Effective radius, calculation

Fig. 9.17 a Energies as a function of radius for amorphous carbon nanotube (a-CNT) structures black crosses and red circles) compared with those for ideal crystalline nanotubes (blue triangles from [66]). The, black crosses show the a-CNT data as a function of the initial radius of the unrelaxed a-CNT, whilst the red circles show the same data as a function of the effective radius, calculated as a mean average radius. The inset shows the same data plotted on a log-log scale in order to highlight the different dependencies of the respective energetics on /J. b Example morphologies in the unrelaxed (upperpanel) and relaxed (lowerpanel) states for an a-CNT of 6.7 A (Reff 6.9 A)... 

General hydrodynamic theory for liquid penetrant testing (PT) has been worked out in [1], Basic principles of the theory were described in details in [2,3], This theory enables, for example, to calculate the minimum crack s width that can be detected by prescribed product family (penetrant, excess penetrant remover and developer), when dry powder is used as the developer. One needs for that such characteristics as surface tension of penetrant a and some characteristics of developer s layer, thickness h, effective radius of pores and porosity TI. One more characteristic is the residual depth of defect s filling with penetrant before the application of a developer. The methods for experimental determination of these characteristics were worked out in [4]. 

Let us consider the calculation of sensitivity threshold in the case when the cracks are revealing by PT method. Constant distance H between crack s walls along the whole defect s depth is assumed for the simplicity. The calculation procedure depends on the dispersity of dry developer s powder [1]. Simple formula has to be used in the case when developer s effective radius of pores IC, which depends mainly on average particle s size, is smaller than crack s width H. One can use formula (1) when Re is small enough being less than the value corresponding maximum sensitivity (0,25 - 1 pm). For example. Re = 0,25 pm in the case when fine-dispersed magnesia oxide powder is used as the developer. In this case minimum crack s width H that can be detected at prescribed depth lo is calculated as... 

Calculate the vapor pressure of water when present in a capillary of 0.1 m radius (assume zero contact angle). Express your result as percent change from the normal value at 25°C. Suppose now that the effective radius of the capillary is reduced because of the presence of an adsorbed film of water 100 A thick. Show what the percent reduction in vapor pressure should now be. 

Hydration and solvation have also been studied by conductivity measurements these measurements give rise to an effective radius for the ion, from which a hydration number can be calculated. These effective radii are reviewed in the next section. 

Physical Properties. The absorption of x-rays by iodine has been studied and the iodine crystal stmcture deterrnined (12,13). Iodine crystallizes in the orthorhombic system and has a unit cell of eight atoms arranged as a symmetrical bipyramid. The cell constants at 18°C (14) are given in Table 1, along with other physical properties. Prom the interatomic distances of many iodine compounds, the calculated effective radius of the covalently bound iodine atom is 184 pm (15). 

Rough quantitative calculations of the energy of interaction of the electron pairs and the phonon can be made with use of the force constants for the bonds19 and the changes in the position of the minimum in the potential functions for a bond, as given by the foregoing values of the change in effective radius. 

Here, A is the contacting surface area of anode electrode facing with electrolyte and P is the porosity of anode electrode. The average effective radius of pore,, could be calculated from the results of the capillary rise method using ethanol, which shows a contact angle of 0° with the anode electrode. And then, the contact angle 0 could be acquired as the slope from the plot of m versus... 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

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]. 

The radii of both orifices can be either on a micrometer or a submicrometer scale. If the device is micrometer-sized, it can be characterized by optical microscopy. The purposes of electrochemical characterization of a dual pipette are to determine the effective radii and to check that each of two barrels can be independently polarized. The radius of each orifice can be evaluated from an IT voltammogram obtained at one pipette while the second one is disconnected. After the outer surface of glass is silanized, the diffusion-limiting current to each water-filled barrel follows Eq. (1). The effective radius values calculated from that equation for both halves of the d-pipette must be close to the values found from optical microscopy. 

With respect to the size and charge selectivity of paracellular pathways, equivalent pore theory has been utilized to calculate an effective radius based on the membrane transport of uncharged hydrophilic molecules, while equivalent circuit theory has been used to separate mediated from paracellular membrane transport of small ions. The term equivalent should be emphasized, as selectivity parameters are obtained from membrane transport data, so phenomenological information is used to quantitate the magnitude of aqueous pathways... 

From these data one can calculate the effective radius of the pores through which solutes diffuse across the junctional strands (Fig. 16). At day 3, the pore radius was —5.5 A. This correlates with pore radii of —10 A for dog alveolar epithelium (Taylor and Gaar, 1970) and 5 and 8 A for rabbit and bullfrog gallblad-... 

Experimental mobility values, 1.2 X 10-2 cm2/v.s. for eam and 1.9 x 10-3 cm2/v.s. for eh, indicate a localized electron with a low-density first solvation layer. This, together with the temperature coefficient, is consistent with the semicontinuum models. Considering an effective radius given by the ground state wave-function, the absolute mobility calculated in a brownian motion model comes close to the experimental value. The activation energy for mobility, attributed to that of viscosity in this model, also is in fair agreement with experiment, although a little lower. 

