Molecular diffusion, calculating

The diffusivity of the vapour of a volatile liquid in air can be conveniently determined by Winkelmann s method, in which liquid is contained in a narrow diameter vertical tube maintained at a constant temperature, and an air stream is passed over the top of the tube sufficiently rapidly to ensure that the partial pressure of the vapour there remains approximately zero. On the assumption that the vapour is transferred from the surface of the liquid to the air stream by molecular diffusion, calculate the diffusivity of carbon tetrachloride vapour in air at 321 K and atmospheric pressure from the following experimentally obtained data ... 

Where, D is the molecular diffusivity calculated from Wilke-Chang equation (1955). Spis the porosity of catalyst and i is the tortosity factor. For the present case the value of 8p was taken as 0.5 and i was taken as 2.4 (Satterfield, 1970). The diffusivity values calculated are presented in Table 2.5. 

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

The mass transfer effect is relevant when the chemical reaction is far faster than the molecular diffusion, i.e. Ha > 1. The rapid formation of precipitate particles should then occur spatially distributed. The relative rate of particle formation to chemical reaction and/or diffusion can as yet be evaluated only via lengthy calculations. 

At a depth l below the liquid surface, the. concentration of A has fallen to one-half of the value at the. surface. What is the. ratio of the. mass transfer rate at this depth t to the. rate, at the surface Calculate the numerical value of the ratio when l /k/D = 0.693, where. D is the molecular diffusivity and k the first-order rate constant. 

Computer simulations therefore have several inter-related objectives. In the long term one would hope that molecular level simulations of structure and bonding in liquid crystal systems would become sufficiently predictive so as to remove the need for costly and time-consuming synthesis of many compounds in order to optimise certain properties. In this way, predictive simulations would become a routine tool in the design of new materials. Predictive, in this sense, refers to calculations without reference to experimental results. Such calculations are said to be from first principles or ab initio. As a step toward this goal, simulations of properties at the molecular level can be used to parametrise interaction potentials for use in the study of phase behaviour and condensed phase properties such as elastic constants, viscosities, molecular diffusion and reorientational motion with maximum specificity to real systems. Another role of ab initio computer simulation lies in its interaction... 

The importance of dilfusion in a tubular reactor is determined by a dimensionless parameter, SiAt/S = QIaLKuB ), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SiAtlB is small, then the elfects of dilfusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin studied the elfects of dilfusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

Optimization requires that a-rtjl have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that 3>AtlR be large. Thus, the method can be used for polymer systems that have thermal diffusivities typical of organic liquids but low molecular diffusivities. The calculations needed to solve the optimization are much longer than those needed to solve the ODEs of Chapter 6, but they are still feasible on small computers. 

Fig. 5.1.2 Non-ideal capillary flow reactor (a) propagators [13] and (b) corresponding RTDs calculated from the propagator data, (a) The propagators indicate the distribution of average velocities over each observation time (A) ranging from 50 ms to 1 s. As the observation time increases the spins exhibit a narrowing distribution of average velocities due to the motional narrowing effect of molecular diffusion across the streamlines. The dashed vertical line represents the maximum velocity that would be present in the absence of molecular...

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Calculations show that after 10-30 years, molecular diffusion begins to transport the first molecules of waste 3 ft downward through a compacted soil liner. Accordingly, even with a perfectly impermeable liner with 0 hydraulic conductivity, in 1-3 decades contaminants will begin to pass through the soil liner due to molecular diffusion. 

PC = partition coefficient values from Adson et at. (1995). Molecular density p 1.0 g/mL. Diffusion coefficient and molecular radius calculated by Eqs. (41) and (42). 

The quantity D, cannot be derived from molecular diffusivities at infinite dilution the calculated ionic diffusivity of Cu2+ is approximately 20% lower than the molecular diffusivity of CuS04. 

Mass-transfer rates from limiting-current measurements in well-supported solutions should invariably be correlated with ionic and not with molecular diffusivities. The former can be calculated from limiting-current measurements, for example, at a rotating-disk electrode. 

Diffusion and mass transfer effects cause the dimensions of the separated spots to increase in all directions as elution proceeds, in much the same way as concentration profiles become Gaussian in column separations (p. 86). Multiple path, molecular diffusion and mass transfer effects all contribute to spreading along the direction of flow but only the first two cause lateral spreading. Consequently, the initially circular spots become progressively elliptical in the direction of flow. Efficiency and resolution are thus impaired. Elution must be halted before the solvent front reaches the opposite edge of the plate as the distance it has moved must be measured in order to calculate the retardation factors (Rf values) of separated components (p. 86). 

The gas A must transfer from the gas phase to the liquid phase. Equation (1) describes the specific (per m2) molar flow (JA) of A through the gas-liquid interface. Considering only limitations in the liquid phase, this molar flow notably depends on the liquid molecular diffusion coefficient DAL (m2 s ). Based on the liquid state theories, DA L can be calculated using the Stokes-Einstein expression, and many correlations have been developed in order to estimate the liquid diffusion coefficients. The best-known example is the Wilke and Chang (W-C) relationship, but many others have been established and compared (Table 45.4) [28-33]. 

Thus, the time that is necessary to attain a certain coverage, 6, or the time necessary to cover the surface completely (9 = 1) is inversely proportional to the square of the bulk concentration (cf. Fig. 4.10b). Assuming molecular diffusion only, 8 is of the order of 2 minutes for a concentration of 10 5 M adsorbate when the diffusion coefficient D is 10 5 cm2 s1 and rmax = 4 1010 mol cm 2 1). Considering that transport to the surface is usually by turbulent diffusion, such a calculation illustrates that the formation of an adsorption layer is relatively rapid at concentrations above 10 6 M. But it can become slow at concentrations lower than 10 6 M. 

When the mean free path is small compared with pore diameter, the dominating experience of molecules is that of collision with other molecules in the gas phase. In that respect, the situation is much the same as that which exists in the bulk gas. The appropriate diffusion coefficient Dm may be obtained from published experimental values, or calculated from a theoretical expression. For molecular diffusion in a binary gas, the Chapman and Enskog equation may be used, as discussed by Bird, Stewart and Lightfoot(32). This takes the form ... 

Experimental measurements of surface diffusion are usually calculated by subtracting from the measured total diffusion that predicted theoretically for Knudsen and molecular diffusion. 

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