Equilibrium and diffusivities

Gates, C. M. and Newman, J. 2000. Equilibrium and diffusion of methanol and water in a Nafion 117 membrane. AlChE Journal 46 2076-2085. 

Table 7.1 Equilibrium and diffusivity parameters used in the simulation...

M1K Mikhailov, Yu.M., Ganina, L.V., Roshchupkin, V.P., and Shapaeva, N.V., Phase equilibrium and diffusion in binaiy systems based on nonyl aciylate, acrylic acid, and their homo- and copolymers, Polym. Sci., Ser. B, 49, 240, 2007. 

T. J. Edwards, Thermodynamics of Aqueous Solutions Containing One or More Volatile Weak Electrolytes, M.S. Thesis, University of California, Berkeley (1974). C. M. Gates and J. Newman, Equilibrium and Diffusion of Methanol and Water in a Nafion 117 Membrane, AIChE Journal, 46, 2076 (2000). 

The general guidelines for developing a gas separation process based on adsorption are reviewed. Two important industrial cases based on adsorption processes are selected the separation of propane/propylene mixtures and n/iso-paraffins mixtures. The 13X zeolite and Ag -Amberlyst were used as adsorbent for propane/propylene mixture taking into account information from the open literature. The 5A zeolite was selected for n/iso-paraffins system the adsorption equilibrium and diffusivity data were obtained from gravimetric and ZLC techniques respectively. A mathematical model for the bulk separation in fixed bed upon non-isothermal non-adiabatic conditions is formulated and solved numerically. The simulated results are compared with the available experimental breakthrough curves. Finally, a cyclic process based in the PSA-VSA and TSA concepts is proposed for these systems. 

Table 2.1 Single-component equilibrium and diffusivity parameters...

The ortho- and meto-isomers are bulkier than the para-iaomer and diffuse less readily in the zeolite pores. The transport restriction favours their conversion into the /lara-isomer, which is fonned in excess of the equilibrium concentration. Because the selectivity is transport influenced, it is dependent on the path length for transport, which is the length of the zeolite crystallites. 

In tills chapter we shall examine how such temporal and spatial stmctures arise in far-from-equilibrium chemical systems. We first examine spatially unifonn systems and develop tlie tlieoretical tools needed to analyse tlie behaviour of systems driven far from chemical equilibrium. We focus especially on tlie nature of chemical chaos, its characterization and the mechanisms for its onset. We tlien turn to spatially distributed systems and describe how regular and chaotic chemical patterns can fonn as a result of tlie interjilay between reaction and diffusion. 

There are many potential advantages to kinetic methods of analysis, perhaps the most important of which is the ability to use chemical reactions that are slow to reach equilibrium. In this chapter we examine three techniques that rely on measurements made while the analytical system is under kinetic rather than thermodynamic control chemical kinetic techniques, in which the rate of a chemical reaction is measured radiochemical techniques, in which a radioactive element s rate of nuclear decay is measured and flow injection analysis, in which the analyte is injected into a continuously flowing carrier stream, where its mixing and reaction with reagents in the stream are controlled by the kinetic processes of convection and diffusion. 

At first glance, the contents of Chap. 9 read like a catchall for unrelated topics. In it we examine the intrinsic viscosity of polymer solutions, the diffusion coefficient, the sedimentation coefficient, sedimentation equilibrium, and gel permeation chromatography. While all of these techniques can be related in one way or another to the molecular weight of the polymer, the more fundamental unifying principle which connects these topics is their common dependence on the spatial extension of the molecules. The radius of gyration is the parameter of interest in this context, and the intrinsic viscosity in particular can be interpreted to give a value for this important quantity. The experimental techniques discussed in Chap. 9 have been used extensively in the study of biopolymers. 

If a sedimentation experiment is carried out long enough, a state of equilibrium is eventually reached between sedimentation and diffusion. Under these conditions material will pass through a cross section perpendicular to the radius in both directions at equal rates downward owing to the centrifugal field, and upward owing to the concentration gradient. It is easy to write expressions for the two fluxes which describe this situation ... 

Influence of Chemical Reactions on Uq and When a chemical reaction occurs, the transfer rate may be influenced by the chemical reac tion as well as by the purely physical processes of diffusion and convection within the two phases. Since this situation is common in gas absorption, gas absorption will be the focus of this discussion. One must consider the impacts of chemical equilibrium and reac tion kinetics on the absorption rate in addition to accounting for the effec ts of gas solubility, diffusivity, and system hydrodynamics. 

