Step—Constant Diffusivity

The simplest case to consider is a single microporous adsorbent particle, such as a zeolite crystal, exposed to a step change in sorbate concentration at the external surface of the particle at time zero. Heat transfer is assumed to be sufficiently rapid, relative to the sorption rate, so that temperature gradients 

If the uptake of sorbate by the adsorbent is small relative to the total quantity of sorbate introduced into the system, the ambient sorbate concentration will remain essentially constant following the initial step change, and the appropriate initial and boundary conditions are 

The solution for the uptake curve is then given by the familiar expression 

At 70% uptake this expression deviates by less than 2% from the full solution [Eq. (6.4)]. 

FIGURE 62. Theoretical uptake curve calculated according to q. (6.4). showing limiting behavior in the long time region. 

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The specific conductivities of molten salts are frequently represented, as a function of temperature by an AtTlrenius equation, but it is unlikely that the unit step in diffusion has a constant magnitude, as in the coiTesponding solids and the results for NaCl may be expressed, within experimental eiTor, by the alternative equations... 

For an aqueous suspension of crystals to grow, the solute must (a) make its way to the surface by diffusion, (b) undergo desolvation, and (c) insert itself into the lattice structure. The first step involves establishment of a stationary diffusional concentration field around each particle. The elementary step for diffusion has an activation energy (AG ), and a molecule or ion changes its position with a frequency of (kBT/h)exp[-AGl,/kBT]. Einstein s treatment of Brownian motion indicates that a displacement of A will occur within a time t if A equals the square root of 2Dt. Thus, the rate constant for change of position equal to one ionic diameter d will be... 

As mentioned, all reaction models will include initially unknown reaction parameters such as reaction orders, rate constants, activation energies, phase change rate constants, diffusion coefficients and reaction enthalpies. Unfortunately, it is a fact that there is hardly any knowledge about these kinetic and thermodynamic parameters for a large majority of reactions in the production of fine chemicals and pharmaceuticals this impedes the use of model-based optimisation tools for individual reaction steps, so the identification of optimal and safe reaction conditions, for example, can be difficult. 

In order to emphasize the critical relationship between the timestep and the spatial step, we consider the one-dimensional diffusion equation with constant diffusion coefficient D = 1 ... 

The experimental method used in TEOM for diffusion measurements in zeolites is similar to the uptake and chromatographic methods (i.e., a step change or a pulse injection in the feed is made and the response curve is recorded). It is recommended to operate with dilute systems and low zeolite loadings. For an isothermal system when the uptake rate is influenced by intracrystalline diffusion, with only a small concentration gradient in the adsorbed phase (constant diffusivity), solutions of the transient diffusion equation for various geometries have been given (ii). Adsorption and diffusion of o-xylene, / -xylene, and toluene in HZSM-5 were found to be described well by a one-dimensional model for diffusion in a slab geometry, represented by Eq. (7) (72) ... 

The diffusion coefficient of an ion in water is 1.5 x 10 cm s". It seems reasonable to take the distance between two steps in diffusion as roughly the diameter of a water molecule (320 pm). With this assumption, calculate the rate constant in for the ion s diffusion. 

The motion of molecules in a liquid has a significant effect on the kinetics of chemical reactions in solution. Molecules must diffuse together before they can react, so their diffusion constants affect the rate of reaction. If the intrinsic reaction rate of two molecules that come into contact is fast enough (that is, if almost every encounter leads to reaction), then diffusion is the rate-limiting step. Such diffusion-controlled reactions have a maximum bimolecular rate constant on the order of 10 ° L mol s in aqueous solution for the reaction of two neutral species. If the two species have opposite charges, the reaction rate can be even higher. One of the fastest known reactions in aqueous solution is the neutralization of hydronium ion (H30 ) by hydroxide ion (OH ) ... 

The first step is diffusion controlled while the second represents the fast formation of the outer sphere complex. The final step involves the conversion of the outer to the inner sphere complex. This is the rate determining step and is dependent on the equilibrium concentration of the outer sphere complex. Consequently, calculations of rate constants by the Eigen model involves estimation of the formation constant of the outer sphere species. 

Since most adsorption and ion exchange separations of commercial significance operate in the nonlinear region of the isotherm, the previous analysis needs to be expanded to nonlinear systems. Nonlinear behavior is distinctly different than linear behavior since one usually observes shock or constant pattern waves during the feed step and diffuse or proportional pattern waves during regeneration. Experimental... 

The shape of the uptake curve does not differ greatly from the shape of the constant diffusivity curve but the uptake rate is modified. The effective diffusivity DJ thus becomes dependent on the step size, lying within the range... 

The small step rotational diffusion model has been employed to extract rotational diffusion constants Dy and D from the measured deuterium spectral densities in liquid crystals [7.25, 7.27, 7.46, 7.49 - 7.53]. Both the single exponential correlation functions [Eq. (7.54)] and the multiexponential correlation functions [Eq. (7.60)] have been used to interpret spectral densities of motion. However, most deuterons in liquid crystal molecules are located in positions where they are rather insensitive to motion about the short molecular axis. Thus, there is a large uncertainty in determining D or Tq (t o) because of Dy > D and the rather small geometric factor [doo( )] for most deuterons in liquid crystal molecules. For 5CB, it is necessary to fix [7.52] the value of D using the known activation... 

Nahir, T.M. Buck, R.P. (1992) Modified Cottrell Behavior in Thin Layers Applied Voltage Steps under Diffusion Control for Constant-Resistance Systems. /, Electroanal. Chem. Vol.341, No.1-2, (December 1992), pp. 1-14, ISSN 1572-6657 Nair, P.R. Alam, M.A. (2010). Kinetic Response of Surfaces Defined by Finite Fractals. 

In its initial application, a triple potential step was applied at a submarine UME placed in the aqueous subphase of a Langmuir trough, close (1-2 pm) to the monolayer. The technique involves generating an electroactive species (Ox) at the UME by diffusion-controlled electrolysis of a precursor (Red) in an initial potential step. Ox diffuses to, and reacts with, the redox-active amphiphile at the water-air interface resulting in the conversion of the solution redox species to its initial form (Red), which then undergoes diffusional feedback to the UME. In this first step, the rate constant for electron transfer between the solution mediator and the surface-confined species can be measured from the UME current-time transient. In the second period, the potential step is reversed to convert the electrogenerated species (Ox) to its initial form (Red). Lateral diffusion of electroactive amphiphile into the interfacial zone probed by the UME occurs simultaneously in this recovery period. 

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