Charged spheres attractive force between

Fig. 8. Attractive force between equal-sized, equally and oppositely charged spheres at various particle clearances.

Equation (50) may also be used to calculate the attractive force between equal sized, equally and oppositely charged spheres that are perfect insulators. For this condition, the value of ksA and of ksg is taken as unity if polarization possibilities are neglected, and if the charge is initially uniformly distributed. Any polarization will tend toward an approach to the conductor condition (which basically represents a condition of infinite polarization). Figure 8 presents a plot of Eq. (50). Thus, if we assume that the maximum possible field intensity is 200 V/micron, the attractive force between equally but... 

Just as a charged sphere in saltwater surrounds itself with a number of mobile ions different from what would occupy the same region in its absence, so does a charged cylinder. As with spheres, there are low-frequency ionic fluctuations that create attractive forces between like cylinders. In the special case of thin cylinders whose material dielectric response is the same as that of the medium and the distance between cylinders is small compared with the Debye screening length, this ionic-fluctuation force has appealing limiting forms. 

How large is an atom We cannot answer this question for an isolated atom. We can, however, devise experiments in which we can find how closely the nucleus of one atom can approach the nucleus of another atom. As atoms approach, they are held apart by the repulsion of the positively charged nuclei. The electrons of the two atoms also repel one another but they are attracted by the nuclei. The closeness of approach of two nuclei will depend upon a balance between the repulsive and attractive forces. It also depends upon the energy of motion of the atoms as they approach one another. If we think of atoms as spheres, we find that their diameters vary from 0.000 000 01 to 0.000 000 05 cm (from 1 X 10-8 to 5 X 10 8 cm). Nuclei are much smaller. A typical nuclear diameter is 10, s cm, about 1/100,000 the atom diameter. 

Outer and Inner Sphere Complexes. Outer sphere complexation involves interactions between metal ions and other solute species in which the co-ordinated water of the metal ion and/or the other solute species are retained. For example, the initial step in the formation of ion pairs, where ions of opposite charge approach within a critical distance and are then held together by coulombic attractive forces, is described as outer sphere complex formation. 

It is well known [4,5] that in the case of hard spheres di = d.2 = 0), the electrostatic force between two dissimilar spheres with charges of unlike sign is attractive for large kH but becomes repulsive at small kH, that is, there is a minimum in the interaction energy except when a lox =1. The case of nonzero Kd and xd2,... 

With his very fine torsion balance, Coulomb was able to demonstrate that the repulsive force between two small spheres electrified with the same type of electricity is inversely proportional to the square of the distance between the centers of the two spheres. At the time, the electron had not yet been discovered, so the underlying reason for this remained a mystery but Coulomb was able to demonstrate that both repulsion and attraction followed this principle. He was not able to make the quantitative step to show that the force was also directly proportional to the product of the charges, but he did complete some experiments exploring this relationship. As a consequence, the law governing one of the four fundamental forces of nature is named Coulomb s law ... 

In the layer with decreased relative permittivity surrounding the ions, the free energy of the solvent is lower than in the absence of the electric field of the ions. The approach of the ions towards one another requires the mutual inter-penetration of the solvate spheres, i.e., the release of a certain amount of solvent from ihc solvate sphere of the ions. This process needs work, and this work appears as a repulsive force between the ions, (This effect lends stability to electrolyte solutions, for in the absence of such repulsive forces, attraction between the charges would favour the precipitation of the solid salts.) By taking into account such repulsive forces, it was possible to interpret the positive deviation of the average activity coefficients of the ions from the Debye-Hiickel limiting law (hypernetted chain equations, HNC, calculation by the Monte Carlo method [Ra 68, Ra 70],... 

The counterions form a diffuse cloud that shrouds each particle in order to maintain electrical neutrality of the system. When two particles are forced together their counterion clouds begin to overlap and increase the concentration of counterions in the gap between the particles. If both particles have the same charge, this gives rise to a repulsive potential due to the osmotic pressure of the counterions which is known as the electrical double layer (EDL) repulsion. If the particles are of opposite charge an EDL attraction will result. It is important to realize that EDL interactions are not simply determined by the Columbic interaction between the two charged spheres, but are due to the osmotic pressure (concentration) effects of the counterions in the gap between the particles. 

Kralchevsky et al. [14] also proved from the energy method that Eqn. (3.49) is valid for calculating the horizontal force between the two floating spherical particles. Since r is always positive, the sign of capillary charge Q. depends solely on y/.. If both particles have the same sign of y/., the lateral capillary force between the two floating spheres is attractive otherwise, the force is repulsive. 

In 1785, the French scientist Charles de Coulomb (Figure 8.1) made very accurate measurements of the force of attraction or repulsion between small charged spheres. He found that the direction of the interaction—that is, attraction or repulsion—is dictated by the types of the charges on the spheres. If two spheres have the same charge, either positive or negative, they repel each other. If, however, the two spheres have different charges, they attract each other. 

Coulomb also found that the magnitude of the interaction between any two spheres is dependent on the distance between the two small spheres. The force of attraction or repulsion, F, between two charged spheres varies inversely with the square of the distance, r, between the spheres ... 

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