Particle free, displacements

Figure 2 shows the form of the displacement vs. time results for different suspension concentrations. For particle-free water (c = 0), the line is straight, since the resistance is just that of the membrane filter, which remains constant. 

Figure 10.16. (a) Schematic representation of the titration of a solution with a standard metal solution at a constant pH. The displacement of the curve (in comparison to a particle-free solution) by the particles reflects the adsorption isotherm and permits the determination of the extent of adsorption, (b) A given metal-ion concentration is titrated with a particle suspension. The slope of this curve also reflects the adsorption isotherm and permits one to determine the extent of adsorption of metal ions to the particles. 

A similar but slower drift of hydroxyl ions moves from the cathode displacing anions adsorbed onto the soil particles. Free anions in the water electrolyte will be in equilibrium with available sites on the soil. Note, however, that these sites compete with OH created at the cathode, and so there will be a net drift of displaced anions to the anode due to electromigration. 

The wavefunction for the lowest energy state of a particle free to oscillate to and fro near a point is e , where x is the displacement from the point (see Section 9.6),... 

It is helpful to consider qualitatively the numerical magnitude of the surface tensional stabilization of a particle at a liquid-liquid interface. For simplicity, we will assume 6 = 90°, or that 7sa = 7SB- Also, with respect to the interfacial areas, J sA = SB, since the particle will lie so as to be bisected by the plane of the liquid-liquid interface, and. AB = rcr - The free energy to displace the particle from its stable position will then be just trr 7AB- For a particle of l-mm radius, this would amount to about 1 erg, for Tab = 40 ergs/cm. This corresponds roughly to a restoring force of 10 dyn, since this work must be expended in moving the particle out of the interface, and this amounts to a displacement equal to the radius of the particle. 

In a solution of molecules of uniform molecular weight, all particles settle with the same value of v. If diffusion is ignored, a sharp boundary forms between the top portion of the cell, which has been swept free of solute, and the bottom, which still contains solute. Figure 9.13a shows schematically how the concentration profile varies with time under these conditions. It is apparent that the Schlieren optical system described in the last section is ideally suited for measuring the displacement of this boundary with time. Since the velocity of the boundary and that of the particles are the same, the sedimentation coefficient is readily measured. 

By locating the anode entirely upstream from the ionized gas volume, collection of long range beta particles is minimized in the displaced coaxial cylinder design, and the direction of gas flow minimizes diffusion and convection of electrons to the collector electrode. However, the free electrons are sufficiently mobile that modest pulse voltages (e.g., 50 V) are adequate to cause the electrons to move against the gas flow and be collected during. this time. 

Relaxation dispersion data for water on Cab-O-Sil, which is a monodis-perse silica fine particulate, are shown in Fig. 2 (45). The data are analyzed in terms of the model summarized schematically in Fig. 3. The y process characterizes the high frequency local motions of the liquid in the surface phase and defines the high field relaxation dispersion. There is little field dependence because the local motions are rapid. The p process defines the power-law region of the relaxation dispersion in this model and characterizes the molecular reorientations mediated by translational displacements on the length scale of the order of the monomer size, or the particle size. The a process represents averaging of molecular orientations by translational displacements on the order of the particle cluster size, which is limited to the long time or low frequency end by exchange with bulk or free water. This model has been discussed in a number of contexts and extended studies have been conducted (34,41,43). 

Fig. 3. Schematic representation of the topological space of hydration water in silica fine-particle cluster (45). The processes responsible for the water spin-lattice relaxation behavior are restricted rotational diffusion about an axis normal to the local surface (y process), reorientations mediated by translational displacements on the length scale of a monomer (P process), reorientations mediated by translational displacements in the length scale of the clusters (a process), and exchange with free water as a cutoff limit.

In general, a dispersed particle is free to move in all three dimensions. For the present, however, we restrict our consideration to the motion of a particle undergoing random displacements in one dimension only. The model used to describe this motion is called a onedimensional random walk. Its generalization to three dimensions is straightforward. 

---