Chemistry can be divided (somewhat arbitrarily) into the study of structures, equilibria, and rates. Chemical structure is ultimately described by the methods of quantum mechanics equilibrium phenomena are studied by statistical mechanics and thermodynamics and the study of rates constitutes the subject of kinetics. Kinetics can be subdivided into physical kinetics, dealing with physical phenomena such as diffusion and viscosity, and chemical kinetics, which deals with the rates of chemical reactions (including both covalent and noncovalent bond changes). Students of thermodynamics learn that quantities such as changes in enthalpy and entropy depend only upon the initial and hnal states of a system consequently thermodynamics cannot yield any information about intervening states of the system. It is precisely these intermediate states that constitute the subject matter of chemical kinetics. A thorough study of any chemical reaction must therefore include structural, equilibrium, and kinetic investigations. 

FIGURE 5A.2 A dialysis experiment. The solution of macromolecules to be dialyzed is placed in a semipermeable membrane bag, and the bag is immersed in a bathing solution. A magnetic stirrer gently mixes the solution to facilitate equilibrium of diffusible solutes between the dialysate and the solution contained in the bag. 

In a liquid that is in thermodynamic equilibrium and which contains only one chemical species, the particles are in translational motion due to thermal agitation. The term for this motion, which can be characterized as a random walk of the particles, is self-diffusion. It can be quantified by observing the molecular displacements of the single particles. The self-diffusion coefficient is introduced by the Einstein relationship... 

Under certain conditions, it will be impossible for the metal and the melt to come to equilibrium and continuous corrosion will occur (case 2) this is often the case when metals are in contact with molten salts in practice. There are two main possibilities first, the redox potential of the melt may be prevented from falling, either because it is in contact with an external oxidising environment (such as an air atmosphere) or because the conditions cause the products of its reduction to be continually removed (e.g. distillation of metallic sodium and condensation on to a colder part of the system) second, the electrode potential of the metal may be prevented from rising (for instance, if the corrosion product of the metal is volatile). In addition, equilibrium may not be possible when there is a temperature gradient in the system or when alloys are involved, but these cases will be considered in detail later. Rates of corrosion under conditions where equilibrium cannot be reached are controlled by diffusion and interphase mass transfer of oxidising species and/or corrosion products geometry of the system will be a determining factor. 

In equilibrium dialysis of a solution of a polyanion (valence Zp negative) with molar concentration Cp against a solution of imi-imivalent electrolyte CA (C = cation, A = anion) with molar concentration Cqa it was shown that the requirement for equal chemical potentials of the salt in the polyanion (a) and diffusate ()) phases results in the following relation... 

Hydrogen may also be determined by both electrochemical and diffusion meters. The electrochemical meter is a hydride-activated concentration cell that employs an electrolyte consisting of a CaH2-CaCl2 mixture. The diffusion meter is based on the equilibrium pressures attained on either side of a thin membrane, usually nickel. 

In this exercise we shall estimate the influence of transport limitations when testing an ammonia catalyst such as that described in Exercise 5.1 by estimating the effectiveness factor e. We are aware that the radius of the catalyst particles is essential so the fused and reduced catalyst is crushed into small particles. A fraction with a narrow distribution of = 0.2 mm is used for the experiment. We shall assume that the particles are ideally spherical. The effective diffusion constant is not easily accessible but we assume that it is approximately a factor of 100 lower than the free diffusion, which is in the proximity of 0.4 cm s . A test is then made with a stoichiometric mixture of N2/H2 at 4 bar under the assumption that the process is far from equilibrium and first order in nitrogen. The reaction is planned to run at 600 K, and from fundamental studies on a single crystal the TOP is roughly 0.05 per iron atom in the surface. From Exercise 5.1 we utilize that 1 g of reduced catalyst has a volume of 0.2 cm g , that the pore volume constitutes 0.1 cm g and that the total surface area, which we will assume is the pore area, is 29 m g , and that of this is the 18 m g- is the pure iron Fe(lOO) surface. Note that there is some dispute as to which are the active sites on iron (a dispute that we disregard here). 

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