Knudsen’s base

Dimethylolnitramine (252) readily participates in Mannich condensation reactions treatment of a aqueous solution of (252) with methylamine, ethylenediamine and Knudsen s base (254) (generated from fresh solutions of ammonia and formaldehyde) yields (253), (255) and (239) (DPT) respectively. The cyclic ether (258) is formed from the careful dehydration of dimethylolnitramine (252) under vacuum. ... 

The base of the nitrate (II) has proved to be identical with Knudsen s base for which an incorrect formula was given by Knudsen [66], and identical with the product (H) of interaction of methyl nitrate with hexamine described by Hahn and Walter [67]. 

Savran, C. A. Knudsen, S. M. Ellington, A. D. Manalis, S. R., Micromechanical detection of proteins using aptamer-based receptor molecules, Anal. Chem. 2004, 76, 3194-3198... 

The factor of appears in equation (21-19) because molecules confined to narrow channels probably collide with the walls of a tube, for example, that are separated by 2(raverage), and the dimensionality of the system is 3 for random Brownian motion in three dimensions. In many cases, the factor of /3 in (21-19) is replaced by the kinetic theory prediction of y/S/jt when Knudsen is based on the average speed of the gas molecules (i.e., (u, ) = SRT/ttMW,). Now the Knudsen diffusion coefficient is given by 92% of (21-19) (see Moore, 1972, p. 124 Bird et al., 2002, pp. 23, 525 Dullien, 1992, p. 293 and Smith, 1970, p. 405). If the average pore size is expressed in angstroms and the temperature... 

The main emphasis in this chapter is on the use of membranes for separations in liquid systems. As discussed by Koros and Chern(30) and Kesting and Fritzsche(31), gas mixtures may also be separated by membranes and both porous and non-porous membranes may be used. In the former case, Knudsen flow can result in separation, though the effect is relatively small. Much better separation is achieved with non-porous polymer membranes where the transport mechanism is based on sorption and diffusion. As for reverse osmosis and pervaporation, the transport equations for gas permeation through dense polymer membranes are based on Fick s Law, material transport being a function of the partial pressure difference across the membrane. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

One method which is known under the name of permeametry [131] or Poiseuille-Knudsen method [124] is based on the law of gas permeability in a porous media in the two flow regimes molecular flow (Knudsen) and laminar or viscous flow (Poiseuille). According to Darcy s law, the gas flux through a membrane with a thickness / can be written as / = KAP/l, where K is the permeability coefficient and AP (AP = Pi - P2) the pressure difference across the membrane. If the membrane pore diameter is comparable to the mean free path of the permeating gas, K can be expressed as a stun of a viscous and a non-vis-cous term... 

Figure 7.] Varialicin of collision frequency function (i(a. 02) with particle size ratio a ju2 for air at 23°C and 1 atm based on Fuchs (1964, p. 294). The value of f i, 2) is. smallest for particles of equal size (01/02 = I) and the spread in value with particle size is smallesl. Forrii/o =. P go< s through a weak maximum for Knudsen number near 5. The value of (0. 03) highest for interacting panicles of very different sizes (large 01/02). The lowest curves correspond to the continuum regime.

The aforementioned preliminary mass spectrometric Knudsen effusion study by Emons et al. [448], which includes also the vapor over KCl(s, 1), is complemented by Hastie s work on KCl(l) [322] using transpiration mass spectrometry. The equilibrium partial pressures of KCl(g) and (KCl)2(g) obtained by Hastie agree reasonably well with those given in the JANAF tables [90] largely based on extrapolated data of KCl(s). Equilibrium pressures of (KCl)3(g) are given for the first time [322]. 

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