Ordinary diffusion

The self-diffusion of carbon dioxide in single pellets of commercial type 5A molecular sieve has been studied using C 02 as a tracer. Experiments were carried out at atmospheric pressure between - -25° and —2S°C. Using a simple model of the pellet structure, it was possible to deduce effective diffusivities for both pore and crystal diffusion. Ordinary gas diffusion occurs in the pores crystal diffusivities have values of the order of 10 cm /sec. 

Mass transport in the pore system of the particle is described by the Dusty-Gas-Model which includes Knudsen diffusion, ordinary diffusion and convection. 

Example 3.1.9 In Example 3.1.8, the phenomenon under consideration was ordinary molecular diffusion. Ordinary molecular diffusion due to a concenttadon gradient or partial pressure gradient (see relations (3.1.40) for example) does result in a molecular velocity V (we assume no bulk motion of the gas or liquid phase). Here we would like to calculate this velocity for arbitrary values of concenttation difference or partial pressure difference over a diffusional path length. 

Disinfection. Ozone is a more effective broad-spectmm disinfectant than chlorine-based compounds (105). Ozone is very effective against bacteria because even concentrations as low as 0.01 ppm are toxic to bacteria. Whereas disinfection of bacteria by chlorine involves the diffusion of HOGl through the ceU membrane, disinfection by ozone occurs with the lysing (ie, mpture) of the ceU wall. The disinfection rate depends on the type of organism and is affected by ozone concentration, temperature (106), pH, turbidity, clumping of organisms, oxidizable substances, and the type of contactor employed (107). The presence of oxidizable substances in ordinary water can retard disinfection until the initial ozone demand is satisfied, at which point rapid disinfection is observed. 

Ordinary diffusion involves molecular mixing caused by the random motion of molecules. It is much more pronounced in gases and Hquids than in soHds. The effects of diffusion in fluids are also greatly affected by convection or turbulence. These phenomena are involved in mass-transfer processes, and therefore in separation processes (see Mass transfer Separation systems synthesis). In chemical engineering, the term diffusional unit operations normally refers to the separation processes in which mass is transferred from one phase to another, often across a fluid interface, and in which diffusion is considered to be the rate-controlling mechanism. Thus, the standard unit operations such as distillation (qv), drying (qv), and the sorption processes, as well as the less conventional separation processes, are usually classified under this heading (see Absorption Adsorption Adsorption, gas separation Adsorption, liquid separation). 

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

We denote by C the value of c(x , t) at any time. Thus, C is a function of time, and differential equations in C are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2,. . . , n (see Fig. 3-52). 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

As a result, collisions with the wall occur more frequently than with other molecules. This is referred to as the Knudsen mode of diffusion and is contrasted with ordinary or bulk diffusion, which occurs by intermolecular collisions. At intermediate pressures, both ordinaiy diffusion and Knudsen diffusion may be important [see Eqs. (5-239) and (5-240)]. 

Few mechanisms of liquid/liquid reactions have been established, although some related work such as on droplet sizes and power input has been done. Small contents of surface-ac tive and other impurities in reactants of commercial quality can distort a reac tor s predicted performance. Diffusivities in liquids are comparatively low, a factor of 10 less than in gases, so it is probable in most industrial examples that they are diffusion controllech One consequence is that L/L reactions may not be as temperature sensitive as ordinary chemical reactions, although the effec t of temperature rise on viscosity and droplet size can result in substantial rate increases. L/L reac tions will exhibit behavior of homogeneous reactions only when they are very slow, nonionic reactions being the most likely ones. On the whole, in the present state of the art, the design of L/L reactors must depend on scale-up from laboratoiy or pilot plant work. 

If the pump is a filter pump off a high-pressure water supply, its performance will be limited by the temperature of the water because the vapour pressure of water at 10°, 15°, 20° and 25° is 9.2, 12.8, 17.5 and 23.8 mm Hg respectively. The pressure can be measured with an ordinary manometer. For vacuums in the range lO" mm Hg to 10 mm Hg, rotary mechanical pumps (oil pumps) are used and the pressure can be measured with a Vacustat McLeod type gauge. If still higher vacuums are required, for example for high vacuum sublimations, a mercury diffusion pump is suitable. Such a pump can provide a vacuum up to 10" mm Hg. For better efficiencies, the pump can be backed up by a mechanical pump. In all cases, the mercury pump is connected to the distillation apparatus through several traps to remove mercury vapours. These traps may operate by chemical action, for example the use of sodium hydroxide pellets to react with acids, or by condensation, in which case empty tubes cooled in solid carbon dioxide-ethanol or liquid nitrogen (contained in wide-mouthed Dewar flasks) are used. 

