Diffusion molar concentration

Pore dijfusion in fluid-filled pores. These pores are sufficiently large that the adsorbing moleciile escapes the force field of the adsorbent surface. Thus, this process is often referred to as macropore dijfusion. The driving force for such a diffusion process can be approximated by the gradient in mole fraction or, if the molar concentration is constant, by the gradient in concentration of the diffusing species within the pores. 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

Equation 10.96 does not apply to either electrolytes or to concentrated solutions. Reid, PRAUSNITZ and Sherwood"7 discuss diffusion in electrolytes. Little information is available on diffusivides in concentrated solutions although it appears that, for ideal mixtures, the product /xD is a linear function of the molar concentration. 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

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... 

Approximation refers to the bringing together of the substrate molecules and reactive functionalities of the enzyme active site into the required proximity and orientation for rapid reaction. Consider the reaction of two molecules, A and B, to form a covalent product A-B. For this reaction to occur in solution, the two molecules would need to encounter each other through diffusion-controlled collisions. The rate of collision is dependent on the temperature of the solution and molar concentrations of reactants. The physiological conditions that support human life, however, do not allow for significant variations in temperature or molarity of substrates. For a collision to lead to bond formation, the two molecules would need to encounter one another in a precise orientation to effect the molecular orbitial distortions necessary for transition state attainment. The chemical reaction would also require... 

Tne molar concentration of pure MEK is ca. 11.2 M. One might question why the concentration of MEK does not reach 11.2 M on the SCP. This is mostly due to the slow process of untangling PMMA chains. For the concentration of MEK to reach 11.2 M, the swollen polymer gel phase has to be untangled and removed from the vicinity of the quartz substrate. This is driven by the entropic force which works rather slowly in the absence of high solvent flow. For example, Mills et al. (22) report, for TCE diffusing into PMMA film, that the SCP of TCE stabilizes at a mole fraction of less than 0.2. By comparison, our results of [MEK] = 3.2 M corresponds to a mole fraction of ca. 0.3. This, again, reflects the better solubility of MEK in PMMA relative to TCE (6 = 9.6). 

In summary, for a non-stationary turbulent reacting flow wherein all scalars can be assumed to have identical molecular diffusivities and the initial and boundary conditions are uniform/constant, the -component molar concentration vector... 

Here Dq and Dl are the diffusivities of A in the gas phase and the coating liquid, respectively, and is the total molar concentration of the shell phase. 

The classical equation for 7 sis provided in Section VII.A of Chapter 2. It depends only on the spin quantum number S, on the molar concentration of paramagnetic metal ions, on the distance d, and on a diffusion coefficient D, which is the sum of the diffusion coefficients of both the solvent molecule (Dj) and the paramagnetic complex (Dm), usually much smaller. The outer-sphere relaxivity calculated with this equation at room temperature and in pure water solution, by assuming d equal to 3 A, is shown in Pig. 25. It appears that the dispersions do not have the usual Lorentzian form. 

In a mixture of two gases A and B provided that the total pressure, and hence the total molar concentration is constant, d[A]/dx and d[B]/dx are equal and opposite and A and B tend to diffuse in opposite directions. In equimolar counter-diffusion, the two components diffuse at equal and opposite rates =- ba- Equation (108) can then be integrated... 

If initial concentration gradients are in two complementing components, such as FeO and MgO (with the sum of FeO and MgO molar concentrations being a constant), and all other components have uniform concentration, the diffusion between the two components may be treated as interdiffusion, or effective binary diffusion. 

C = liquid-phase molar concentration of cesium D — liquid-phase diffusivity of cesium Dv = gas-phase diffusivity f = fractional release J = mass flux... 

Similar considerations apply to the role of spin equilibria in electron transfer reactions. For many years spin state restrictions were invoked to account for the slow electron exchange between diamagnetic, low-spin cobalt(III) and paramagnetic, high-spin cobalt(II) complexes. This explanation is now clearly incorrect. The rates of spin state interconversions are too rapid to be competitive with bimolecular encounters, except at the limit of diffusion-controlled reactions with molar concentrations of reagents. In other words, a spin equilibrium with a... 

The reaction of azide ions with carbocations is the basis of the azide clock method for estimating carbocation lifetimes in hydroxylic solvents (lifetime = 1 lkiy where lq, is the first-order rate constant for attack of water on the carbocation) this is analogous to the radical clock technique discussed in Chapter 10. In the present case, a rate-product correlation is assumed for the very rapid competing product-forming steps of SN1 reactions (Scheme 2.24). Because the slow step of an SN1 reaction is formation of a carbocation, typical kinetic data do not provide information about this step. Furthermore, the rate constant for the reaction of azide ion with a carbocation (kaz) is assumed to be diffusion controlled (ca. 5 x 109 M 1 s 1). The rate constant for attack by water can then be obtained from the mole ratio of azide product/solvolysis product, and the molar concentrations of azide (Equation 2.18, equivalent to Equation 2.14) [48]. The reliability of the estimated lifetimes was later... 

Dj is the mass diffusion coefficient, and cgas is the total molar concentration of the gas mixture. Although Equations (3.9a) and (3.9b) can be used for a free-path gas (e.g. gas channel), when a gas is moving within a porous media (i.e. electrode), Equation (3.9) may not be the most appropriate. Different constitutive laws can be employed for describing the diffusive flux within a porous medium. The choice of the most appropriate law depends on the operating conditions and the porous media properties, as further explained in Section 3.3.2. 

Let d be the volumetric molar concentration of Mi in Q and C12 the surface molar concentration of M12 on TG. Assuming that both types of ions have the same diffusion coefficient D > 0 and since crystalline is immobile, mass conservation for M% gives... 

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