The Dissolving Sphere

Consider the case of a solid sphere falling through a stagnant fluid, in which the sphere is soluble. This is a problem that could be solved on the computer, taking into consideration the change of shape of the sphere due to the differences of mass transfer at different locations. Even if that were the objective, we should do a modeling study before the more detailed analysis and simulation. We assume that the sphere falls under gravity and attains the Stokes velocity at all times we shall return to examine this assumption later. Thus its downward velocity is 

6 Dedimensionalization is an ugly word, fit only to be used in Washington. I am sure 1 have fallen into using it occasionally, but repent in favor of this title in spite of its length. Where there is no possibility of confusion, I will merely use reduction. 

71 learned this the hard way when teaching a course in the old Humble Lectures at Baytown. When I was dealing with the nonisothermal stirred tank, I had one equation for the concentration of the reactant and one for the temperature. To compare them, I expressed both in moles per unit volume. The notion of a temperature in moles per unit volume was not acceptable to the class. 

The number of moles of the solid chemical in the sphere is N = (4/3)ttPpslm, and this is diminished by the mass transfer from the falling sphere into the stagnant fluid. At the surface of the sphere the concentration of the solute is 5, the saturation solubility, and far from the sphere it is zero. Thus, 

We will use the empirical formula for the mass transfer coefficient (checking later on the range of its validity8) 

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To illustrate what is meant by scaling, let us return to the problem of the dissolving sphere and try to make sure that all the important dependent variables are in the interval [0, 1]. The independent variable, time, must be allowed to run its course. We have certainly done this with the radius of the sphere, for r can only diminish so, if the initial radius is R,x = r/R is obviously the correct choice. With an eye to extending the model later, we define U as the terminal velocity of a sphere of radius R, and because this decreases with decreasing radius, v = u/U is certainly in [0,1]. The very simple relationship v = x2 holds as long as our assumption of the validity of Stokes law is true. 

Finally, calculations (not comparable with experiment) were periormed which better approximate the physical dissolution process. In these calculations, the dissolved uranium is allowed to spread throve the water solution in layers of increasing ra us and decreasii uranium concentration until the condition of total homogenization is reached. It is found that systems are more reactive when the dissolved material is in close proximity to the dissolving sphere. 

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]. 

Multiparticle collision dynamics provides an ideal way to simulate the motion of small self-propelled objects since the interaction between the solvent and the motor can be specified and hydrodynamic effects are taken into account automatically. It has been used to investigate the self-propelled motion of swimmers composed of linked beads that undergo non-time-reversible cyclic motion [116] and chemically powered nanodimers [117]. The chemically powered nanodimers can serve as models for the motions of the bimetallic nanodimers discussed earlier. The nanodimers are made from two spheres separated by a fixed distance R dissolved in a solvent of A and B molecules. One dimer sphere (C) catalyzes the irreversible reaction A + C B I C, while nonreactive interactions occur with the noncatalytic sphere (N). The nanodimer and reactive events are shown in Fig. 22. The A and B species interact with the nanodimer spheres through repulsive Lennard-Jones (LJ) potentials in Eq. (76). The MPC simulations assume that the potentials satisfy Vca = Vcb = Vna, with c.,t and Vnb with 3- The A molecules react to form B molecules when they approach the catalytic sphere within the interaction distance r < rc. The B molecules produced in the reaction interact differently with the catalytic and noncatalytic spheres. 

In the broadest sense, coordination chemistry is involved in the majority of steps prior to the isolation of a pure metal because the physical properties and relative stabilities of metal compounds relate to the nature and disposition of ligands in the metal coordination spheres. This applies both to pyrometallurgy, which produces metals or intermediate products directly from the ore by use of high-temperature oxidative or reductive processes and to hydrometallurgy, which involves the processing of an ore by the dissolution, separation, purification, and precipitation of the dissolved metal by the use of aqueous solutions. 4... 

Fig. 7.5 TEM image of microcapsules prepared theinsetcorrespondsto800nm. PLL/PGAlayers by LbL assembly of three bilayers of a PLL/PGA were assembled from a 0.05 M MES, pH 5.5 shell on catalase-loaded BMS spheres, following buffer. The MS spheres were dissolved usingHF/ removal ofthe BMS particle template (A). CLSM NH4F at pH 5. (Adapted from [82] with per-images of (PLL/PGA)3 microcapsules loaded mission of Wiley-VCH). with FITC-labeled catalase (B). The scale bar in...

When an ionic compound is dissolved in a solvent, the crystal lattice is broken apart. As the ions separate, they become strongly attached to solvent molecules by ion-dipole forces. The number of water molecules surrounding an ion is known as its hydration number. However, the water molecules clustered around an ion constitute a shell that is referred to as the primary solvation sphere. The water molecules are in motion and are also attracted to the bulk solvent that surrounds the cluster. Because of this, solvent molecules move into and out of the solvation sphere, giving a hydration number that does not always have a fixed value. Therefore, it is customary to speak of the average hydration number for an ion. 

