Equation, Boltzmann, generalized thermodynamic

Caution should be taken when calculating the doublelayer force between two parallel plates. It is clear that the force is not proportional to the excess concentration of ions at the middle distance (with respect to the concentration of ions at infinity), since this Langmuir equation involved the assumption of ions of negligible sizes. We will use instead the procedure introduced by Verwey and Overbeek,18 which is based on general thermodynamic principles, and does not imply the Boltzmann distribution of ions.19 The force, per unit area, between two parallel plates separated by a distance l is given by... 

The Boltzmann equation is generally used to obtain an expression for AS of simple mixtures (mixtures of solvent-solvent or solvent-simple solute molecules) from the number of different arrangements ft (or the thermodynamic probabilities) of the solute and solvent molecules in the system For simple systems, the volume elements of solution are modeled by a three-dimensional lattice, where solute or solvent molecules can occupy any cell within the... 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

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. 

The solution procedure for a particle covered by a membrane is similar to that for a rigid surface, except that the membrane phase needs additional treatments. In this section, we introduce three methods for recovering the solution of a Poisson-Boltzmann equation. As in the case of a rigid surface, obtaining the exact analytical solution for a Poisson-Boltzmann equation is almost impossible, in general, and only approximate results are available. Procedures for the estimation of the basic thermodynamic properties for the problem under consideration are also discussed. 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

The DH and MSA theory, that are linear in charge can be considered in the framework of linearized Poisson-Boltzmann (PB) equation. The concept of ion association entails nonlinearity in the treatment of electrostatic interactions by the formulation of appropriate thermodynamic equilibrium constants between free ions and ion clusters [14], In general, this formulation can be considered as the division of ion-ion interaction potentials into an associative part responsible for the ion association, and nonassociative part which is more or less arbitrary. In order to optimize this division in the framework of associative hypernetted chain approximation (AHNC), the division of energy and distance were considered [17] with the parameters calculated from the condition of sta-... 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

In general, the thermodynamic definition of entropy (Equations 8.6 to 8.8) yields the same value for the entropy change of a process as Boltzmann s statistical definition (Equation 8.3) for the same process. Consider, for example, the entropy change in the reversible and isothermal (constant teinperature) mmqn n i n i les of an ideal gas from an initial volume Vi to a inB39B9tft0-6K ile heat... 

The third law is a consequence of the statistical nature of entropy as reflected in Boltzmann s formula (Equation 8.1), which relates the entropy to W, the number of molecular quantum states (microstates) consistent with the macroscopic conditions. At r = 0, there is no available thamal energy, and the thermodynamically most stable state is the lowest possible energy state (the ground state). In general, this state is unique, so W = 1 for a system at 0 K. From Boltzmann s formula we then have... 

Au contraire to the empirical equation of Tait for EOS predictions, theoretical models can be used but generally require an understanding of forces between the molecules. These laws, strictly speaking, need be derived from quantum mechanics. However, Lenard-Jones potential and hard-sphere law can be used. The use of statistical mechanics is an intermediate solution between quantum and continuum mechanics. A canonical partition function can be formulated as a sum of Boltzmann s distribution of energies over all possible states of the system. Necessary assumptions are made during the development of the partition function. The thermodynamic quantities can be obtained by use of differential calculus. For instance, the thermodynamic pressure can be obtained from the partition function Q as follows ... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

The concept of entropy was developed so chemists could understand the concept of spontaneity in a chemical system. Entropy is a thermodynamic property that is often associated with the extent of randomness or disorder in a chemical system. In general if a system becomes more spread out, or more random, the system s entropy increases. This is a simplistic view, and a deeper understanding of entropy is derived from Ludwig Boltzmann s molecular interpretation of entropy. He used statistical thermodynamics (which uses statistics and probability) to link the microscopic world (individual particles) and the macroscopic world (bulk samples of particles). The connection between the number of microstates (arrangements) and its entropy is expressed in the Boltzmann equation, S = kin W, where W is the number of microstates and k is the Boltzmann constant, 1.38 x 10 JK L... 

We have repeatedly used the term hydrodynamic, and we now give it a more precise definition. By a hydrodynamic process we mean one for which the local thermodynamic variables, temperature, chemical potential (or density), and velocity, are determined by the past history of their boundary values. The normal solution to the Boltzmann equation, as well as its generalization obtained in the previous Sections, then clearly corresponds to a hydrodynamic process. The significance of the term hydro-dynamic may be clarified by the consideration of some processes of non-hydrodynamic type. A process of relaxation in momentum space in a spatially uniform gas is clearly non-hydrod5mamic, since the local thermod5mamic variables are not at all pertinent to its description. Another example is provided by processes in a Knudsen gas. Here there is an essential dependence on the particular form of the boundary forces. An insensitivity to the nature of the boundary forces is implied in the definition of a hydrod5mamic process, for which it is immaterial whether a thermal reservoir is constructed of, say, copper or aluminum, and... 

From a general point of view, the distribution of counterions is imposed by that of the electrostatic potential. Our purpose is to treat the thermodynamic behaviour of a cylindrical system without excess added salt the polyelectrolyte is always considered as a thin rod characterized by its linear charge density. Two approaches are investigated the first one needs the resolution of a Poisson-Boltzmann equation without... 

Perhaps the most elegant and exact but therefore the most limited approach to the transport properties is through a general equation such as the Boltzmann or Fokker-Planck equation describing the time-dependent distribution functions of the system. The results of the zeroth-order theories can be conveniently cast in the form of the friction constants of irreversible thermodynamics. 

We justified the statement above by using Boltzmann s equation for S. However, before Boltzmann presented his famous equation, scientists had already deduced the third law of thermodynamics by studying isothermal processes, such as phase transitions or reactions, at very low temperatures. They found that for a wide variety of isothermal processes, AS approached a value of zero as the temperature approached 0 K. This general observation suggests that the entropies of all substances must be the same at 0 K. When Boltzmann presented his equation for S, it was clear that S = 0 is the correct value for entropy at 0 K. 

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