Potential energy of repulsion

The last entry in Table 10.1 is the least well defined of those listed. This is of little importance to us, however, since our interest is in attraction, and the final entry in Table 10.1 always corresponds to repulsion. The reader may recall that so-called hard-sphere models for molecules involve a potential energy of repulsion that sets in and rises vertically when the distance of closest approach of the centers equals the diameter of the spheres. A more realistic potential energy function would have a finite (though steep) slope. An inverse power law with an exponent in the range 9 to 15 meets this requirement. For reasons of mathematical convenience, an inverse 12th-power dependence on the separation is frequently postulated for the repulsion between molecules. 

A potential energy of repulsion may extend appreciable distances from surfaces, but its range is compressed (i.e., reduced) by increasing the electrolyte content of the system. 

The total potential energy of repulsion is given by integrating over all values of z. This is most readily done by assuming that z varies between s and infinity. This upper limit is justified because the potential function drops off exponentially with distance. Therefore large separations make a negligible contribution to the total. Integrating between these limits, we obtain... 

That these different compounds produce the same effect at so nearly the same concentration argues that the principal cause of the effect is electrostatic. Use the average of these concentrations to calculate (a) the value of k at which this system coagulates, (b) the force of repulsion (Equation (82)), and (c) the potential energy of repulsion (Equation (86)) when two planar surfaces are separated by a distance of 10 nm. For the purpose of calculation in parts (b) and (c), i/ o may be taken as 100 mV. Comment on the applicability of these equations to the physical system under consideration. 

The first term on the right is the operator for the electrons kinetic energy the second term is the operator for the potential energy of attraction between the electrons and the nucleus (r, being the distance between electron i and the nucleus) the third term is potential energy of repulsion between all pairs of electrons ru being the distance between electrons / and j) the last term is the spin-orbit interaction (discussed below). In addition, there are other relativistic terms besides spin-orbit interaction, which we neglect. 

Several repulsive and attractive forces operate between colloidal species and determine their stability [12,13,15,26,152,194], In the simplest example of colloid stability, dispersed species would be stabilized entirely by the repulsive forces created when two charged surfaces approach each other and their electric double layers overlap. The overlap causes a coulombic repulsive force acting against each surface, which will act in opposition to any attempt to decrease the separation distance (see Figure 5.2). One can express the coulombic repulsive force between plates as a potential energy of repulsion. There is another important repulsive force causing a strong repulsion at very small separation distances where the atomic electron clouds overlap, called Born repulsion. 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

There are three energy contributions that must be considered in the description of the helium atom (1) the kinetic energy of the electrons as they move around the nucleus, (2) the potential energy of attraction between the nucleus and the electrons, and (3) the potential energy of repulsion between the two electrons. 

The potential energy of repulsion Vr depends on the size and shape of the dispersed particles, the distance between them, their surface potential To, the dielectric constant sr of the dispersing liquid, and the effectiveness thickness of the electrical double layer 1 /k (Chapter 2, Section I), where... 

The potential energy of repulsion is always positive, since its value at infinity is zero and increases as the particles approach each other. 

The term, l/ri2, is the potential energy of repulsion between the two electrons. Since the electrons repel one another, this effect will tend to keep them apart thus r 2 will be as large as possible under the constraints that ri and r2 must be small, since the electrons are both attracted to the nucleus. 

A similar calculation can be carried out for the potential energy of repulsive forces between two identical spherical particles. In the case when the thickness of the electrical double layer around these particles is small compared to their radii, interaction of electrical double layers of these spheres may, according to Derjaguin, be considered as a superposition of interactions of infinitely narrow parallel rings (Fig. 10.2) [53]. 

Fig. 37. Potential energy of repulsion between two spherical particles at constant surface potential. t — xa% s R/fl. r (5—2) = x (R —2fl) measures the distance between the surfaces of the particles in the thickness of the double layer l/ as a unit.

Therefore we define the potential energy of repulsion (repulsive potential) Vn per cm cross section of the plates... 

This is the so-called Lennard-Jones potential 6-12, offered for the interaction description of nonpolar molecules. Values a and b for different molecules are different. The first term expresses the potential energy of attraction whereas the second term expresses the potential energy of repulsion. In Figure 1.32 both curves are represented by dotted lines and the solid line is a resulting curve (eq. (1.5.8)). 

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