Potential energy of electrostatic

According to Coulomb s law, the potential energy of electrostatic interaction U between two point charges, q and q2 in vacuum, is given by... 

A considerable advance was made in the 1940 s when in the theories of Derjaguin and Landau (11) and Verwey and Overbeek (12) a theory of the stability of lyophobic colloids was obtained by assuming the pairwise additivity of the potential energy of electrostatic repulsion, VR, and the van der Waals attraction,... 

On insertion of the field (34) into equation (55) one obtains the general expression for the potential energy of electrostatic interaction between two electric multipolar systems ... 

With the general definition of the electric multipole (40) and n-th order field (53), the potential energies of electrostatic interaction between electric multipoles and fields take the general form ... 

Energy of Induced Multipole Interaction.—Interaction between a Multipolar System and External Fields. We calculated above the potential energy of electrostatic interaction between a multipole system and external fields, or thefirst-orderenergyduetothefirstpowerofthefield. Besides that energy, which took account only of the reoriratation of permanent multipoles, we have to take into consideration contributions due to the drcum-stance that an external electric field induces higher-order multipole moments given by the expressions (72) and (79). 

Work done and potential energy of electrostatic interactions... 

Hence to obtain an expression for the potential energy of electrostatic interactions between the two charges requires that an expression for the work done in bringing the two charges from a... 

The distribution of the ions in the ionic atmosphere of the y -ion is governed by the total potential energy of electrostatic interactions at all positions, r, from the central reference y -ion. For any ion of type, i, in the ionic atmosphere of the y -ion, this potential energy is given by Zie j/j. Here is the charge on the ion of type i, and x]/j is the potential at this ion due to the central y -ion and all the other ions of the ionic atmosphere, but it does not include a contribution from the ion which is at position, r. 

An expression for the force of electrostatic repulsion between two charged crossed hemicylindrical surfaces is given in Equation (3.5) and in fact this turns out to be equivalent to interaction between a sphere and a flat plate. Integration of this expression leads directly to the potential energy of electrostatic repulsion, namely. 

The potential energy of electrostatic interaction between ft and the reaction field is ft Er. The corresponding quantum-mechanical operator in atomic units is... 

Table 10.5 Approximate expressions for the potential energy of electrostatic interactions, Mr, as a function of closest distance of approach, H. In the table, z is the counter-ion change number (e.g. with NaCI as electrolyte, z is either I or -1, depending on whether Na or Ct is the counter-ion to the surface). The "ionic valency" is the same as the "charge number" (both terms include the sign of the charge). More expressions are available in Israelachvili (2011), e.g. for electric double layer interaction between two cylinders or a cylinder and a flat surface...

Fig. 5. A representation of ranitidine displaying four layers of the Connolly solvent-accessible dot surface normally color-coded in this process to correspond with the energies of electrostatic potential (color not shown here). Thus, the highest charge density would be indicated by red dots representing points where the attraction to an atom is strongest, and conversely, purple points would signify regions of maximal positive charge.

Adsorption Forces. Coulomb s law allows calculations of the electrostatic potential resulting from a charge distribution, and of the potential energy of interaction between different charge distributions. Various elaborate computations are possible to calculate the potential energy of interaction between point charges, distributed charges, etc. See reference 2 for a detailed introduction. 

Electrostatic Repulsive Forces. As the distance between two approaching particles decreases, their electrical double layers begin to overlap. As a first approximation, the potential energy of the two overlapping double layers is additive, which is a repulsive term since the process increases total energy. Electrostatic repulsion can also be considered as an osmotic force, due to the compression of ions between particles and the tendency of water to flow in to counteract the increased ion concentration. 

To go from experimental observations of solvent effects to an understanding of them requires a conceptual basis that, in one approach, is provided by physical models such as theories of molecular structure or of the liquid state. As a very simple example consider the electrostatic potential energy of a system consisting of two ions of charges Za and Zb in a medium of dielectric constant e. 

In a solution of a solute in a solvent there can exist noncovalent intermolecular interactions of solvent-solvent, solvent-solute, and solute—solute pairs. The noncovalent attractive forces are of three types, namely, electrostatic, induction, and dispersion forces. We speak of forces, but physical theories make use of intermolecular energies. Let V(r) be the potential energy of interaction of two particles and F(r) be the force of interaction, where r is the interparticle distance of separation. Then these quantities are related by... 

Some force fields make special provision for the mutual electrostatic potential energy of pairs of atoms that have different electronegativities. If atom A has a formal charge of i2a and atom B (distant J ab from Qa) has a formal charge of (2b, then their mutual potential energy is... 

The spheres represent (roughly) the 2p atomic orbitals on C and N, and half an electron resides in each sphere. The mutual potential energy of this charge distribution can be easily calculated from elementary electrostatics. For small distances, a polynomial fit was used instead. 

Let the initial distance between the particles AH and B be denoted by r. The mutual potential energy of the two charged particles is —c2/r, as in the simple ionic dissociation depicted in Fig. 8a. If the value of r is sufficiently great, the energy associated with the electrostatic fields will not depend appreciably on r. For the proton transfer there is thus a characteristic quantity similar to D or. 

This is the mutual electrostatic potential energy when the dipole lies as shown in Fig. 22. We see that the value is more than four times as large as the value obtained above for the mutual potential energy of two H20 dipoles in their most favorable position. 

Note 1. The Free Energy Lost by a Polar Dielectric in an Electrostatic Field. Let Fig. 76 depict a permanent rigid dipole whose axis makes an angle 0 with a uniform field of intensity E. If y is the dipole moment, the potential energy of the dipole in the field is — Ey cos 0. If the dipole is held in this fixed position, any increment dE in the intensity of the field will clearly mean a change in the potential energy of the dipole, equal to — y cos 0 dE. 

The total potential energy of the charged particles (again per mole, with charge ZjF) is the snm of a chemical and an electrostatic component ... 

The energy of an ion in a given medium depends not only on chemical forces but also on the electrostatic held hence the chemical potential of an ion j customarily is called its electrochemical potential and labeled fi. The electrostatic potential energy of an ion j when reckoned per mole is given by ZjF, where / is the electrostatic (inner) potential of the phase containing the ion a plus sign for cations and a minus sign for anions. Hence, the electrochemical potential can be written as the sum of two terms ... 

The DLVO theory, with the addition of hydration forces, may be used as a first approximation to explain the preceding experimental results. The potential energy of interaction between spherical particles and a plane surface may be plotted as a function of particle-surface separation distance. The total potential energy, Vt, includes contributions from Van der Waals energy of interaction, the Born repulsion, the electrostatic potential, and the hydration force potential. [Israelachvili (13)]. 

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