Molecules characterization, diffusion equation

Dimensional analysis of the coupled kinetic-transport equations shows that a Thiele modulus (4> ) and a Peclet number (Peo) completely characterize diffusion and convection effects, respectively, on reactive processes of a-olefins [Eqs. (8)-(14)]. The Thiele modulus [Eq. (15)] contains a term ( // ) that depends only on the properties of the diffusing molecule and a term ( -) that includes all relevant structural catalyst parameters. The first term introduces carbon number effects on selectivity, whereas the second introduces the effects of pellet size and pore structure and of metal dispersion and site density. The Peclet number accounts for the effects of bed residence time effects on secondary reactions of a-olefins and relates it to the corresponding contribution of pore residence time. 

The impingement rate discussed here characterizes the maximum intrinsic rate at which gas molecules immediately above a solid can strike the surface of a solid. The impingement rate does not take concentration gradients into effect, and thus it represents the maximum rate of transport for a gas-phase species to a surface when diffusion is unimportant. When calculating the rate of transport of a gas-phase species to a solid surface, the decision to use the impingement rate versus the diffusion equation depends on the length scale of the transport relative to the mean free path of the transporting gas molecules. 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Small-step rotational diffusion is the model universally used for characterizing the overall molecular reorientation. If the molecule is of spherical symmetry (or approximately this is generally the case for molecules of important size), a single rotational diffusion coefficient is needed and the molecular tumbling is said isotropic. According to this model, correlation functions obey a diffusion type equation and we can write... 

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

The haphazard rotational motions of molecules or one or more segments of a molecule. This diffusional process strongly influences the mutual orientation of molecules (particularly large ones) as they encounter each other and proceed to form complexes. Rotational diffusion can be characterized by one or more relaxation times, t, describing the motion of a molecule or segment of volume, V, in a medium of viscosity, 17, as shown in the following equation ... 

Equation (42) cannot be used if NO concentrations approach their equilibrium values, since the net production rate then depends on the concentration of NO, thereby bringing bivariate probability-density functions into equation (40). Also, if reactions involving nitrogen in fuel molecules are important, then much more involved considerations of chemical kinetics are needed. Processes of soot production similarly introduce complicated chemical kinetics. However, it may be possible to characterize these complex processes in terms of a small number of rate processes, with rates dependent on concentrations of major species and temperature, in such a way that a function w (Z) can be identified for soot production. Rates of soot-particle production in turbulent diffusion flames would then readily be calculable, but in regions where soot-particle growth or burnup is important as well, it would appear that at least a bivariate probability-density function should be considered in attempting to calculate the net rate of change of soot concentration. 

Knudsen flow is characterized by the mean free path (A) of the molecules, which is larger than the pore size, and hence collisions between the molecules and the pore walls are more frequent than intermolecular collisions. A lower limit for the significance of the Knudsen mechanism has usually been set at dp> 20 A [28]. The classical Knudsen equation for diffusion of gas is... 

For many purposes, it is more convenient to characterize the rotary Brownian movement by another quantity, the relaxation time t. We may imagine the molecules oriented by an external force so that the a axes are all parallel to the x axis (which is fixed in space). If this force is suddenly removed, the Brownian movement leads to their disorientation. The position of any molecule after an interval of time may be characterized by the cosine of the angle between its a axis and the x axis. (The molecule is now considered to be free to turn in any direction in space —its motion is not confined to a single plane, but instead may have components about both the b and c axes.) When the mean value of cosine for the entire system of molecules has fallen to ile(e — 2.718... is the base of natural logarithmus), the elapsed time is defined as the relaxation time r, for motion of the a axis. The relaxation time is greater, the greater the resistance of the medium to rotation of the molecule about this axis, and it is found that a simple reciprocal relation exists between the three relaxation times, Tj, for rotation of each of the axes, and the corresponding rotary diffusion constants defined in equation (i[Pg.138]

Generalizing the hydrod3mamical equations derived by Stokes for spheres, Edwardes (35) calculated the coefficients fj, Ca and for ellipsoids as a function of their axial ratios. The general equations are complicated but for ellipsoids of revolution, which may be characterized by only two values of f, they assume a simpler form, and have been employed by Gans 47) and F. Perrin 92) to evaluate the rotary diffusion constants of molecules which may be treated as ellipsoids of revolution. The formulas of Gans and Perrin are not identical, but the numerical values of 0 calculated from them are nearly so, so that the formulas of either author may be used in practice. In the following discussion we shall employ Perrin s equations. 

Diffusion is the macroscopic result of the sum of all molecular motions involved in the sample studied. Molecular motions are described by the general equation of dynamics. However, because of the enormous difference in the orders of magnitude of the masses, sizes, and forces that characterize molecules and macroscopic solids, it can be shown [1] that, when a force field (e.g., an electric field to an ionic solution) is applied to a chemical system, the acceleration of the molecules or ions is nearly instantaneous, molecules drift at a constant velocity, and, in the absence of an external field and of internal forces acting on the feed components, which is the case in chromatography, the diffusional flux, /, of a chemical species i in a gradient of chemical potential is given by... 

P (Q Q) and Q(Q -> n) are the probabilities that a molecule will rotate in the redistribution processes of trans cis optical transition and cis =i trans thermal recovery respectively. The orientational hole burning is represented by a probability proportional to cos 9. The last terms on the right-hand side of Eq. 11 describe the rotational diffusion due to Brownian motion. This is a Smoluchowski equation for the rotational diffusion characterized by a constant of diffusion for the cis (trans) configur-... 

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