The Helmholtz Equation

Equation (3) is often called the Helmholtz equation on account of its close relationship with the original Helmholtz equation (Chapter VI. p. 186(22)). It can be derived directly from equation (21) (p. 186), since — is the change in the thermodynamic potential and —Qp the change in the heat content H produced by the isothermal interaction of unit mass in mols. of each of the reacting substances. 

This equation demonstrates the falseness of Berthelot s principle (p. 128), which states that the affinity and the heat of reaction are equal to one another. We see further that A and Qp approach one another as the temperature is diminished to the absolute dA 

We can deduce the above equations directly by means of the thermodynamic potential (see Chapter VI. p. 174). 

Equation (3 ) is strictly true for all chemical reactions, and is not confined to reactions between perfect gases. The same is true of the Helmholtz equation (3), which may be deduced directly fiom (3a). By equation (12), Chapter VI. p. 184, we have 

It is worthy of note that A may be either greater or smaller 

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Electropositive adsorbates cause a decrease in the work function, of surfaces while electronegative adsorbates increase O. The variation in with the coverage, 0j, of an adsorbate is described by the Helmholtz equation ... 

Using the Helmholtz equation (2.21) and the initial AO vs 9aik slopes of Fig. 2.4 one computes alkali initial dipole moments P as high as 15 D. 

On the basis of the dipole moment, Paik, values computed from the Helmholtz equation (2.21) and the alkali ion radius one can estimate the effective positive charge, q, on the alkali adatom, provided its coordination on the surface is known. Such calculations give q values between 0.4 and 0.9 e (e.g. 0.86e for K on Pt(lll) at low coverages) which indicate that even at very low coverages the alkali adatoms are not fully ionized.6 This is confirmed by rigorous quantum mechanical calculations.27,28... 

In general Pj is coverage-dependent and also the Helmholtz equation has to be written in its general form ... 

Monochromatic Waves (1.14) A monochromatic e.m. wave Vcj r,t) can be decomposed into the product of a time-independent, complex-valued term Ucj r) and a purely time-dependent complex factor expjojt with unity magnitude. The time-independent term is a solution of the Helmholtz equation. Sets of base functions which are solutions of the Helmholtz equation are plane waves (constant wave vector k and spherical waves whose amplitude varies with the inverse of the distance of their centers. 

The surface potential change, besides the surface pressure, is the most important quantity describing the surface state in the presence of an adsorbed substance. However, the significance in molecular terms of this very useful experimental parameter still remains unclear. It is common in the literature to link A% with the properties of the neutral adsorbate by means of the Helmholtz equation" ... 

It is common in the literature to link with the properties of the neutral adsorbate by means of the Helmholtz equation [6,11,15,17,18]... 

Probably most popular for the modelling of field propagation in 10-design is BPM, which applies to z-variant geometries, too. The method is derived from the Helmholtz-equation, here for brevity depicted in scalar notation for one transverse direction, only,... 

The Helmholtz equation is widely used as the starting point for much theoretical acoustics in fluids (Morse and Ingard 1987). 

For their Hertz potential, Rodrigues and Vaz chose the factor < )(f,x) = 4>(x) exp (j fl t). Now, since II satisfies the wave equation, we conclude that the factor (<))x) in turn satisfies the Helmholtz equation ... 

An eigenfunction ijr of the Schrodinger equation satisfies the homogeneous integral equation b = f 3 Govt/r. In cell r this defines a locally regular solution of the Helmholtz equation... 

In a true scattering problem, an incident wave is specified, and scattered wave components of ifr are varied. In MST or KKR theory, the fixed term x in the full Lippmann-Schwinger equation, f = x + / GqVms required to vanish, x is a solution of the Helmholtz equation. In each local atomic cell r of a space-filling cellular model, any variation of i// in the orbital Hilbert space induces an infinitesimal variation of the KR functional of the form 8 A = fr Govi/s) + he. This... 

For energy k2 in Rydberg units, the Green function of the Helmholtz equation, with given boundary conditions, satisfies... 

To verify this expansion, consider Go(r, r ) for a fixed point r. For r < r, this function is a regular solution of the Helmholtz equation and must have an expansion... 

When coordinate x = X/x + r lies in cell r/x and x = Xv + r lies in a different cell r , a two-center expansion of the Green function is needed. Since N /fx — X/x) is a regular solution of the Helmholtz equation in a sphere of radius X/( — Xv about the origin of a displaced cell v,... 

The Helmholtz equation resembles the spatial part of the classical wave equation for matter waves (waves in ocean, sound waves, vibrations of a string, electromagnetic waves in vacuum, etc.) of amplitude F = F(r, f) ... 

The optical field distribution, E, can also be obtained by solving the Helmholtz equation using finite-difference method. (108)... 

The Helmholtz equation was confirmed experimentally by Jahn, and somewhat later by Bugarsky.t Their results are given in the following table. 

The values for the Daniel cell are taken from the latest and most accurate measurements of Cohen, Chattaway, and TombrokJ In dilute solutions the Daniel cell has a very small temperature coefficient, so that nE is nearly equal to Q. This accounts for the apparent confirmation of the old erroneous equation Q = nE. It is important to note that the Helmholtz equation is strictly accurate, as it depends solely on the two laws of thermodynamics. It is, of course, only applicable to reversible cells. 

Comparing (3) and (4) with the Helmholtz equation for the E.M.F. of a galvanic cell, we find that the Peltier effect corresponds to the latent heat of the cell and the values of cr (the... 

The total energy of unit area of the surface is y+gf. y, on the other hand, is the free energy of the surface which can be completely converted into mechanical work (see p. 373). Using the notation of Chapter VI. p. 186, we have TJ=y- -q and = y. Thus, from the Helmholtz equation,... 

U is equal to uV, and hence, from the Helmholtz equation =V+T, it follows that... 

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