Expressions for the potential

This function is a solution of Laplace s equation regardless of the values of constants, and our goal is to find such of them that the potential satisfies the boundary condition on the surface of the given ellipsoid of rotation and at infinity. In order to solve this problem we have to discuss some features of Legendre s functions. First of all, as was shown in Chapter 1, the Legendre s function of the first kind P (t]) has everywhere finite values and varies within the range 

Examples of this function are given below, (Chapter 1) 

The Legendre s function of the second kind g ( ) has completely different behavior in particular, it tends to infinity when = 1. In accordance with Equation (2.121), this happens at points of the z-axis. Since the potential has everywhere a finite value the function Qn(ri) cannot describe the attraction field and has to be removed from Equation (2.132). This first simplification gives  

The latter can be also simplified, and with this purpose in mind consider the behavior of Legendre s functions with an imaginary argument. As follows from 

Making use of recursive relations between them it is easy to see that all the above-mentioned functions, except Pq, increase unlimitedly with the distance from masses, 8 00 which contradicts the condition at infinity. Therefore, terms with E (/ ) have to be discarded too, and this gives one more simplification of Equation (2.132) 

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The parameters used in the above expressions for the potential energy surfaces and the calculations are given in Table 1 of [60],... 

A.30 The expression EP = mgh applies only close to the surface of the Earth. The general expression for the potential energy of a mass m at a distance R from the center of the Earth (of mass wzE) is EP = —GmEm/R. Write R = R( + h, where Rr is the radius of the Earth and show that, when h <3C RF, this general expression reduces to the special case, and find an expression for g. You will need the expansion (1+x) 1 = 1 — x + . 

To find the wavefunctions and energy levels of an electron in a hydrogen atom, we must solve the appropriate Schrodinger equation. To set up this equation, which resembles the equation in Eq. 9 but allows for motion in three dimensions, we use the expression for the potential energy of an electron of charge — e at a... 

In the last step, we have used the relation 1— + f — +. .. = ln2. Finally, we multiply T. by 2 to obtain the total energy arising from interactions on each side of the ion and then by Avogadro s constant, NA, to obtain an expression for the potential energy per mole of ions. The outcome is... 

All interionic and almost all intermolecular interactions can be traced to the coulombic interaction between two charges (Section 2.4), and throughout the discussion of intermolecular interactions we shall build on Eq. 5 from Section A, the expression for the potential energy Ef of two charges qx and q1 separated by a distance r ... 

The expression for the potential energy of a potassium ion and a chloride ion, for example, is similar to that of Equation 6, but is still more complicated. 

Next, applying the principle of superposition, we arrive at an expression for the potential caused by a volume distribution of masses ... 

Now we return to Poisson s and Laplace s equations, which describe the behavior of the potential inside and outside masses, respectively. Earlier we have already derived an expression for the potential ... 

Our task is to derive an explicit expression for the potential U proceeding from this equation. This means that we have to take the function U out of this integral. With this purpose in mind consider the limiting value of the second integral, when the radius of the spherical surface r tends to zero. Since both the potential and its derivatives are continuous functions inside the volume, we have... 

This is the most important result in our derivations since it permits us to obtain an expression for the potential at any point of the volume in terms of boundary conditions. Substitution of Equation (1.106) into Equation (1.104) gives... 

Here C is some constant. Thus, the expression for the potential is... 

Substituting the expression for the potential If, derived above, we see that it is a solution of Poisson s equation. In other words, the integral... 

Certainly, the expression for the potential is much simpler than that for the field, and this is a very important reason why we pay special attention to the behavior of this function U(p). As follows from the behavior of the gravitational field, the potential U has a maximum at the earth s center and with an increase of the distance from this point it becomes smaller, since the first derivative in the radial direction, that is, the component of the gravitational field, is negative. At very large distances from the earth the function U has a minimum and then it starts to increase, but this range is beyond our interest. In the first chapter we demonstrated that the potential of the attraction field obeys Poisson s and Laplace s equations inside and outside the earth, respectively ... 

Thus, the expression for the potential of the attraction field is greatly simplified and in place of an infinite sum, Equation (2.134), we have... 

