Diffusion kinetic theory

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating mmticomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simnlta-neonsly by Stefan and Maxwell. The problem is to determine the diffusion coefficient D, . The Stefan-Maxwell equations are simpler in principle since they employ binary diffnsivities ... 

The traditional unipolar diffusion charging model is based on the kinetic theory of gases i.e., ions are assumed to behave as an ideal gas, the properties of which can described by the kinetic gas theory. According to this theory, the particle-charging rate is a function of the square of the particle size dp, particle charge numbers and mean thermal velocity of tons c,. The relationship between particle charge and time according White s... 

Kinetic theories of adsorption, desorption, surface diffusion, and surface reactions can be grouped into three categories. (/) At the macroscopic level one proceeds to write down kinetic equations for macroscopic variables, in particular rate equations for the (local) coverage or for partial coverages. This can be done in a heuristic manner, much akin to procedures in gas-phase kinetics or, in a rigorous approach, using the framework of nonequihbrium thermodynamics. Such an approach can be used as long as... 

As we have implied, diffusion is a rather complex process so far as molecular motion is concerned. Effusion, the flow of gas molecules at low pressures through tiny pores or pinholes, is easier to analyze using kinetic theory. 

Ludwig Boltzmann (1844-1906) was born in Vienna. His work of importance in chemistry became of interest in plastics because of his development of the kinetic theory of gases and rules governing their viscosity and diffusion. They are known as the Boltzmann s Law and Principle, still regarded as one of the cornerstones of physical science. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

The molecular diffusivity D may be expressed in terms of the molecular velocity um and the mean free path of the molecules Xrn. In Chapter 12 it is shown that for conditions where the kinetic theory of gases is applicable, the molecular diffusivity is proportional to the product umXm. Thus, the higher the velocity of the molecules, the greater is the distance they travel before colliding with other molecules, and the higher is the diffusivity D. 

Figure 4. The Brownian ratchet model of lamellar protrusion (Peskin et al., 1993). According to this hypothesis, the distance between the plasma membrane (PM) and the filament end fluctuates randomly. At a point in time when the PM is most distant from the filament end, a new monomer is able to add on. Consequently, the PM is no longer able to return to its former position since the filament is now longer. The filament cannot be pushed backwards by the returning PM as it is locked into the mass of the cell cortex by actin binding proteins. In this way, the PM is permitted to diffuse only in an outward direction. The maximum force which a single filament can exert (the stalling force) is related to the thermal energy of the actin monomer by kinetic theory according to the following equation ...

In bulk diffusion, the predominant interaction of molecules is with other molecules in the fluid phase. This is the ordinary kind of diffusion, and the corresponding diffusivity is denoted as a- At low gas densities in small-diameter pores, the mean free path of molecules may become comparable to the pore diameter. Then, the predominant interaction is with the walls of the pore, and diffusion within a pore is governed by the Knudsen diffusivity, K-This diffusivity is predicted by the kinetic theory of gases to be... 

Diffusion Molecules of a fluid already inside a polymer at a high-concentration region compared with surrounding regions will diffuse over a finite time away from the high concentration until an equilibrium situation is achieved. If the high concentration is at the surface, diffusion occurs into the bulk. The diffusant molecules move stepwise into free volume holes as they form according to kinetic theory. 

Specific heat of each species is assumed to be the function of temperature by using JANAF [7]. Transport coefficients for the mixture gas such as viscosity, thermal conductivity, and diffusion coefficient are calculated by using the approximation formula based on the kinetic theory of gas [8]. As for the initial condition, a mixture is quiescent and its temperature and pressure are 300 K and 0.1 MPa, respectively. 

The ordinary multicomponent diffusion coefficients D j and the viscosity and thermal conductivity are computed from appropriate kinetic theory expressions. First, pure species properties are computed from the standard kinetic theory expressions. For example, the binary diffusion coefficients are given in terms of pressure and temperature as... 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

Chapman, S., Cowling, T, The Mathematical Theory of Non-uniform Gases An account of the Kinetic Theory of Viscosity, Thermal Gonduction and Diffusion in Gases, Cambridge University Press, Cambridge (1970). 

Shape of the polarographic curve. The kinetic theory of electrolysis (Section 3.2) for a redox system at a static inert electrode for partial and full exhaustion at the electrode under merely diffusion-controlled conditions leads, for ox + ne - red, to the relationship... 

By comparing the relative magnitude of the mean free path (z) and the pore diameter (27), it is possible to determine whether bulk diffusion or Knudsen diffusion may be regarded as negligible. Using the principles of the kinetic theory... 

If ordinary molecular diffusion is the dominant mass transfer process, the kinetic theory of gases indicates that the diffusivity is proportional to T3/2 and it is easily shown that... 

Hikosaka presented a chain sliding diffusion theory and formulated the topological nature in nucleation theory [14,15]. We will define chain sliding diffusion as self-diffusion of a polymer chain molecule along its chain axis in some anisotropic potential field as seen within a nucleus, a crystal or the interface between the crystalline and the isotropic phases . The terminology of diffusion derives from the effect of chain sliding diffusion, which could be successfully formulated as a diffusion coefficient in our kinetic theory. 

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. 

This approximation may be considered to be an alternative to the hybrid theory. The value of di can be found by equating the attachment coefficients for the diffusion and kinetic theories (d x = 8D/v). 

The average attachment coefficient 3 can be calculated for the diffusion and the kinetic theory from the equation ... 

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