Potential energy distance

Fig. 6. Van der Waals potential energy-distance curve showing regions of operation of contact, noncontact, and intermittent contact or tapping-mode afm...

Fig. 20.17 Potential energy-distance curves for a cathodic reaction showing how the potential energy barrier is lowered by when E < p,z.c. The barrier is assumed to be symmetrical so that /S => yi, where 5 is the distance of the O.H.P. from the surface of the electrode. Full curve—no field across double layer dashed curve-potential diflcrence is E and is negative...

Fig. 20.24 Potential energy-distance from metal surface curves, illustrating (a) an M /M system in which, owing to the relative position of the energy wells, the initial ionisation reaction occurs more rapidly than the discharge reaction, resulting in an excess negative charge on the surface of the metal, (b) equilibrium at which the energy wells are approximately the same and...

FIGURE 14.1 Potential energy-distance curves for reactants and products in a chemical reaction. 

FIGURE 14.2 Potential energy-distance curves for two reactions of the same type. 

When the potential energy-distance curves for the reactants and products are symmetric and have the same slope, we have a = a = 0.5. 

FIGURE 14.3 Schematic potential energy-distance curves of reactants in (1) a normal, (2) a barrierless, (3) an activationless reaction, and (4) potential energy-distance curve for the products. 

The principle of this method is that the initial slope (time = zero) of the optical density-time curve is proportional to the rate of flocculation. This initial slope increases with increasing electrolyte concentration until it reaches a limiting value. The stability ratio W is defined as reciprocal ratio of the limiting initial slope to the initial slope measured at lower electrolyte concentration. A log W-log electrolyte concentration plot shows a sharp inflection at the critical coagulation concentration (W = 1), which is a measure of the stability to added electrolyte. Reerink and Overbeek (12) have shown that the value of W is determined mainly by the height of the primary repulsion maximum in the potential energy-distance curve. 

Fig. 7.75. The effect of the variation of the AH (A AH) on the potential energy distance diagram. (Reprinted from J. O M. Bockris and S. U. M. Khan, Surface Electrochemistry, Plenum, 1993, p. 264.)...

The plot is of the potential energy of a representative point in the reaction and its variation with distance as the proton converts to an adsorbed H atom. There are two components in it. Component X is the potential energy-distance relation of the proton in the hydroxonium ion ... 

A different (second) approach may be adopted. The main point in this new approach is that the value of P will be shown to depend on the relative slopes of the potential energy-distance curves representing the energies of the particles (rather than... 

Is this result a feature of barriers at interfaces or merely a consequence of shifting a linear curve It is clear that once a linear curve is displaced vertically, it cannot but yield a parallel shift of the curve and therefore a constant P- The apparent constancy of P with potential is a result of the linearization of the potential energy-distance... 

It can be seen at once that no simple relation (in particular not Ux = kip, a simple harmonic relation) can represent these potential energy-distance relations. As known since the 1930s, from gas phase spectroscopy, curves with the appearances of those shown in Figs. 9.15 and 9.16 can be represented in form by an empirical relation, the Morse equation ... 

Consider a linear analogue of Fig. 9.9. This figure is a simplification23 of the potential energy-distance relations when there is a vibrational stretching of theH+-0 bond in the system M(e) + H+-OH2 or the M-H bond in the system M-H + H20. It is concerned with proton stretching as a precondition for electron tunneling. It is obvious (Fig. 9.28) that... 

Interesting possibilities arise in these potential energy-distance curves if they are not drawn in the extreme simplification of straight lines (see Fig. 9.33), but with the natural curvature that potential energy-distance relations have. Thus, Fig. 9.33 shows the simplified situation where, from the formula given above for P it can be seen that with approximately equal slopes of the potential energy curves near the intersection point, equal slopes of the value of P will be about one-half, as is often observed. 

One equation that does express the shape of the potential-energy-distance curve for the interaction of two particles is the Morse equation24 ... 

Figure 3. Diagram of a potential energy-distance curve to illustrate the condition for defining the critical coagulation concentration...

FIGURE 4.32 Schematic form of the total potential energy-distance curves. 

Fig. 1. Schematic potential energy-distance relations for the ions in the metal and in the solution, (a) On immediate immersion, (b) After ionic equilibrium has been established.

The potential energy distance relations are shown schematically in Pig. 7. Tne three possibilities for the siow, rate controlhug step may occur at the metal-film contact [Pg.346]

Fig. 7. Schematic potential energy-distance curves for ionic transitions when film is present on metal surface.

E. R. Lippincott and R. Schroeder. J. Chem. Phys. 23, 1131-41 (1955). Theory potential energy-distance relation. 

Rgure 7.8 Entropic (steric) stabilisation the potential energy-distance plots for (a) particles with no electrostatic repulsion, = Vs + a i (b) with electrostatic repulsion, V ,oi = Vs + + Vr. 

Fig. 4. Linear analog of potential energy-distance relation for a simple charge transfer reaction.

Due to the nonlinearity of the potential energy-distance relations, one may expect the symmetry factor to depend on potential. Its effect will be examined in this section using Morse cmrves for the stretching of the bonds A—B+ and M—A (19). In Fig. 5, cimve D-D represents the potential energy-distance relation for stretching of the A—B+ bond, which is expressed by the Morse equation... 

---