Diffusion motion, Brownian

Behavior. Diffusion, Brownian motion, electrophoresis, osmosis, rheology, mechanics, and optical and electrical properties are among the general physical properties and phenomena that are primarily important in coUoidal systems (21,24—27). Of course, chemical reactivity and adsorption often play important, if not dominant, roles. Any physical and chemical feature may ultimately govern a specific industrial process and determine final product characteristics. 

Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle in a fluid of the same density and viscosity settling under laminar flow conditions. Correction for deviation from Stokes law may be necessary at the large end of the size range. Sedimentation methods are limited to sizes above a [Lm due to the onset of thermal diffusion (Brownian motion) at smaller sizes. 

Brownian diffusion (Brownian motion) The diffusion of particles due to the erratic random movement of microscopic particles in a disperse phase, such as smoke particles in air. 

Kinetics is concerned with many-particle systems which require movements in space and time of individual particles. The first observations on the kinetic effect of individual molecular movements were reported by R. Brown in 1828. He observed the outward manifestation of molecular motion, now referred to as Brownian motion. The corresponding theory was first proposed in a satisfactory form in 1905 by A. Einstein. At the same time, the Polish physicist and physical chemist M. v. Smolu-chowski worked on problems of diffusion, Brownian motion (and coagulation of colloid particles) [M. v. Smoluchowski (1916)]. He is praised by later leaders in this field [S. Chandrasekhar (1943)] as a scientist whose theory of density fluctuations represents one of the most outstanding achievements in molecular physical chemistry. Further important contributions are due to Fokker, Planck, Burger, Furth, Ornstein, Uhlenbeck, Chandrasekhar, Kramers, among others. An extensive list of references can be found in [G.E. Uhlenbeck, L.S. Ornstein (1930) M.C. Wang, G.E. Uhlenbeck (1945)]. A survey of the field is found in [N. Wax, ed. (1954)]. 

Here, coR is the frequency of motion in the reactant well, and Eb is the height of the transition-state barrier. Xr is the effective barrier frequency with which the reactant molecule passes, by diffusive Brownian motions through the barrier region and is given by the following self-consistent relation... 

The two contrasting approaches, the macroscopic viewpoint which describes the bulk concentration behavior (last chapter) versus the microscopic viewpoint dealing with molecular statistics (this chapter), are not unique to chromatography. Both approaches offer their own special insights in the study of reaction rates, diffusion (Brownian motion), adsorption, entropy, and other physicochemical phenomena [2]. 

Advanced Monte Carlo algorithms, developed in other fields (e.g., for electron transport studies in solids), automatically switch from a kinetic Boltzmann-like to a diffusive ( Brownian motion-like ) description, depending on the local background medium conditions, in order to improve statistical performance at high collisionality. 

The first theory giving the tj -induced decrease of the rate constant is the Kramers theory presented as early as in 1940. He explicitly treated dynamical processes of fluctuations in the reactant state, not assuming a priori the themud equilibrium distribution therein. His reaction scheme can be understood in Fig. 1 which shows, along a reaction coordinate X, a double-well potential VTW composed of a reactant and a product well with a transition-state barrier between them. Reaction takes place as a result of diffusive Brownian motions of reactants surmounting... 

Figure 2. Two-dimensional potential surface for reaction in the Sumi-Maicus scheme along the coordinate X for diffusive Brownian motions and the coordinate q for much faster intramolecular vibrational motions, and an example of reactive trajectories there.

Next, let us discuss whether the Grote-Hynes theory can be fitted to the rate constants kgi,. In this thoery. kgf,Jkfsr is related to the frequency (that is, the speed) n with which the reactant passes, by diffusive Brownian motions, through Ae transition-state-barrier region in Fig. 1, as... 

There are three major sinks that act to remove particles from the atmosphere diffusion (Brownian motion), wet deposition, and gravitational settling. The relative importance of each mechanism depends primarily upon particle size (Seinfeld and Pandis 1998 Kreidenweis et al. 1999 Friedlander 2000). As shown in Figure 4, diffusion is the... 

The principle of filtration combines many of the individual mechanisms of collection on which other methods are based. Thus, diffusion (Brownian motion), inertia, interception, charge, and sedimentation may all contribute to deposition of particles on filters. The inertial and interception effects are illustrated in Fig. 3. 

Particle concentration gradient Particle diffusion, Brownian motion, Concentration fluctuations ... 

Fig. 12 Schematic diagram showing a self-diffusion, b tracer diffusion, and c self-diffusion (Brownian motion). From Karger and Ruthven [37]...

We must now relate the energetics to the kinetics. The theory of rapid flocculation was first proposed by Smoluchowski , who treated the problem as one of diffusion (Brownian motion) of the spherical particles of an initially monodisperse dispersion, with every collision, in the absence of a repulsive force, leading to a permanent contact. In general, we may express the rate of flocculation, i.e. the rate of decrease of the total number of particles, as... 

Much has been written on the various mechanisms by which particles are arrested by unused filter media. These are normally explained in terms of the effect of a spherical particle on a single fibre and may be summarised as gravitational, impaction, interception, diffusion (Brownian motion), and electrostatic. These mechanisms are shown diagrammatically in Fig. 3.1. 

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