This expression for the potential is valid for any distribution of masses, provided that R>Ri- Now we focus our attention on the potential of the normal field caused by regular part of masses. As was assumed before, their density is independent of the longitude, and the equator is a plane of symmetry. For this reason this part of the potential of the earth has a similar behavior and, correspondingly, Equation (2.213) is greatly simplified. Let us rewrite it in the form ... 

This result was taken as an experimental eonfirmation of the model developed by Sehmiekler [7]. However, it appeared somehow eontradictory with other results obtained with SECM. It was also suggested that eoneentration polarization phenomena occurring at the aqueous side are negligible as the whole potential drop is presumably developed in the benzene phase. This assumption can be qualitatively verified by evaluating a simplified expression for the potential distribution based on a back-to-back diffuse double layer [40,113],... 

In order to arrive at values of the virtually intrinsic acidity, i.e., an acidity expression independent of the solvent used (Tremillon12 called it the absolute acidity), Schwarzenbach13 used the normal acidity potential as an expression for the potential of a standard Pt hydrogen electrode (1 atm H2), immersed in a solution of the acid and its conjugate base in equal activities analogously to eqn. 2.39 for a redox system and assuming n = 1 for the transfer of one proton, he wrote for the acidity potential... 

The expression for the potential of electrodes of the second kind on the hydrogen scale can be derived from the affinity of the reaction occurring in a cell with a standard hydrogen electrode. For example, for the silver chloride electrode with the half-cell reaction... 

The introduction of the concept of the micropotential permits derivation of various expressions for the potential difference produced by the adsorbed anions, i.e. for the potential difference between the electrode and the solution during specific adsorption of ions. It has been found that, with small coverage of the surface by adsorbed species, the micropotential depends almost linearly on the distance from the surface. The distance between the inner and outer Helmholtz planes is denoted as xx 2 and the distance between the surface of the metal and the outer Helmholtz plane as jc2. The micropotential, i.e. the potential difference between the inner and... 

In the framework of the force field calculations described here we work with potential constants and Cartesian coordinates. The analytical form of the expression for the potential energy may be anything that seems physically reasonable and may involve as many constants as are deemed feasible. The force constants are now derived quantities with the following definition expressed in Cartesian coordinates (x ) ... 

The expression for the potential variation with x within the Donnan phase is approximated by [17] ... 

Derivation of expressions for the potential, A V, and steady state currents,... 

An explicit expression for the potential energy U exists, and this function can be differentiated to give dU, whereas no explicit expression for W that leads to DW can be obtained. The function for the potential energy is a particularly simple one for the gravitational field because two of the space coordinates drop out and only the height h remains. That is,... 

Most "molecular mechanics" expressions for the potential energy U of a molecule contain no explicit terms in qext) and these are frequently described as "vacuum" potentials, implying rigorous applicability of the potential only in the absence of the numerous and, hence, troublesome, solvent molecules.A typical such function is shown in eqn. 6... 

Expression (9.49) can be evaluated by substituting the expression for the potential in terms of the charge distribution [Eq. (8.2)], which gives... 

This results in the following expression for the potential 4-vector = ( , )... 

In general terms, the interactions between the colloidal particles with surfaces covered by adsorbed biopolymer layers can be described qualitatively and quantitatively using the appropriate expression for the potential of mean force W(r). Extending the formalism of equation (3.2), at least four separate contributions to W(r) can contribute to the total free energy of interaction between a pair of colloidal particles in the aqueous dispersion medium (a biopolymer solution) (Snowden et al, 1991 Dickinson, 1992 Israelachvili, 1992 Vincent, 1999 de Kruif, 1999 Praus-nitz, 2003) ... 

The purpose of this chapter is to introduce the basic ideas concerning electrical double layers and to develop equations for the distribution of charges and potentials in the double layers. We also develop expressions for the potential energies and forces that result from the overlap of double layers of different surfaces and the implication of these to colloid stability. 

To evaluate A we proceed as follows. In the limit of infinite dilution — that is, as k - 0— the potential around the charged particle is given by the expression for the potential of an isolated charge. Elementary physics gives this as... 

A general expression for the potentials of these cells is given by... 

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