Mean diffusion coefficient, definition

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Although the assignment of a value to DM is arbitrary, it is by no means unrestricted. From its definition (Equation 20.5) it is clear that DM must be positive. To find the upper limit for DM, one need only substitute the Einstein definition [6] of the diffusion coefficient,... 

Other definitions of chemical diffusion coefficients were also suggested for various particular cases (e.g., see [iii, vi-viii]). In all cases, however, their physical meaning is related either to the ambipolar diffusion or to diffusion in non-ideal systems where the activity coefficients differ from unity. 

The following symbols are used in the definitions of the dimensionless quantities mass (m), time (t), volume (V area (A density (p), speed (u), length (/), viscosity (rj), pressure (p), acceleration of free fall (p), cubic expansion coefficient (a), temperature (T surface tension (y), speed of sound (c), mean free path (X), frequency (/), thermal diffusivity (a), coefficient of heat transfer (/i), thermal conductivity (/c), specific heat capacity at constant pressure (cp), diffusion coefficient (D), mole fraction (x), mass transfer coefficient (fcd), permeability (p), electric conductivity (k and magnetic flux density ( B) ... 

Undeniably, the speed vector, by its size and directional character, masks the effect of small displacements of the particle. Another difference comes from the different definition of the diffusion coefficient, which, in the case of the property transport, is attached to a concentration gradient of the property it means that there is a difference in speed between the mobile species of the medium. A second difference comes from the dimensional point of view because the property concentration is dimensional. When both equations are used in the investigation of a process, it is absolutely necessary to transform them into dimensionless forms [4.6, 4.7, 4.37, 4.44]. 

These, and all other equations for concentration-dependent diffusion, consist of an infinite dilution diffusivity and a thermodynamic correction term. The thermodynamie correction term in all cases is equivalent to the derivative dGildx. The definition of the thermodynamic metastable limit (the spinodal curve) is the locus of points where dG2ldx = 0. This means that concentration-dependent diffusion theory predicts a diffusivity of zero at the spinodal. Thermodynamics tells us that the diffusivity goes from some finite value at saturation to zero at the spinodal. Unfortunately, it does not tell us how the diffusion coefficient declines. In addition, lack of thermodynamic data makes prediction of the spinodal difficult. We are, therefore, left with only the fact that as the concentration is increased in the supersaturated region, the diffusivity should decline towards zero but we do not know at what concentration the diffusivity becomes zero. 

The water photolysis under low O2 pressure always led to a loss of hydrogen into space. The diffusion rate of the H2 (or H after it has been broken down by photolysis) through the homopause and exobase is limited. The definition of the homopause (80-90 km altitude) is the point at which the molecular and eddy diffusion coefficients are equal or, in other words, the critical level below which an atmosphere is well-mixed. The exobase ( 550 km) is the height at which the atmosphere becomes collisionless above that height the mean free path of the molecules exceeds the local scale height (RTIg). 

High total pressures lead to k whereas in vacuum adsorbers Dy is valid. Note that the Knudsen diffusion coefficient is changing with the pore diameter and depends on the definition of a mean pore diameter. 

Diffusion means a local non-convective flux of matter under the action of a chemical or - in the case of charged particles - an electrochemical potential gradient. By expressing the flux of particles of type i per unit of the concentration gradient dc /dx (i. e. by forming the quotient of the measurable parameters ji and dCijdx), we arrive at a definition of the partial chemical diffusion coefficient of the particles of type r. 

Here, Dt is the diffusion coefficient of species i in the gas phase, Em is the porosity of the membrane having thickness Sm and pore tortuosity and xsgim is the logarithmic mean mole fraction difference of the inert gas species B along the diffusion path (see definition (3.1.132)). Such an expression for k g is valid for conditions of ordinary diffusion of A through a stagnant gas film of B in the pores of the membrane. If the mean free path conditions are such that Knudsen diffusion or other convection mechanisms are valid, appropriate equations have to be used from Section 3.4.2.4. 

In order to obtain a definite breakthrough of current across an electrode, a potential in excess of its equilibrium potential must be applied any such excess potential is called an overpotential. If it concerns an ideal polarizable electrode, i.e., an electrode whose surface acts as an ideal catalyst in the electrolytic process, then the overpotential can be considered merely as a diffusion overpotential (nD) and yields (cf., Section 3.1) a real diffusion current. Often, however, the electrode surface is not ideal, which means that the purely chemical reaction concerned has a free enthalpy barrier especially at low current density, where the ion diffusion control of the electrolytic conversion becomes less pronounced, the thermal activation energy (AG°) plays an appreciable role, so that, once the activated complex is reached at the maximum of the enthalpy barrier, only a fraction a (the transfer coefficient) of the electrical energy difference nF(E ml - E ) = nFtjt is used for conversion. 

The relaxation rates calculated from Eq. (15) are smaller than the measured ones at low field, while they are larger at high field. OST is thus obviously unable to match the experimental results. However, water protons actually diffuse around ferrihydrite and akaganeite particles and there is no reason to believe that the contribution to the rate from this diffusion would not be quadratic with the external field. This contribution is not observed, probably because the coefficient of the quadratic dependence with the field is smaller than predicted. This could be explained by an erroneous definition of the correlation length in OST, this length is the particle radius, whilst the right definition should be the mean distance between random defects of the crystal. This correlation time would then be significantly reduced, hence the contribution to the relaxation rate. 

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