Kinetic theory small particles

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

White s equation is widely used mainly because it is easy to use and because it gives values which are in reasonable agreement with the experimental ones. However, because this model is based on the kinetic theory of gases, it should be used for small particles only. This model (as many others) assumes that particle charge can be described with a continuous function. Especially in the case of small particles, only the lowest charge numbers (0, 1, 2) are possible, and therefore the model—which does not take into account the discrete charge numbers—is somewhat questionable. 

Besides these chemical effects, which are understood in terms of the established theories in semiconductor physics and chemical kinetics, new physico-chemical phenomena are observed in the case of extremely small particles. The metal or semiconductor behavior is gradually lost with decreasing size, the consequences being drastic changes in the optical properties of the materials and also in their photocatalytic effects. 

The two fundamental theories for calculating the attachment coefficient, 3, are the diffusion theory for large particles and the kinetic theory for small particles. The diffusion theory predicts an attachment coefficient proportional to the diameter of the aerosol particle whereas the kinetic theory predicts an attachment coefficient proportional to the aerosol surface area. The theory... 

When the radius of an aerosol particle, r, is of the order of the mean free path, i, of gas molecules, neither the diffusion nor the kinetic theory can be considered to be strictly valid. Arendt and Kallman (1926), Lassen and Rau (1960) and Fuchs (1964) have derived attachment theories for the transition region, r, which, for very small particles, reduce to the gas kinetic theory, and, for large particles, reduce to the classical diffusion theory. The underlying assumptions of the hybrid theories are summarized by Van Pelt (1971) as follows 1. the diffusion theory applies to the transport of unattached radon progeny across an imaginary sphere of radius r + i centred on the aerosol particle and 2. kinetic theory predicts the attachment of radon progeny to the particle based on a uniform concentration of radon atoms corresponding to the concentration at a radius of r + L... 

For small particle sizes the kinetic theory is applicable, whereas for large particle sizes the diffusion theory applies. A useful approximation is therefore to use the kinetic theory in the small particle range and the diffusion theory in the large size region. 

Einstein A. (1905) The motion of small particles suspended in static liquids required by the molecular kinetic theory of heat. Ann. Phys. 17, 549-560. 

Pai, S. I. (1974). Fundamental equations of a mixture of gas and small spherical particles from simple kinetic theory. Revue Roum. Sci. Mech. Tech. Appl., 19 605-621. 

Carbon Monoxide Oxidation. Analysis of the carbon monoxide oxidation in the boundary layer of a char particle shows the possibility for the existence of multiple steady states (54-58). The importance of these at AFBC conditions is uncertain. From the theory one can also calculate that CO will bum near the surface of a particle for large particles but will react outside the boundary layer for small particles, in qualitative agreement with experimental observations. Quantitative agreement with theory would not be expected, since the theoretical calculations, are based on the use of global kinetics for CO oxidation. Hydroxyl radicals are the principal oxidant for carbon monoxide and it can be shown (73) that their concentration is lowered by radical recombination on surfaces within a fluidized bed. It is therefore expected that the CO oxidation rates in the dense phase of fluidized beds will be suppressed to levels considerably below those in the bubble phase. This expectation is supported by studies of combustion of propane in fluidized beds, where it was observed that ignition and combustion took place primarily in the bubble phase (74). More attention needs to be given to the effect of bed solids on gas phase reactions occuring in fluidized reactors. 

When the particle size is very small compared to the mean free path, i.e., Kn > 10, the particle can be regarded as a large spherical molecule undergoing independent binary collisions with the gas molecules and the coagulation coefficient is obtained from the kinetic theory of rarefied gases as (5)... 

Small particles suspended in a gas undergo random translational motion because they are being buffeted by collisions with swiftly moving gas molecules. This motion appears almost as a vibration of the ensemble of particles, although there is a net displacement with time of any given particle. Observation of this motion in a liquid was first made in 1828 by the British naturalist Robert Brown (1828), and the phenomenon thus has been called brownian motion (also known as brownian movement). Bodaszewski (1883) studied the brownian motion of smoke particles and other suspensions in air and likened these movements to the movements of gas molecules as postulated by the kinetic theory. The principles governing brownian motion are the same, whether the particles are suspended in a gas or in a liquid. 

In the derivation of the kinetic relations it was assumed that free radicals enter the particles one by one the initiation process just described satisfies this condition. This is not the case when radicals are formed by thermal decomposition of an oil-soluble initiator. Such decomposition produces pairs of radicals in the hydrocarbon phase. One would expect a pair of radicals, confined to the extremely small volume of a latex particle, to recombine rapidly. The kinetics of this type of polymerization have been described above. It is recalled here that the subdivision factor, z, and hence rate and degree of polymerization are smaller than 1 and decrease with a. These predictions from kinetic theory are in contradiction to experimental observations. Although some oil-soluble initiators, which are good catalysts in solution systems, are poor initiators in emulsion polymerizations—e.g., benzoyl peroxide—other thermally decomposing peroxides and azo compounds produce polymer in emulsion at rates comparable to those observed in polymerization initiated by water-soluble catalysts, where the radicals enter the particles one by one. Such is the case for cumene hydroperoxide, which at low concentrations yields a rate of polymerization per particle equal to that of a persulfate-initiated reaction. It must therefore be concluded that, although oil-soluble initiators may decompose into radical pairs within the particles, polymer radicals are formed one by one. The following mechanisms are consistent with formation of polymer radicals singly. 

This is precisely the result from kinetic theory of self-diffusion, neglecting the small Somine corrections, and also applies when the test particle is identical to the solvent particle. At intermediate densities the packing effects of the solvent around the test particle become important and Eq. (3.7) collapses to the Enskog result... 

It is common practice in soil science to use kinetic theory of homogeneous reactions as an approximation for reactions occurring in soil-solution systems. The rationale is that the very fine subdivision of the soil particles (clays, minerals, oxides, zeolites, humic acids, etc.) allows the system to appear homogeneous except on a very small scale. 

Consider an aerosol composed of very small particles in the size range 10 to 100 nm such as might be produced during a welding process. TTie average velocity with which particles of mass m collide with a stationary surface exposed to the gas due to the thermal (Brownian) motion is (based on the kinetic theory of ga.ses) given by... 

The limiting case of very small particles only a few of which become charged is easiest to treat using a kinetic theory analysis. The rate of successful collisions between ions of unit charge and concentration with uncharged particles of concetitration A o is... 

For small particles, i.e. when Kn>10 2, one can apply the relationships established in the molecular kinetic theory of gases, according to which the resistance to the particle motion is proportional to the cross-sectional area of particles, and the velocity of their motion, u, due to the applied force, F, is given by... 

With very large particles the liquid interlayer thinning process is complicated by the deformation of the bubble surface by an inertia impact of the particle. It was shown by Derjaguin et al. (1977) and Dukhin Rulyov (1977) that in the inertia-free deposition of small particles on a bubble surface its deformation under the influence of the hydrodynamic pressing force is insignificant. This third important feature facilitates the development of a quantitative kinetic theory of flotation of small particles. 

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