Kinetic Theory of Diffusion

We saw in the last chapter that the rate of steady-state creep, 55, varies with temperature as 

In this chapter we discuss the origin of Arrhenius s Law and its application to diffusion. In the next, we examine how it is that the rate of diffusion determines that of creep. 

Here / is the number of ink molecules diffusing down the concentration gradient per second per unit area it is called the flux of molecules (Fig. 18.3). The quantity c is the concentration of ink molecules in the water, defined as the number of ink molecules per unit volume of the ink-water solution and D is the diffusion coefficient for ink in water - it has units of m s .  

This diffusive behaviour is not just limited to ink in water - it occurs in all liquids, and more remarkably, in all solids as well. As an example, in the alloy brass - a mixture 

Why is this relevant to the diffusion of zinc in copper Imagine two adjacent lattice planes in the brass with two slightly different zinc concentrations, as shown in exaggerated form in Fig. 18.5. Let us denote these two planes as A and B. Now for a zinc atom to diffuse from A to B, down the concentration gradient, it has to squeeze between the copper atoms (a simplified statement - but we shall elaborate on it in a moment). This is another way of saying the zinc atom has to overcome an energy barrier 

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Diffusion is the intermingling of the atoms or molecules of more than one species it is the inevitable result of the random motions of the individual molecules that are distributed throughout space. The development of a rigorous kinetic theory to describe this intermingling in gas mixtures is one of the major scientific achievements of the nineteenth century. A simplified kinetic theory of diffusion, adapted from Present (1958), is the main theme of Section 2.1. More rigorous (and complicated) developments are to be found in the books by Hirschfelder et al. (1964), Chapman and Cowling (1970), and Cunningham and Williams (1980). An extension to cover diffusion in nonideal fluids is developed thereafter. 

Within the control volume the molecules of species 1 may lose (or gain) momentum each time they collide with the atoms or molecules of the other species. Accounting for the momentum exchange on collision is one part of the elementary kinetic theory of diffusion that follows. 

Given this failure of the continuum model, it is evidently necessary to treat the solvent as an assembly of molecules. A hard-sphere model is the first approximation. Kinetic theory of diffusion in dilute gases, where the mean free path is much greater than the collision diameter, is well established it can be extended with some success to dense gases, where the two quantities are more nearly equal, and (more speculatively) to hard-sphere models of liquids, where they are comparable. For these highly mathematical theories the reader may consult more specialised works [14]. Analytical solutions are not always to be expected numerical solutions may be required. Computer-simulation calculations have had considerable success, and with the advent of fast computers have become a major source of understanding of real systems (cf., e.g.. Section 7.3.4.5). 

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... 

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. 

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. 

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... 

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. 

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... 

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... 

The variation of attachment coefficient with Og for CMD =0.2 ym and 0.3 ym is shown in Figures 6 and 7. Again it is apparent that the kinetic theory or diffusion theory are correct only at certain CMD and og. Neither is applicable under all circumstances. It is also evident that the kinetic-diffusion theory is a good approximation to the hybrid theory under all circumstances. 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic theory of gases, and their interrelationship through A, and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests that this should be a dependence on T1/2, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to r3/2 for the case of molecular inter-diffusion. The Arrhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, then an activation enthalpy of a few kilojoules is observed. It will thus be found that when the kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation enthalpy will be a few kilojoules only (less than 50 kJ). 

Figure 16. Schematic illustration of envelopes of gas species i, in this case Mg, surrounding a volatilizing molten chondrule in space. The size of the gas envelope is a function of ambient background pressure P by virtue of the effect that pressure has on the gas molecule diffusivity D,. The diffusion coefficient can be calculated from the kinetic theory of gases, as shown. The level of isotopic fractionation associated with volatilization of the molten chondrule depends upon the balance between the evaporative flux J vap and the condensation flux Tom When the fluxes are equal (i.e., when = 0), there is no mass-dependent isotope fractionation associated with volatilization. This will be the case when the local partial pressure P, approaches the saturation partial pressure P,...

The species diffusivity, varies in different subregions of a PEFC depending on the specific physical phase of component k. In flow channels and porous electrodes, species k exists in the gaseous phase and thus the diffusion coefficient corresponds with that in gas, whereas species k is dissolved in the membrane phase within the catalyst layers and the membrane and thus assumes the value corresponding to dissolved species, usually a few orders of magnitude lower than that in gas. The diffusive transport in gas can be described by molecular diffusion and Knudsen diffusion. The latter mechanism occurs when the pore size becomes comparable to the mean free path of gas, so that molecule-to-wall collision takes place instead of molecule-to-molecule collision in ordinary diffusion. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows... 

Because the theory of the liquid state is not nearly so well developed as the kinetic theory of gases, estimation methods for liquid diffusion coefficients are not as reliable as those used for gases. For dilute solutions of non-electrolytes, one widely used correlation is that due to Wilke and Chang[48]... 

The kinetic theory of gases has therefore given us a fairly simple equation for the diffusion coefficient of a molecule. All that remains to be determined is the mean velocity of the molecule and the mean free path of the molecule. 

Although the kinetic theory of gases is not generally used, as is, to estimate the diffusion coefficients of gases, it does provide a framework for the characterization of data through predictive equations. This simple kinetic theory has shown us the following ... 

The Molecular Origins of Mass Diffusivity. In a manner directly analogous to the derivations of Eq. (4.6) for viscosity and Eq. (4.34) for thermal conductivity, the diffusion coefficient, or mass diffusivity, D, in units of m /s, can be derived from the kinetic theory of gases for rigid-sphere molecules. By means of summary, we present all three expressions for transport coefficients here to further illustrate their similarities. 

Alberty, R. A., and Hammes, G. G. (1958). Application of the theory of diffusion-con-trolled reactions to enzyme kinetics. J. Phys. Chem. 62, 154-159. 

For a two-component mixture the multicomponent diffusion coefficients D, become the ordinary binary diffusion coefficients Sh,. For these quantities 2D,-, = 2D,- and 2D = 0. For a three-component system the multicomponent diffusion coefficients are not equal to the ordinary binary diffusion coefficients. For example, it has been shown by Curtiss and Hirschfelder (C12) in their development of the kinetic theory of multicomponent gas mixtures that... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

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