There is another use of the Kapustinskii equation that is perhaps even more important. For many crystals, it is possible to determine a value for the lattice energy from other thermodynamic data or the Bom-Lande equation. When that is done, it is possible to solve the Kapustinskii equation for the sum of the ionic radii, ra + rc. When the radius of one ion is known, carrying out the calculations for a series of compounds that contain that ion enables the radii of the counterions to be determined. In other words, if we know the radius of Na+ from other measurements or calculations, it is possible to determine the radii of F, Cl, and Br if the lattice energies of NaF, NaCl, and NaBr are known. In fact, a radius could be determined for the N( )3 ion if the lattice energy of NaNOa were known. Using this approach, which is based on thermochemical data, to determine ionic radii yields values that are known as thermochemical radii. For a planar ion such as N03 or C032, it is a sort of average or effective radius, but it is still a very useful quantity. For many of the ions shown in Table 7.4, the radii were obtained by precisely this approach. 

To calculate the release through diffusion of an entrapped residue, Barraclough et al. (2005) considered the size of organic matter particles (effective radius 10" to 10 cm) and the effective diffusion coefficient of small organic molecules in a sorbing medium (D 10 cm s )- The time for 50% of the material in a sphere to diffuse out is given by... 

Wortman (W8) has given an approximate method of extending results for spheres to other shapes which undergo random tumbling. The method requires calculation of an effective radius of curvature, using kinetic theory (Hll) to define an equivalent cross section. The only restriction, aside from Ma > 1, is that the flight must last for a sufficiently long period that there is no statistically... 

Gas adsorption data may be analyzed for the distribution of pore sizes. What is generally done is to interpret one branch of the isotherm and use an appropriate equation to calculate the effective pore radius at a given pressure. The amount of material adsorbed or desorbed for each increment or decrement in pressure measures the volume of pores with that effective radius. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Strictly speaking, it is correct in the case of complete particle recombination at the black sphere only partial particle reflection is discussed by Doktorov and Kotomin [50]. Incorporation of the back reactions into the kinetics of geminate recombination has been presented quite recently by [74, 75]. The effective radius for an elastic interaction of defects in crystals, (3.1.4), was calculated by Schroder [3], Kotomin and Fabrikant [76],... 

The variational procedure was developed in [61, 63] for calculating Tieff for anisotropic potentials. Employing different trial functions, it is shown that for a strong elastic interaction the effective radius is... 

Figure 6. Results of applying the LUT algorithm to synthetic extinction spectra calculated from measured pre- and post-Pinatubo size distributions obtained from Pueschel el al. [9], Goodman el al [10] and Deshler el al. [11,12]. R,/bimodal) is the effective radius of the measured bimodal size distribution, and R /uni modal) is the corresponding effective radius returned by the LUT. The dots are results obtained when a range of distribution widths are considered in the LUT calculations and the crosses are results obtained when at is restricted to the value that yields the best fit between calculated and measured extinction spectra. The solid and dashed curves are second order polynomial fits to the dots and crosses, respectively.

Comparing the rate constants in the foregoing table with an approximate calculated value of the encounter rate is of interest. Taking the effective radius of the solvated electron as slightly less than 3 A., and the diffusion coefficient in water as 10 4 cm.2/sec., it appears (14) that most of these rate constants are only very slightly lower than diffusion controlled. Only the reaction with triphenyl methanol is substantially slower than diffusion controlled. 

Diffusion-Controlled Reactions. The specific rates of many of the reactions of elq exceed 10 Af-1 sec.-1, and it has been shown that many of these rates are diffusion controlled (92, 113). The parameters used in these calculations, which were carried out according to Debye s theory (41), were a diffusion coefficient of 10-4 sec.-1 (78, 113) and an effective radius of 2.5-3.0 A. (77). The energies of activation observed in e aq reactions are also of the order encountered in diffusion-controlled processes (121). A very recent experimental determination of the diffusion coefficient of e aq by electrical conductivity yielded the value 4.7 0.7 X 10 -5 cm.2 sec.-1 (65). This new value would imply a larger effective cross-section for e aq and would increase the number of diffusion-controlled reactions. A quantitative examination of the rate data for diffusion-controlled processes (47) compared with that of eaq reactions reveals however that most of the latter reactions with specific rates of < 1010 Af-1 sec.-1 are not diffusion controlled. 

The radius thus calculated from the theory of Smith and Symons does not correspond to any known property of halide ions. However, when the acceptable physical model of Franck and Platzman is combined with the concept of a variable radius, as proposed by Smith and Symons, both absolute value and environmental effects can be accounted for. This was done in the theory of Stein and Treinin (18, 19, 47), using an improved energetic cycle to obtain absolute values of r, the spectroscopically effective radius of the cavity containing the X ion. These values were then found to correspond to the known partial ionic radii in solution, as did values of dr/dT to values obtained from other experiments. The specific effects of temperature, solvents, and added salts could be used to differentiate between internal and such CTTS transitions where the electron interacts in the excited state strongly with the medium. These spectroscopic aspects of the theory were examined later in detail and compared with experiment by Treinin and his co-workers (3, 4, 32, 33, 42,48). 

In their investigation Halliwell and Nyburg (39) employed cation radii due to Ahrens (7) and anion radii due to Pauling (8) as the appropriate values for R. In the present work the calculation is repeated using the radii in col. 2 of Table 1. As in the original study, the effective radius of the water molecule has been taken as 1.38 A this is the magnitude found in ice. 

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