Entropy microscopic interpretation

MSN.91. I. Prigogine and A. Grecos, On the dynamical theory of irreversible processes and the microscopic interpretation of nonequihbrium entropy. Proceedings, I3th lUPAP Conference on Statistical Physics, Ann. Israel Phys. Soc. 2, 84—97, 1978. 

Entropy is interpreted as the number of microscopic arrangements included in the macroscopic definition of a system. The second law is then used to derive the distribution of molecules and systems over their states. This allows macroscopic state functions to be calculated from microscopic states by statistical methods. 

If, at 0 K, entropies of compounds are the sum of nonzero values assigned to each atom, conservation of atoms would ensure that the entropy change for all chemical reactions would be zero. This way of satisfying the third law would not be in agreement with the microscopic interpretation of entropy given in Chapter 5. 

Because the molecules of an ideal gas do not interact, its internal energy resides with individual molecules. This is not trae of the entropy. The microscopic interpretation of entropy is based on an entirely different concept, as suggested by the following example. 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

Use the microscopic interpretation of entropy from Section 13.2 to explain why the entropy change of the system in Problem 28 is positive. 

When a gas undergoes a reversible adiabatic expansion, its entropy remains constant even though the volume increases. Explain how this can be consistent with the microscopic interpretation of entropy developed in Section 13.2. Hint Consider what happens to the distribution of velocities in the gas.)... 

Up to this point the essential equations have been presented. Now, it is possible to analyze the work carried out in connection with the classical thermodynamic approach. The first systematic study of a thermodynamic adsorption quantity was perhaps the work done by de Boer and coworkers [10] on the determination, interpretation and significance of the enthalpy and entropy of adsorption. Their papers analyzed almost all aspects of the experimental determination of the entropy and how to interpret the values obtained in terms of two extreme models, i.e., those of mobile and locahzed adsorption, which today have lost much of their usefulness. To catalog the behavior of the adsorbed film as localized or mobile is a very simplistic solution and it has been demonstrated [9] that in most cases the adsorbed film is neither completely localized nor completely mobile. This approach also is somehow outdated because numerical simulations provide a better microscopic interpretation of the system s behavior. 

Thus we should not interpret entropy as a measure of disorder. We must look elsewhere for a satisfactory microscopic interpretation of entropy. 

The contenet of the third law of thermodynamics is summarized in Fig. 2.4. The third law is particularly easy to understand if one combines the macroscopic entropy definition of entropy with its statistical, microscopic interpretation through the Boltzmann equation, Eq. (11). The symbol k is the Boltzmaim constant, the gas constant R divided by Avogadro s number and W is the thermodynamic probability, representing the number of ways a system can be arranged on a microscopic level. One can state the third law, as proposed by Nernst and formulated by Lewis and Randall, as follows "If... 

Until now diffusion has been treated as a macroscopic physical process driven by entropy, however, the diffusion equation implies a microscopic interpretation in terms of stochastic trajectories. Since much of the work in this thesis uses and develops simulation methods at this microscopic level, it is necessary to introduce the fundamental concepts of this theory. 

The configurational entropy model describes transport properties which are in agreement with VTF and WLF equations. It can, however, predict correctly the pressure dependences, for example, where the free volume models cannot. The advantages of this model over free volume interpretations of the VTF equation are numerous but it lacks the simplicity of the latter, and, bearing in mind that neither takes account of microscopic motion mechanisms, there are many arguments for using the simpler approach. 

Zero-Temperature Relaxation. This interpretation rationalizes the aging behavior found in exactly solvable entropy barrier models that relax to the ground state and show aging at zero temperature [190, 191]. At T = 0, the stimulated process is suppressed (microscopic reversibility, Eq. (8), does not hold), and Eq. (204) holds by replacing the free energy of a CRR by its energy, F = E. In these models a region corresponds to just a... 

