Potential energy of dipole

Complete and Incomplete Ionic Dissociation. Brownian Motion in Liquids. The Mechanism of Electrical Conduction. Electrolytic Conduction. The Structure of Ice and Water. The Mutual Potential Energy of Dipoles. Substitutional and Interstitial Solutions. Diffusion in Liquids. 

An important point to note is that, like the potential energy of dipole-dipole interactions between rotating molecules, the potential energy of the London interaction also decreases very rapidly with distance (as 1/r6 see Fig. 5.1). 

We will restrict our consideration to weak fields for which pEy CKT (the potential energy of dipole interaction with an external field U is significantly less than the energy... 

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. 

Using the method given in Note 2 of the Appendix, obtain an expression for the mutual potential energy of the two dipoles shown in Fig. 21... 

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 mutual potential energy of the rigid dipole and the charge q... 

FIGURE 5.1 The distance dependence of the potential energy of the interaction between ions (red, lowest line), ions and dipoles (brown), stationary dipoles (green), and rotating dipoles (blue, uppermost line). 

Closely related to the London interaction is the dipole-induced-dipole interaction, in which a polar molecule interacts with a nonpolar molecule (for example, when oxygen dissolves in water). Like the London interaction, the dipole—induced-dipole interaction arises from the ability of one molecule to induce a dipole moment in the other. However, in this case, the molecule that induces the dipole moment has a permanent dipole moment. The potential energy of the interaction is... 

Once again, the potential energy is inversely proportional to the sixth power of the separation. Notice that the potential energies of the dipole-dipole interaction of rotating polar molecules in the gas phase, the London interaction, and the dipole-induced-dipole interaction all have the form... 

The reorganization of the solvent molecules can be expressed through the change in the slow polarization. Consider a small volume element AC of the solvent in the vicinity of the reactant it has a dipole moment m = Ps AC caused by the slow polarization, and its energy of interaction with the external field Eex caused by the reacting ion is —Ps Eex AC = —Ps D AC/eo, since Eex = D/eo- We take the polarization Ps as the relevant outer-sphere coordinate, and require an expression for the contribution AU of the volume element to the potential energy of the system. In the harmonic approximation this must be a second-order polynomial in Ps, and the linear term is the interaction with the external field, so that the equilibrium values of Ps in the absence of a field vanishes ... 

Figure 2 Correlation of potential energy and dipole moment change with dihedral angle, ic.11 Reprinted with permission from El-Sayed, I. Hatanaka, Y. Onozawa, S. Tanaka, M. J. Am. Chem. Soc. 2001, 123, 3597-3598. 2001 American Chemical Society.

From the studies made by I.R. and Raman spectra on 1,2-dihaloethanes, the conclusion has been that the gauche form is more important in polar than in non polar solvents. This is because that the gauche form has a considerable dipole moment while the anti has nearly none. Solvation by polar solvents reduces the potential energy of a dipole and so makes the gauche form more stable relative to the anti. 

In Fig. 5, the agreement of the calculated absolute intensity values with the corresponding experimental values is an indication of the high quality of the ab initio dipole moment surfaces employed in the calculation. The qualitatively correct appearance of the bands indicates that our solution of the rotation-vibration Schrodinger equation and the potential energy surface employed are satisfactory. It should be emphasized that the ab initio potential energy and dipole moment surfaces have not been adjusted to fit experiment. 

The potential energy of interaction W of an electric dipole in an external electric field is given by. 

The ground-state energy terms of this form contribute to the potential energy of the Hamiltonian from Eq. 6, where they act as effective ferroelectric interactions between neighboring PO4 dipoles, thus being responsible for phase transition in KDP and DKDP. In the case of KDP (H atom and Ro = 2.50 A) calculated values of parameters h and I are = llOmeV and = 0.22 A, while in the case of DKDP (D atom and Ro-o = 2.52 A), they are = 58 meV and P = I = 0.22 A. 

It was assumed that values of other model parameters (A, B, K, and M) are the same for both KDP and DKDP. These values were chosen to reproduce some experimental results for KDP. More precisely, KDP and DKDP systems have the Hamiltonian defined by Eqs. 5-7. In the case of classical PO4 dipoles both systems show a ferroelectric ordered state at T = 0 K where /u, = yu, holds for each PO4 dipole. It follows from Eq. 6 that the potential energy per dipole in this state is given by ... 

By analogy to the SDPC model, the parameters A, Aj, K, K , M, and M of the MSDPC model have the same value for KDP and DKDP. For the KDP and DKDP system of classical PO4 dipoles in ferroelectric ordered state at T = 0 K, all PO4 dipoles have components fit = (0> F-)- The potential energy per dipole and the ferroelectric-mode frequency in this state are given by the same relations from the SDPC model (Eq. 8 and fe = ( ( (/Usl/Af) / ), where B = 0. To reproduce the same saturated dipole value Fs = and the same saturated force value Fs = 2Kfs = 0.6 evA for KDP, the constants A and K have the same values as do the constants A + B and K in the SDPC model, respectively. Therefore, the isotope effect on Fs and Fs has the same... 

The potential energy of the second dipole due to this field is given by - n2E or -ajs2. To this must be added the energy necessary to induce the dipole (1 /2)a2E2 since the second is not a permanent dipole. Therefore the total potential energy 4> of the second dipole is... 

The second dipole acts on the original dipole in a similar fashion, giving a second contribution to the interaction energy that is identical to Equation (10) except that the subscripts are interchanged. The total potential energy of attraction is the sum of these two contributions ... 

The potential energy of a dipole with charges e and — e at a distance d from one another, and an ion with charge e at a distance r from the dipole, Figure 33> will be... 

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