When the solution is formed in the places of atom-components contact, the unified electron density has to be established. The dissolving process is accompanied by the redistribution of this density between valence areas of both particles and transition of some electrons from external spheres to the neighboring ones. 

The influence of hydration on the reactivity of anions is much more evident in the case of OH. In the chlorobenzene-aqueous NaOH system the hydration sphere of tetrahexylammonium hydroxide dissolved in the organic phase progressively decreases from 11 to 3.3 water molecules when the base concentration is raised from 15 to 63%. This leads to an enhanced reactivity of OH which was measured in the Hofmann elimination (Equation 3). In the examined ranges of NaOH concentrations the reactivity increased up to more than four orders of magnitude (Table I). Although the dehydration of OH is... 

The surface chemistry of coesite and stishovite was studied by Stiiber (296). The packing density of hydroxyl groups was estimated from the water vapor adsorption. More adsorption sites per unit surface area were found with silica of higher density. Stishovite is especially interesting since it is not attacked by hydrofluoric acid. Coesite is dissolved slowly. The resistance of stishovite is ascribed to the fact that silicon already has a coordination number of six. Dissolution of silica to HaSiFg by hydrogen fluoride is a nucleophilic attack. It is not possible when the coordination sphere of silicon is filled completely. In contrast, stishovite dissolves with an appreciable rate in water buffered to pH 8.2. The surface chemistry of. stishovite should be similar to that of its analog, rutile. 

To explain this different fractionation behavior, Taube (1954) postulated different isotope effects between the isotopic properties of water in the hydration sphere of the cation and the remaining bulk water. The hydration sphere is highly ordered, whereas the outer layer is poorly ordered. The relative sizes of the two layers are dependent upon the magnitude of the electric field around the dissolved ions. The strength of the interaction between the dissolved ion and water molecules is also dependent upon the atomic mass of the atom to which the ion is bonded. 

Figure 9-11 Concentration profiles around growing or dissolving sphere. The dissolving case implies a product...

Figure 6.5 Temperature dependence of the characteristics of sodium k-carrageenan particles dissolved in an aqueous salt solution (0.1 M NaCl). The cooling rate is 1.5 °C min-1, (a) ( ) Weight-average molar weight, Mw, and (A) second virial coefficient, A2. (b) ( ) Specific optical rotation at 436 nm, and ( ) penetration parameter, y, defined as tlie ratio of the radius of the equivalent hard sphere to the radius of gyration of the dissolved particles (see equation (5.33) in chapter 5). See the text for explanations of different regions I, II, III and IV.

Note that in the notation of Fig. 18.6, CA becomes the aqueous concentration in the pores of the particle aggregate, C, while Cg becomes the time-constant fluid concentration C°. For simplicity we consider only the case where the initial concentration in the sphere is zero (CA = 0). The dissolved concentration in the sphere, C (r,t), follows from Eq. 18-33 by putting CA = 0 and replacing C by C°. The half-saturation time tm is given by Eq. 18-37 (D replaced by the effective diffusivity... 

Until now we have tacitly assumed that the initial concentration in the fluid outside the sphere, c° > remains constant during the process of sorptive equilibration between fluid and particle. This situation is called the infinite bath case (Wu and Gsch-wend, 1988), since it requires an infinite fluid volume. In the case where the fluid volume is finite and thus a significant portion of the dissolved load is transferred to the particle, things become more complicated. The finite bath situation can be characterized by the nondimensional number ... 

FIGURE 8.26 When a nonpolar compound (the yellow sphere) dissolves in water, the water molecules may become organized around it. As a result, the entropy of the solvent is reduced when the solution forms. 

These observations show that ions bound in a coordination sphere are not free to react. When the red complex dissolves in water bromide ions become free to move through the solution (Fig. 16.23a), and when they encounter a silver ion, they form a precipitate. Conversely, when the violet complex dissolves in water the bromide ion remains attached to the cobalt ion and is not free to react with silver ions (Fig. 16.23b). However the sulfate ion is not a part of the coordination sphere and is free to react with added barium ions. 

The other pathway leading to the formation ofoxogroups in the coordination sphere of the metal atom is provided by uncontrolled oxidation of the basic alkoxides such as alkali, alkaline earth metal, and quite probably the rare earth metal ones by oxygen dissolved in solvents and present in the atmosphere. The primary oxidation products are peroxides and hydroperoxides — M(OOR)n and M(OOH)n, whose decomposition gives water among the other... 

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