmetry defined by the photon polarization, as... 

Diffusion is the process by which solute molecules are transported from one part of a system to another as a result of random molecular motion [2], It can be observed with the naked eye when a drop of dye is carefully and slowly placed at the bottom of a beaker filled with water. At first the colored part is separated from the clear by a sharp, well-defined boundary. Later the upper part turns colored, and the color becomes fainter toward the top while the lower part becomes correspondingly less intensely colored. After sufficient time, the whole solution has a uniform color. There is evidently, therefore, a net transfer of dye molecules from the lower part to the upper part of the beaker. The dye molecules have diffused into the water. This diffusion process is primarily due to random molecular motion. 

Nucleation can occur either homogeneously or heterogeneously. Homogeneous nucleation occurs when random molecular motion in the molten state results in the alignment of a sufficient number of chain segments to form a stable ordered phase, known as a nucleus. The minimum number of unit cells required to form a stable nucleus decreases as the temperature falls. Thus, the rate of nucleation increases as the temperature of the polymer decreases. The rate of homogeneous nucleation also increases as molecular orientation in the molten polymer increases. This is because the entropy difference between the molten and crystalline states diminishes as molecular alignment in the molten state increases. 

Fig. 6 (a) SQ and (b) DQ rotor-synchronized 2H MAS NMR spectra of sodium tetrathionate dihydrate-d4 (solid lines). The dashed line in (a) represents the exact numerical simulation of the SQ spectrum for random molecular motion with the rate constant k given in the figure, (c) The corresponding experimental and simulated static 2H quadrupolar-echo spectra, (d) Simulated SQ (solid line) and DQ (dashed line) linewidths as functions of k. (Reproduced with permission from [88])... 

It is of interest to consider first what is happening in pipe flow. Random molecular movement gives rise to a mixing process which can be described by Fick s law (given in Volume 1, Chapter 10). If concentration differences exist, the rate of transfer of a component is proportional to the product of the molecular diffusivity and the concentration gradient. If the fluid is in laminar flow, a parabolic velocity profile is set up over the cross-section and the fluid at the centre moves with twice the mean velocity in the pipe. This... 

The relaxation mechanisms require some kind of nuclear interaction subject to stochastic fluctuations, typically due to random molecular motions. 

Our procedure depends on a new computer program, RAMM (RAndom Molecular Mechanics), which is applicable to any kind of biomolecule. It is described in detail elsewhere (KoS r, T./ Petrak, F. Galova, Z. TvaroSka, I. Carbohvdr. Res.. in Press). Only the basic characteristics of RAMM and its application to conformational analysis of disaccharides are discussed here, concentrating on the effect of the orientations of pendant groups on the energy values at the various < ) and f torsion angles. 

Navard and Haudin studied the thermal behavior of HPC mesophases (87.88) as did Werbowyj and Gray (2), Seurin et al. (Sp and, as noted above, Conio et al. (43). In summary, HjPC in H2O exhibits a unique phase behavior characterized by reversible transitions at constant temperatures above 40 C and at constant compositions when the HPC concentration is above ca. 40%. A definitive paper has been recently published by Fortin and Charlet ( who studied the phase-separation temperatures for aqueous solutions of HPC using carefully fractionated HPC samples. They showed the polymer-solvent interaction differs in tiie cholesteric phase (ordered molecular arrangement) from that in the isotropic phase (random molecular arrangement). 

The process by which random molecular motions result in a net flow of molecules from regions of high concentration to those of low concentration. 

Einstein loG, oit.) showed further that the actual mean distance X travelled by a particle in a short time t under the influence of random molecular collisions was related to the diffusion constant jD, by the following equation ... 

Practically speaking, this concept explains the basis for the establishment of partial pressure equilibrium of anesthetic gas between the lung alveoli and the arterial blood. Gas molecules will move across the alveolar membrane until those in the blood, through random molecular motion, exert pressure equal to their counterparts in the lung. Similar gas tension equilibria also will be established between the blood and other tissues. For example, gas molecules in the blood will diffuse down a tension gradient into the brain until equal random molecular motion (equal pressure) occurs in both tissues. 

Diffusion - In this text, we will dehne diffusion (and most other processes) from an engineering perspective, in that we will go to the level of detail that suits our objective. Diffusion can then be defined as the mixing of chemicals by random molecular motion. Diffusion coefficients in dilute solutions will be discussed in detail in Chapter 3. 

Ordinary diffusion is the result of random molecular movement in first one direction and then another and thus, resembles the Random Walk Model. Uhlenbeck and Ornstein (8), derived the following expression for the overall standard deviation (o) arising from diffusion process,... 

Sol-Gel Measurements. The relative rates of crosslinking and scission may be estimated from soluble-fraction measurements. For an initial random molecular weight distribution and random chain scission, extrapolating a curve of S + S vs. 1/D, where S = sol fraction and D —... 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

Around 1800, experimental challenges to caloric theory were being presented by Count Rumford (cannon boring) and Humphrey Davy (melting of ice by friction). It became apparent that heat could be produced from a body in unlimited quantity by friction, further stretching its credibility as a substance. By about 1840, caloric theory was overturned by the modem kinetic molecular theory of heat (Sidebar 2.7), which identified heat with the energy of random molecular motions. 

The constant motion and high velocities of gas particles lead to some important practical consequences. One such consequence is that gases mix rapidly when they come in contact. Take the stopper off a bottle of perfume, for instance, and the odor will spread rapidly through the room as perfume molecules mix with the molecules in the air. This mixing of different gases by random molecular motion with frequent collisions is called diffusion. A similar process in which gas molecules escape without collisions through a tiny hole into a vacuum is called effusion (Figure 9.13). 

Heat and work can be distinguished in terms of random molecular motion versus directed or coordinated motion. In muscle cells, from organisms as simple as yeast to those as complex as humans, the hydrolysis of ATP provides the driving force for the interactions and conformation of two cellular proteins, myosin and actin. Conformational changes associated with the binding and release of ATP and ADP provide the means by which a coordinated movement of these muscle cells is possible to do mechanical work. 

An alternative (but equivalent) approach is the so-called fluctuation theory, in which light scattering is treated as a consequence of random non-uniformities of concentration and, hence, refractive index, arising from random molecular movement (see page 26). Using this approach, the above relationship can be written in the quantitative form derived by Debye140 for dilute macromolecular solutions ... 

Diffusion, the basis of the solution-diffusion model, is the process by which matter is transported from one part of a system to another by a concentration gradient. The individual molecules in the membrane medium are in constant random molecular motion, but in an isotropic medium, individual molecules have no preferred direction of motion. Although the average displacement of an individual molecule from its starting point can be calculated, after a period of time nothing can be said about the direction in which any individual molecule will move. However, if a concentration gradient of permeate molecules is formed in the medium, simple statistics show that a net transport of matter will occur... 

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