Equation Boltzmann, for entropy

The skeptical reader may reasonably ask from where we have obtained the above rules and where is the proof for the relation with thermodynamics and for the meaning ascribed to the individual terms of the PF. The ultimate answer is that there is no proof. Of course, the reader might check the contentions made in this section by reading a specialized text on statistical thermodynamics. He or she will find the proof of what we have said. However, such proof will ultimately be derived from the fundamental postulates of statistical thermodynamics. These are essentially equivalent to the two properties cited above. The fundamental postulates are statements regarding the connection between the PF and thermodynamics on the one hand (the famous Boltzmann equation for entropy), and the probabilities of the states of the system on the other. It just happens that this formulation of the postulates was first proposed for an isolated system—a relatively simple but uninteresting system (from the practical point of view). The reader interested in the subject of this book but not in the foundations of statistical thermodynamics can safely adopt the rules given in this section, trusting that a proof based on some... 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

As a first approximation, assume that the total entropy change, ASy, is made up solely of configurational entropy. This can be determined by using the Boltzmann equation for the entropy of a disordered system ... 

The second statement of the third law (which bears Planck s name) is that as the temperature goes to zero, AS goes to zero for any process for which a reversible path could be imagined, provided the reactants and products are perfect crystals. Here, perfect crystals are defined as those which are non-degenerate, that is, they have only a single quantum state in which they can exist at absolute zero. This statement follows rigorously from Boltzmann s equation for entropy,... 

The Boltzmann equation for the relation between entropy and thermodynamic probability is the basis for this ... 

Boltzmann s equation for entropy, S, is S = ft In W. In this equation, the term W represents the number of possible arrangements of molecules. In is the logarithm to the base e, and k is the constant universally referred to as Boltzmann s constant. It is equal to where R is the gas constant and Mis Avogadro s number (6.02 x 10 ), the number of molecules in a mole. 

EXAMPLE 7.1 The ideal gas law derived from the lattice modeL Take the definition of pressure, pjT = dSjdV)y,u, from Equation (7.6). Into this expression, insert the function 5(V) from the lattice model in Example 2.2. For a lattice of M sites with N particles, use the Boltzmann expression for entropy and Equation (2.3) to get... 

Within the assumptions used all contributions arising from the charge interactions are included in F which may also be written as a contribution of an internal energy, defined by (52) and of an entropy given by (54). Further simplification by the use of the so-called Boltzmann equation for the total charge density will only be possible in some particular cases. 

The thermodynamic probability is converted to an entropy through the Boltzmann equation [Eq. (3.20)] so we can write for the entropy of the mixture (subscript mix)... 

Application of the Boltzmann equation to Eq. (8.33) gives the entropy of the mixture according to this model for concentrated solutions ... 

The critical size of the stable nucleus at any degree of under cooling can be calculated widr an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n atoms widr o atoms of the liquid per unit volume, is given by the Boltzmann equation... 

The procedures described so far imply that So = 0, but do not rigorously prove that this is so. The final proof comes from a comparison of Sr for the ideal gas, obtained from the integration of Cp data assuming the Third Law is valid combined with the entropies of transition, with values obtained from a calculation of St for the ideal gas by statistical methods. The procedure, to be described in detail in Chapter 10, starts with the Boltzmann equation... 

Using the lattice model, the approximate value of W in the Boltzmann equation can be estimated. Two separate approaches to this appeared in 1942, one by P. J. Rory, the other by M. L. Huggins, and though they differed in detail, the approaches are usually combined and known as the Rory-Huggins theory. This gives the result for entropy of mixing of follows ... 

The details of the derivation are complicated, but the essence of this equation is that the more possible descriptions the system has, the greater is its entropy. The equation states that entropy increases in proportion to the natural logarithm of W, the proportionality being given by the Boltzmann constant, k — 1.3 806 x lO V/r. Equation also establishes a starting point for entropy. If there is only one way to describe the system, it is fully constrained and W — 1. Because ln(l)=0,S = 0 when W — 1. 

The Boltzmann equation, S = kXxiW, establishes the zero point for entropies. Because ln(l) = 0, this equation predicts that the entropy will be zero for a system with only one possible description. Figure 14-1 la shows a system with W =, nine identical marbles placed in nine separate compartments. The figure also shows two types of conditions where > 0. Figure 14-llZ) shows a condition when the marbles are not identical. If one marble is a different color than the other eight, there are nine different places to place the different marble. Figure I4-IIc shows a condition where there are more compartments than marbles. If there are more places to put the marbles than there are marbles, there are multiple ways to place the marbles in the compartments. 

A solvated MD simulation is performed to determine an ensemble of conformations for the molecule of interest. This ensemble is then used to calculate the terms in this equation. Vm is the standard molecular mechanics energy for each member of the ensemble (calculated after removing the solvent water). G PB is the solvation free energy calculated by numerical integration of the Poisson-Boltzmann equation plus a simple surface energy term to estimate the nonpolar free energy contribution. T is the absolute temperature. S mm is the entropy, which is estimated using... 

In the limit of an infinite micellar radius, i.e. a charged planar surface, the salt dependence of Ge is solely due to the entropy factor. A difficult question when applying Eq. (6.13) to the salt dependence of the CMC is if Debye-Hiickel correction factors should be included in the monomer activity. When Ge is obtained from a solution of the Poisson-Boltzmann equation in which the correlations between the mobile ions are neglected, it might be that the use of Debye-Hiickel activity factors give an unbalanced treatment. If the correlations between the mobile ions are not considered in the ionic atmosphere of the micelle they should not be included for the free ions in solution. 

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. 

Both the classical and statistical equations [Eqs. (5.22) and (5.23)] yield absolute values of entropy. Equation (5.23) is known as the Boltzmann equation and, with Eq. (5.20) and quantum statistics, has been used for calculation of entropies in the ideal-gas state for many chemical species. Good agreement between these calculations and those based on calorimetric data provides some of the most impressive evidence for the validity of statistical mechanics and quantum theory. In some instances results based on Eq. (5.23) are considered more reliable because of uncertainties in heat-capacity data or about the crystallinity of the substance near absolute zero. Absolute entropies provide much of the data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 15. 

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 Statistical Rate Theory (SRT) is based on considering the quantum-mechanical transition probability in an isolated many particle system. Assuming that the transport of molecules between the phases at the thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between the two phases 1 and 2 , R 2, was developed by using the first-order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. 

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