Azobenzene [103-33-3] M 182.2, m 68", pK 2.48. Ordinary azobenzene is nearly all in the transform. It is partly converted into the cw-form on exposure to light [for isolation see Hartley J Chem Soc 633 1938, and for spectra of cis- and /ran5-azobenzenes, see Winkel and Siebert Chem Ber 74B 6707947]. trans-Azobenzene is obtained by chromatography on alumina using 1 4 benzene/heptane or pet ether, and crystd from EtOH (after refluxing for several hours) or hexane. All operations should be carried out in diffuse red light or in the dark. 

Several methods have been employed to study chemical reactions theoretically. Mean-field modeling using ordinary differential equations (ODE) is a widely used method [8]. Further extensions of the ODE framework to include diffusional terms are very useful and, e.g., have allowed one to describe spatio-temporal patterns in diffusion-reaction systems [9]. However, these methods are essentially limited because they always consider average environments of reactants and adsorption sites, ignoring stochastic fluctuations and correlations that naturally emerge in actual systems e.g., very recently by means of in situ STM measurements it has been demon-... 

The trapped radicals, most of which are presumably polymeric species, have been used to initiate graft copolymerization [127,128]. For this purpose, the irradiated polymer is brought into contact with a monomer that can diffuse into the polymer and thus reach the trapped radical sites. This reaction is assumed to lead almost exclusively to graft copolymer and to very little homopolymer since it can be conducted at low temperature, thus minimizing thermal initiation and chain transfer processes. Moreover, low-molecular weight radicals, which would initiate homopolymerization, are not expected to remain trapped at ordinary temperatures. Accordingly, irradiation at low temperatures increases the grafting yield [129]. 

The exchange process in a chelating resin is generally slower than in the ordinary type of exchanger, the rate apparently being controlled by a particle diffusion mechanism. 

Carbon dioxide devices were originally developed by Severinghaus and Bradley (59) to measure the partial pressure of carbon dioxide in blood. This electrode, still in use today (in various automated systems for blood gas analysis), consists of an ordinary glass pH electrode covered by a carbon dioxide membrane, usually silicone, with an electrolyte (sodium bicarbonate-sodium chloride) solution entrapped between them (Figure 6-17). When carbon dioxide from the outer sample diffuses through the semipermeable membrane, it lowers the pH of the inner solution ... 

The temperature dependence of a diffusion-controlled rate constant is very small. Actually, it is just the temperature coefficient of the diffusion coefficient, as we see from the von Smoluchowski equation. Typically, Ea for diffusion is about 8-14 kJ mol"1 (2-4 kcal mol-1) in solvents of ordinary viscosity. 

Mechanisms of micellar reactions have been studied by a kinetic study of the state of the proton at the surface of dodecyl sulfate micelles [191]. Surface diffusion constants of Ni(II) on a sodium dodecyl sulfate micelle were studied by electron spin resonance (ESR). The lateral diffusion constant of Ni(II) was found to be three orders of magnitude less than that in ordinary aqueous solutions [192]. Migration and self-diffusion coefficients of divalent counterions in micellar solutions containing monovalent counterions were studied for solutions of Be2+ in lithium dodecyl sulfate and for solutions of Ca2+ in sodium dodecyl sulfate [193]. The structural disposition of the porphyrin complex and the conformation of the surfactant molecules inside the micellar cavity was studied by NMR on aqueous sodium dodecyl sulfate micelles [194]. 

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... 

Tubular reactors are used for some polycondensations. Para-blocked phenols can be reacted with formalin to form linear oligomers. When the same reactor is used with ordinary phenol, plugging will occur if the tube diameter is above a critical size, even though the reaction stoichiometry is outside the region that causes gelation in a batch reactor. Polymer chains at the wall continue to receive formaldehyde by diffusion from the center of the tube and can crosslink. Local stoichiometry is not preserved when the reactants have different diffusion coefficients. See Section 2.8. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

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