Helmholtz differential equation

Thus, we now have a reasonable model of the interface in terms of the classical Helmholtz model that can explain the parabolic dependence of y on the applied potential. The various plots predicted by equation (2.18) are shown in Figures 2.5(a) to (c). The variation in the surface tension of the mercury electrode with the applied potential should obey equation (2.18). Obtaining the slope of this curve at each potential V (i.e. differentiating equation (2.18)), gives the charge on the electrode, [Pg.49]

Equation (7.85) frequently is called the Gibbs-Helmholtz equation. From it, the temperature coefficient of the free energy change (0AGm /. 7-/0T)p can be obtained if AGiji and AH are known. By differentiating Equation (7.83), we obtain... 

In the MSF theory, the function,/, in addition to simple reptation, is also related to both the elastic effects of tube diameter reduction, through the Helmholtz free energy, and to dissipative, convective molecular-constraint mechanisms. Wagner et al. arrive at two differential equations for the molecular stress function/ one for linear polymers and one for branched. Both require only two trial-and-error determined parameters. 

Here V denotes the outward normal gradient on a. It can easily be verified from the defining differential equation that the Helmholtz Green function is given by the expansion... 

Taking the divergence of Equation (35) and substituting it into Equation (37), a second-order partial differential equation of the Helmholtz type is finally obtained for the PI approximation... 

The above relations are known as Gibbs-Helmholtz equations. The solution of these partial differential equations, after insertion for E or H from (1.13.13)-(1.13.17), yields A or G. We have thus achieved a complete specification of the various thermodynamic functions of interest in terms of experimental information. We shall later discuss simpler experimental techniques for determining these quantities. 

All three derivations of the Clausius equation (3) are identical in principle, as they all make use of the second law of thermodynamics. In giving them all in detail we merely wished to show in what diverse ways the second law can be made to lead to concrete experimental results. The most diverse methods have been employed by various investigators in deriving such results. The choice of method depends partly on the nature of the problem and partly also on the task of the investigator. Van t Hoif, for example, generally used reversible cycles in his classical researches. Other physical chemists prefer Helmholtz s equation or the thermodynamic potential, while partial differential equations, as used in the first of the above derivations, are generally found in physical papers. 

Differential equation (15.48) can be transformed into an integral equation for the scattered field, using the 1-D Green s function g (z z oj) for the 1-D Helmholtz equation, which is dependent on the position of the source z, the observation point z, and the frequency ur. This function satisfies the 1-D Helmholtz equation... 

Equations [. 2.7 and 8] define interfacial tensions phenomenologically as differential surface Helmholtz or Gibbs energies. 

Activity coefficient models are equations representing the Gibbs or the Helmholtz energy of solutions. Activity coefficients and related properties are derived form these energy functions by proper differentiation (Equation (1)). 

Differentiating the Helmholtz energy, one obtains other thermodynamic properties, like the compressibility factor, pressure, and internal energy. To clarify, consider the example of a single component with a single acceptor and a single donor. Noting that X = X°, by symmetry in this instance, a simple quadratic equation results in... 

Wertheim found and in his perturbation theory. Upon differentiating the Helmholtz energy with respect to volume, the equation of state is obtained. 

Ideal-gas free energies A and G are, however, p or V dependent. Consider Helmholtz energy first. From the fundamental differential Equation (4.79),... 

We will begin the derivation with Helmholtz energy, as it is the natural energy function for the independent variables T and V of equations of state. By the fundamental differential equation for A, Equation (4.81)... 

The second inhomogeneous equation (80) splits into three, one for each component of the current density j. Maxwell s equations accordingly constitute a system of six coupled first-order differential equations for the components of E and B. A more compact formulation is obtained with the introduction of potentials. We invoke Helmholtz theorem [37] stating that the fields, in view of the boundary conditions stated above, can be written as V = -I-V-, where the... 

By using a gauge condition the differential equation becomes a Helmholtz equation for the vector potential A ... 

Substituting this expression into Helmholtz equation and by separating variables we obtain two normal differential equations of the second order. It is important to emphasize here that the method of separation of variables is applicable only for some orthogonal curvilinear coordinate systems. 

In the thermodynamics of critical phenomena one prefers alternative thermodynamic variables and alternative variable-dependent thermodynamic potentials, namely, the density of the Helmholtz energy A/V as a function of temperature and molar density p = n/V, or the pressure P as a function of temperature and chemical potential p = G/n = d A/V)/dp r [1, 2]. The corresponding differential equations for the density-dependent potential A/V and for the field-dependent potential P read... 

There are several differential equations that are related to Laplace s equation, e.g. the Poisson equation for the distribution of electric potential in the presence of electric charges, the wave equation for the propagation of a disturbance or the Helmholtz differential equation for the time-invariant distribution of harmonic fields. The latter is of particular relevance for scattering phenomena it has the form ... 

Spherical harmonics Y 9,(l>) are the angular contribution to the solution of Laplace s equation (or Helmholtz differential equation) in spherical coordinates (i.e. Eqs. (C.9) and (C.IO)). They are hence the product of the associated Legendre polynomial of cos0 and the general sine of the azimuth (/> ... 

The spherical vector wave functions (SVWF) are the general solution of the vectorial Helmholtz differential equation in spherical coordinates (Xu 1995) ... 

Several different partial differential equations are available for implementation in the computer models with varjdng degree of success. The classic depth averaged shallow water equation originaUy used by Leendertse has been used by many to solve for the water surface elevation and velocity field. The classic Laplace s equation or Helmholtz equation was used by many to solve for the velocity potential. 

Derive relations for the physical properties of materials whose magnetization follows the Curie-Weiss law M = AxM-ol T + ), where is a parameter, called the Weiss constant. Check on the expression (5.8.14) by basing your derivation on the Gibbs-Helmholtz relation and solving the first-order differential equation. 

The lattice PB equation is subsequently solved again until self-consistency is achieved (i.e., the differential equation is solved and the output Tjr field corresponds to a distribution with the desired total number of simple ions). The Helmholtz free energy can be extracted from the converged if field and the appropriate values of y using standard thermodynamic relations, namely ... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

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