Helmholtz-Boltzmann

The Helmholtz-Boltzmann Equation (based on Fermi-Dirac Statistics)... 

At this time, Stockman, et. al134. and Lamb135 and Baylor et. al.136 are using empirical expressions for the absorption spectrums of human chromophores. Stockman, et. al. are using a conventional arithmetic series in even powers of the variable. They make no claim to a physical foundation for their series. Lamb says It needs to be emphasized that the above (his) represents no more than an exercise in curve-fitting, and that neither equation (1) nor equation (2) has any known physical significance.. . These equations (his equation 2 in particular) are basically attempts to approximate the Helmholtz-Boltzmann equation as it is derived from the Fermi-Dirac equation, by empirical... 

At the third stage the equilibrium thermodynamics was created by Clausius, Helmholtz, Boltzmann and Gibbs. Since that time the equilibrium principles started to develop as applied to macroscopic systems of any physical nature. The main, second law of thermodynamics was discovered by Clausius (Clausius, 2008). He found out the existence of the state function, entropy (S), that can change in the isolated systems exclusively towards increase. The inequality that shows such monotonicity of change... 

However, the ptractical apphcation of the second law in the analysis of equilibrium irreversible trajectories faced great difficulties. Clausius and then Helmholtz, Boltzmann, J. Thomson, Planck and other researchers tried to harmonize the second law of thermodynamics with the principle of the least action and derive the equation that meets this painciple similar to the equations (3) or (4) for dissipative macroscopic systems (in which the organized energy forms turn into a non-organized form, i.e. heat, due to friction). As is known their attempts were unsuccessful and resulted in understanding the necessity to statistically substantiate thermodynamics (Polak, 2010). 

When calculating free energies, one generates, either by molecular dynamics or MC, configuration space samples distributed according to a probability distribution function (e.g., the Boltzmann distribution in the case of the Helmholtz free energy). 

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

As in the case of the Helmholtz model, the plane AA will be negative due to the adsorbed R-species. Therefore, the Na+ and CL ions will be distributed nonuni-formly due to electrostatic forces. The concentrations of the ions near the surface can be given by the Boltzmann distribution, at distance x from the plane AA, as... 

Figure 2.13 illustrates what is currently a widely accepted model of the electrode-solution interphase. This model has evolved from simpler models, which first considered the interphase as a simple capacitor (Helmholtz), then as a Boltzmann distribution of ions (Gouy-Chapman). The electrode is covered by a sheath of oriented solvent molecules (water molecules are illustrated). Adsorbed anions or molecules, A, contact the electrode directly and are not fully solvated. The plane that passes through the center of these molecules is called the inner Helmholtz plane (IHP). Such molecules or ions are said to be specifically adsorbed or contact adsorbed. The molecules in the next layer carry their primary (hydration) shell and are separated from the electrode by the monolayer of oriented solvent (water) molecules adsorbed on the electrode. The plane passing through the center of these solvated molecules or ions is referred to as the outer Helmholtz plane (OHP). Beyond the compact layer defined by the OHP is a Boltzmann distribution of ions determined by electrostatic interaction between the ions and the potential at the OHP and the random jostling of ions and... 

Factor 1 The exponential of the empty water lattice Helmholtz free energy divided by kT, where k is Boltzmann s constant,... 

Stern combined the ideas of Helmholtz and that of a diffuse layer [64], In Stern theory we take a pragmatic, though somewhat artificial, approach and divide the double layer into two parts an inner part, the Stern layer, and an outer part, the Gouy or diffuse layer. Essentially the Stern layer is a layer of ions which is directly adsorbed to the surface and which is immobile. In contrast, the Gouy-Chapman layer consists of mobile ions, which obey Poisson-Boltzmann statistics. The potential at the point where the bound Stern layer ends and the mobile diffuse layer begins is the zeta potential (C potential). The zeta potential will be discussed in detail in Section 5.4. 

The decrease of the concentration of the electroactive species with increasing potential has to be attributed to double layer effects. As first pointed out by Frumkin [58], in dilute solutions the electron transfer rate is affected by variations of the potential in the double layer in two ways. The potential in the outer Helmholtz plane, fa, is due to the extension of the double layer not identical to the potential in the solution (at the end of the double layer), so that the effective driving force of the reaction is DL — fa. Furthermore, the concentration of ionic reactants in the reaction plane, c, is influenced by electrostatic effects and differs from the concentration just outside the double layer, c0, by a Boltzmann term ... 

In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduction, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of "equilibrium" and "reversibility," "equilibrium processes" and "reversible processes" started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the "time arrow."... 

In proving that SQ T is a perfect differential, both Boltzmann and Gibbs always use ergodically distributed ensembles of systems. It is only in connection with his criticism of the Helmholtz... 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

This transformed the nonlinear Poisson-Boltzmann equation into the linear Helmholtz-type equation... 

FIGURE 9.15 Potential near the surface of a flat platelet particle using linear and nonlinear Poisson-Boltzmann equation with a surface potential of % = 2.0, (51.4 mV), which is the potential at the outer Helmholtz plane in Figure 9.14. Also showing the shear plane where the zeta potential is measured. 

In the following we give a short account on this theory for demonstrating that interactions between dipolar ion pairs and free ions and/or other pairs can be incorporated in a natural and transparent way [31, 32], The theories rest on the Poisson-Boltzmann equation. With the presumption of electro neutrality, the expansion in first order of f3 yields the Helmholtz equation or linearized Poisson-Boltzmann equation,... 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

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