Spheres dilute

This hard sphere dilute solution radial distribution function gives rise to a virial coefficient series in the volume fraction, cf), for the compressibility of the suspension... 

For a hard-sphere dilute solution of colloidal particles, the osmotic pressure Posm is given by the ideal gas expression... 

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

For dilute dispersions of hard spheres, Einstein s viscosity equation predicts... 

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... 

In the derivation of both Eqs. (9.4) and (9.9), the disturbance of the flow streamlines is assumed to be produced by a single particle. This is the origin of the limitation to dilute solutions in the Einstein theory, where the net effect of an array of spheres is treated as the sum of the individual nonoverlapping disturbances. When more than one sphere is involved, the same limitation applies to Stokes law also. In both cases contributions from the walls of the container are also assumed to be absent. 

UfQ = terminal velocity of a single sphere (infinite dilution) c = volume fraction sohd in the suspension n = function of Reynolds number Re = dpUto /[L as given Fig. 6-58... 

For a chemical reaction such as combustion to proceed, mixing of the reactants on a molecular scale is necessary. However, molecular diffusion is a very slow process. Dilution of a 10-m diameter sphere of pure hydrocarbons, for instance, down to a flammable composition in its center by molecular diffusion alone takes more than a year. On the other hand, only a few seconds are required for a similar dilution by molecular diffusion of a 1-cm sphere. Thus, dilution by molecular diffusion is most effective on small-scale fluctuations in the composition. These fluctuations are continuously generated by turbulent convective motion. 

We are concerned with bimolecular reactions between reactants A and B. It is evident that the two reactants must approach each other rather closely on a molecular scale before significant interaction between them can take place. The simplest situation is that of two spherical reactants having radii Ta and tb, reaction being possible only if these two particles collide, which we take to mean that the distance between their centers is equal to the sum of their radii. This is the basis of the hard-sphere collision theory of kinetics. We therefore wish to find the frequency of such bimolecular collisions. For this purpose we consider the relatively simple case of dilute gases. 

Equation (2) is valid only for very dilute suspensions of nondeformable, smooth, uniform spheres. It assumes a Newtonian liquid phase and neglects interaction between particles, a plausible condition when the volume of the solid phase is small compared with the liquid phase. 

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

The picture of a very dilute solution that we must adopt is shown schematically in Fig 2 Each ion is enclosed in its own co-sphere, while the remainder of the solvent between the ions docs not differ in any way from ordinary pure solvent. As a result of recent progress in atomic physics, we now know in great detail the structure and properties of different species of atomic ions in a vacuum and at the same time we know the physical properties of the pure solvent. In order to understand the properties of a very dilute solution, we need to discuss the portions of solvent that lie in the co-spheres of the ions. 

The term Brownian motion was originally introduced to refer to the random thermal motion of visible particles. There is no reason why we should not extend its use to the random motion of the molecules and ions themselves. Even if the ion itself were stationary, the solvent molecules in the outer regions of the co-sphere would be continually changing furthermore, the ion itself executes a Brownian motion. We must use the term co-sphere to refer to the molecules which at any time are momentarily in that region of solvent which is appreciably modified by the ion. In this book we are primarily interested in solutions that are so dilute that the co-spheres of the ions do not overlap, and we are little concerned with the size of the co-spheres. In studying any property... 

Consider 1 mole of a completely dissociated uni-univalent solute in aqueous solution at extreme dilution at 25°C, each positive and each negative ion having a diameter equal to 3.0 angstroms. Find from (19), in calories per mole per degree, what would be the total amount of entropy lost by the solvent in the fields of all these ions, if (19) could correctly be used for a sphere as small as 3 angstroms. 

In a very dilute solution, between the co-spheres of the ions the interstitial solvent is unmodified and has the same properties as in the pure. solvent,. The co-sphere of each positive ion and the co-sphere of each negative ion, however, may contribute toward a change in the viscosity. We should expect to find, in a very dilute solution, for each species of ion present, a total contribution proportional to the number of ions of that species present in unit volume. At the same time, we may anticipate that the electrostatic forces between the positively and the negatively charged ions must be taken into account. 

If the B-coefficients represent the contributions from the co-spheres of the ions, we should expect that in very dilute solution the contributions from the positive and the negative ions would be independent, and... 

At the same time, since there has been, in each co-sphere, an increment in the density of the solvent, we must expect some modification in other properties of the solvent, such as its compressibility. In a very dilute solution it may be difficult to detect such a change by measuring the compressibility of the solution. At higher concentrations, however, when a sufficient fraction of the total solvent lies within the ionic co-spheres, the sum of these local modifications can be detected by measuring the compressibility of the whole solution. 

Let us now consider the same charged sphere immersed in various liquids with widely different values of n. By diluting water with dioxane at constant temperature, we can reduce n from 3.3 X 1022 toward zero. Clearly when n, the number of dipoles per unit volume, approaches zero, the total entropy lost per unit volume must approach zero. From this point of view the expression (170) is seen to have a somewhat paradoxical appearance, since e, which, according to Table 32, is roughly proportional to n, occurs in the denominator. This means that, as the number of dipoles per cubic centimeter decreases, the total amount of entropy lost progressively increases. The reason for this is that, when... 

If we consider a shear flow of a diluted suspension of noninteracting particles, then substitution of spheres by particles of ellipsoidal form leads only to a variation of... 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

Theory presented earlier in this chapter led to the expectation that the frictional coefficient /o for a polymer molecule at infinite dilution should be proportional to its linear dimension. This result, embodied in Eq. (18) where P is regarded as a universal parameter which is the analog of of the viscosity treatment, is reminiscent of Stokes law for spheres. Recasting this equation by analogy with the formulation of Eqs. (26) and (27) for the intrinsic viscosity, we obtain ... 

A determination of dimethyl sulphoxide by Dizdar and Idjakovic" is based on the fact that it can cause changes in the visible absorption spectra of some metal compounds, especially transition metals, in aqueous solution. In these solutions water and sulphoxide evidently compete for places in the coordination sphere of the metal ions. The authors found the effect to be largest with ammonium ferric sulphate, (NH4)2S04 Fe2(S04)3T2H20, in dilute acid and related the observed increase in absorption at 410 nm with the concentration of dimethyl sulphoxide. Neither sulphide nor sulphone interfered. Toma and coworkers described a method, which may bear a relation to this group displacement in a sphere of coordination. They reacted sulphoxides (also cyanides and carbon monoxide) with excess sodium aquapentacyanoferrate" (the corresponding amminopentacyanoferrate complex was used) with which a 1 1 complex is formed. In the sulphoxide determination they then titrated spectrophotometrically with methylpyrazinium iodide, the cation of which reacts with the unused ferrate" complex to give a deep blue ion combination product (absorption maximum at 658 nm). 

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

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