In this section, we will present and discuss results from Sc2 C84, which is the most widely studied dimetallofullerene to date. Early scanning tunnelling microscopy [26] and transmission electron microscopic [27] investigations provided evidence in favour of the endohedral structure of this system, which was later confirmed by x-ray diffraction experiments utilising maximum entropy methods [28]. Before experimental data from this system were available, the Sc ions were predicted to be divalent from quantum chemical calculations [29]. Subsequent data from vibrational spectroscopy [30,31], core-level photoemission [32] and further theory [33] on this system were indeed interpreted in terms of divalent Sc ions. 

We note that the classical equilibrium entropy (i.e., the eta-function evaluated at equilibrium states) acquires in the context of the Microcanonical Ensemble an interesting physical interpretation. The entropy becomes a logarithm of the volume of the phase space that is available to macroscopic systems having the fixed volume, fixed number of particles and fixed energy. If there is only one microscopic state that corresponds to a given macroscopic state, we can put the available phase space volume equal to one and the entropy becomes thus zero. The one-to-one relation between microscopic and macroscopic thermodynamic equilibrium states is thus realized only at zero temperature. 

Classical thermodynamics is based on a description of matter through such macroscopic properties as temperature and pressure. However, these properties are manifestations of the behavior of the countless microscopic particles, such as molecules, that make up a finite system. Evidently, one must seek an understanding of the fundamental nature of entropy in a microscopic description of matter. Because of the enormous number of particles contained in any system of interest, such a description must necessarily be statistical in nature. We present here a very brief indication of the statistical interpretation of entropy, t... 

Detailed studies of the temperature dependence of the kinetics of cation-pseudobase equilibration have been reported92 for three cations, and activation parameters have been evaluated for each of k0H, kH20, kt, and k2 in these cases. Whereas entropies of pseudobase formation from the cation are positive (Section III), the entropies of activation associated with k0H are quite negative (-11 to -17 cal mol-1 deg-1). For direct hydroxide ion attack on the cation via transition state A, one would predict an entropy of activation similar to the entropy of formation of the pseudobase from heterocyclic cation and hydroxide ion. Interpretation of k0H in terms of transition state B in which hydroxide ion acts as a general-base catalyst for the attack of a water molecule seems to be more consistent with the observed entropies of activation. The k2 step, which is the microscopic reverse of fcoH would then be interpreted as general-acid-catalyzed (by a water molecule) decomposition of the neutral pseudobase to the cation. These interpretations of /cqH and k2 are also consistent with the observed solvent isotope effects for the reactions in HzO and DzO and with the presence of general... 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

Note that interpretations of the time-reversal experiments are only valid in strictly euclidean space-time. This condition is rarely emphasized by authors who state that all laws of physics are time-reversible, except for the law of entropy. Fact is that entropy is the only macroscopic state function which is routinely observed to be irreversible. One common explanation is to hint that entropy is an emergent property of macro systems and hence undefined for microsystems. Even so, the mystery of the microscopic origin of entropy remains. A plausible explanation may be provided if the assumed euclidean geometry of space-time is recognized as an approximate symmetry as demanded by general relativity. 

Provide an interpretation in terms of microscopic phenomena why the imposition of an electric field invariably lowers the entropy of a system. 

Thus, a system at equilibrium remains in the same macroscopic state, even though its microscopic state is changing rapidly. There are an enormous number of microscopic states consistent with any given macroscopic state. This concept leads us at once to a molecular interpretation of entropy entropy is q measure of how many different nncroscqpii stqte arje, agivenmqavscopic state. 

At the end of nineteenth century, based on a concept of probabilistic distribution of energy states, Ludwig Boltzmann (1844-1906) presented an innovative interpretation that the entropy S is the most feasible number of microscopic energy states... 

A rigorous interpretation is provided by the discipline of statistical mechanics, which derives a precise expression for entropy based on the behavior of macroscopic amounts of microscopic particles. Suppose we focus our attention on a particular macroscopic equilibrium state. Over a period of time, while the system is in this equilibrium state, the system at each instant is in a microstate, or stationary quantum state, with a definite energy. The microstate is one that is accessible to the system—that is, one whose wave function is compatible with the system s volume and with any other conditions and constraints imposed on the system. The system, while in the equilibrium state, continually jumps from one accessible microstate to another, and the macroscopic state functions described by classical thermodynamics are time averages of these microstates. 

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