Measuring diffusivities

When the concentration of the solute is low, the measured diffusion profiles are predictable from Eick s second law of diffusion ... 

By way of example, Volume 26 in Group III (Crystal and Solid State Physics) is devoted to Diffusion in Solid Metals and Alloys, this volume has an editor and 14 contributors. Their task was not only to gather numerical data on such matters as self- and chemical diffusivities, pressure dependence of diffusivities, diffusion along dislocations, surface diffusion, but also to exercise their professional judgment as to the reliability of the various numerical values available. The whole volume of about 750 pages is introduced by a chapter describing diffusion mechanisms and methods of measuring diffusivities this kind of introduction is a special feature of Landolt-Bornstein . Subsequent developments in diffusion data can then be found in a specialised journal. Defect and Diffusion Forum, which is not connected with Landolt-Bdrnstein. 

It is seen that the value of (H) is completely dependent on the diffusivity of the solute in the mobile phase, the column radius and the linear velocity of the mobile phase. The simple uncoated open tube can clearly be used to determine the diffusivity of any solute in any given solvent (the mobile phase). This technique for measuring diffusivities will be discussed in a later chapter. 

Equations (4-5) and (4-7) are alternative expressions for the estimation of the diffusion-limited rate constant, but these equations are not equivalent, because Eq. (4-7) includes the assumption that the Stokes-Einstein equation is applicable. Olea and Thomas" measured the kinetics of quenching of pyrene fluorescence in several solvents and also measured diffusion coefficients. The diffusion coefficients did not vary as t) [as predicted by Eq. (4-6)], but roughly as Tf. Thus Eq. (4-7) is not valid, in this system, whereas Eq. (4-5), used with the experimentally measured diffusion coefficients, gave reasonable agreement with measured rate constants. 

Typical approaches for measuring diffusivities in immobilised cell systems include bead methods, diffusion chambers and holographic laser interferometry. These methods can be applied to various support materials, but they are time consuming, making it onerous to measure effective dififusivity (Deff) over a wide range of cell fractions. Owing to the mathematical models involved, the deconvolution of diffusivities can be very sensitive to errors in concentration measurements. There are mathematical correlations developed to predict DeS as... 

Meylan S, Odzak N, Behra R, Sigg L (2004) Speciation of copper and zinc in natural freshwater comparison of voltammetric measurements, diffusive gradients in thin Aims (DGT) and chemical equilibrium models. An Chim Acta 510 91... 

Tokita et al. [394] measured diffusion of different molecules with molecular weights varying from 18 to 342 in polyacrylamide gels with constant percentage cross-linker and varying total acrylamide concentration. They found the data to be in good agreement with the stretched exponential of the form... 

Johnson et al. [186] measured diffusion of fluorescein-labeled macromolecules in agarose gels. Their data agreed well with Eq. (85), which combined the hydrodynamic effects with the steric hindrance factors. Gibbs and Johnson [131] measured diffusion of proteins and smaller molecules in polyacrylamide gels using pulsed-field gradient NMR methods and found their data to fit the stretched exponential form... 

In the data compiled by Janz and Bansal, various methods for measuring diffusion coefficients in molten salts are mentioned. The methods may be broadly classified as electrochemical and analytical. However, some other methods have occasionally been employed. Various electrochemical methods were reviewed by Lesko. Tracer diffusion in molten salts was reviewed by Spedding in 1971, where some other methods were also mentioned. 

Methods Used to Measure Diffusion Coefficients in Molten Salts ... 

Unlike solid electrodes, the shape of the ITIES can be varied by application of an external pressure to the pipette. The shape of the meniscus formed at the pipette tip was studied in situ by video microscopy under controlled pressure [19]. When a negative pressure was applied, the ITIES shape was concave. As expected from the theory [25a], the diffusion current to a recessed ITIES was lower than in absence of negative external pressure. When a positive pressure was applied to the pipette, the solution meniscus became convex, and the diffusion current increased. The diffusion-limiting current increased with increasing height of the spherical segment (up to the complete sphere), as the theory predicts [25b]. Importantly, with no external pressure applied to the pipette, the micro-ITIES was found to be essentially flat. This observation was corroborated by numerous experiments performed with different concentrations of dissolved species and different pipette radii [19]. The measured diffusion current to such an interface agrees quantitatively with Eq. (6) if the outer pipette wall is silanized (see next section). The effective radius of a pipette can be calculated from Eq. (6) and compared to the value found microscopically [19]. 

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

As noted by Liu et al. [42], there is a resurgence of interest in measuring diffusion coefficients using NMR spectroscopy, particularly in complex biological fluids which contain a number of molecules having a broad range of molecular weights and concentrations. These authors demonstrated the use of two-dimensional diffusion-edited total correlation NRM spectroscopy to measure diffusion coefficients in human blood samples. 

A Poison. New method for measuring diffusion constants of biologically active substances. Nature 154 823, 1944. 

Politzer, P., J. S. Murray, and P. Flodmark. 1996. Relationship Between Measured Diffusion Coefficients and Calculated Molecular Surface Properties. J. Phys. Chem. 100, 5538. 

To continue the derivation, the next step is to determine the variation of the absorbance readings starting with the definition of absorbance. The extension we present here, of course, is based on Beer s law, which is valid for clear solutions. For other types of measurements, diffuse reflectance for example, the derivation should be based on a suitable function of T that applies to the situation, for example the Kubelka-Munk function for diffuse reflectance should be used for that case ... 

The aim of many of the studies of diffusion is to relate the measured diffusion coefficient to a mechanism of diffusion. By this is meant a model of atomic jumps that accurately reproduces the diffusion coefficient and the measured concentration profile over a wide range of temperatures. This objective has been most pronounced... 

As discussed in the previous section, a number of diffusion mechanisms might operate at the same time. In such cases, Eq. (5.7) needs to take correlation into account, and the measured diffusion coefficient, D, can be written ... 

Equation (5.11) ignores the correction factors included in Eqs. (5.7) and (5.8). When these are added in, the measured diffusion coefficient can be written ... 

The likelihood that Fick s laws will be obeyed in a crystal containing dislocations is dependent upon the spacing between the defects. Provided that this spacing is much greater than the diffusion length (Dt)1/2, where D is the bulk diffusion coefficient, Fick s laws are obeyed, with an effective (measured) diffusion coefficient, Deff, given by... 

A common method to measure diffusion coefficients consists in the welding of two samples with different concentrations of the element of which the diffusion coefficient is to be known. Upon heating, diffusion asymmetrical profiles are quite commonly obtained which show that different species diffuse at different rates. The concentration-... 

Immobilized enzymes used in conjunction with ion-selective electrodes provide very convenient methods of analysis. The immobilized enzyme may be held in a gel or membrane around the electrode and the substance to be measured diffuses into the enzyme gel. Its conversion to the product alters the ionic equilibrium across the ion-selective membrane (Figure 8.23). It is important that the enzyme layer is thin, to minimize any problems caused by slow diffusion rates through the layer. 

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