Entropy statistical thermodynamics definition

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

The next important thermodynamic function that we must obtain is the entropy S. The statistical thermodynamic definition of entropy is... 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

Traditional thermodynamics gives a clear definition of entropy but unfortunately does not tell us what it is. An idea of the physical nature of entropy can be gained from statistical thermodynamics. Kelvin and Boltzmann recognised diat there was a relationship between entropy and probability (cf., disorder) of a system with the entropy given by... 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

For solutions comprised of species of equal molecular volume in which all molecular interactions are the same, one can show by the methods of statistical thermodynamics that the lowest possible value of the entropy is given by an equation analogous to Eq. (10.7). Thus we complete the definition of an ideal solution by specifying that its entropy be given by the equation ... 

The successful development of the thermodynamics of irreversible phenomena depends on the possibility of an explicit evaluation of the production of entropy, and for this it is necessary to assume that the thermodynamic definition of entropy can be applied equally to systems which are not in equilibrium, that is to states whose mean lifetime is limited. We are thus confronted immediately with the problem of the domain of validity of the thermodynamic treatment of irreversible phenomena, which can be determined only by a comparison of the results of the thermodynamic treatment with those obtained by the use of statistical mechanics. This problem wall be dealt with in more detail in the third volume of this work meanwhile the main conclusions can be summarized as follows. 

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. 

From the scientific definition point of view, there is a slight difference between our continuum thermodynamics definition of the Second Law and its statistical mechanical version so that the continuum thermodynamics definition of the Second Law states that an observation of decreased universal entropy is impossible in isolated systems however the statistical mechanical definition says that an observation of universal increased entropy is not probable. 

In this section we shall briefly review the thermodynamic description of chemical lasers. The analysis will be based on the fundamental laws of thermodynamics and the statistical-molecular definitions of entropy and energy. The approach outUned below does not intend to yield new detailed results, these have been supplied adequately in the kinetic analyses. On the contrary, it attempts to compact the detailed data by focusing attention on a few macroscopically significant observables, and by applying general thermodynamic relationships to shed a different light on the phenomena described in the previous sections. 

Classical thennodynamics deals with the interconversion of energy in all its forms including mechanical, thermal and electrical. Helmholtz [1], Gibbs [2,3] and others defined state functions such as enthalpy, heat content and entropy to handle these relationships. State functions describe closed energy states/systems in which the energy conversions occur in equilibrium, reversible paths so that energy is conserved. These notions are more fully described below. State functions were described in Appendix 2A however, statistical thermodynamics derived state functions from statistical arguments based on molecular parameters rather than from basic definitions as summarized below. 

For large numbers of particles, then, probability favors random arrangements. Using this insight, we can tentatively define entropy as a measure of the randomness or disorder of a system. However, we still have to establish a definition that can be used quantitatively and from a molecular perspective. To do this, we turn to a branch of physical chemistry called statistical mechanics, or statistical thermodynamics, where we find a subtle addition to the definition. The probability of events that must be counted is not the number of ways particles can be arranged... 

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... 

In this model, classical statistical thermodynamics cannot be applied, and we abandon the accounting of sites and configurational entropy. Instead we enter simply the volume concentrations of defects in the widespread definition of the... 

In statistic thermodynamics the entropy of the equilibrious system equals the logarithm of the probability of its definite macrostate ... 

Then we can easily see that the left-hand side of Eq. 90 is the thermodynamical definition of the local entropy production as given by Eq. 17, although, in the present case, it includes tlie second summation corresponding to the production due to the viscous flow, but does not include the term for the chemical reactions. Thus the condition Eq. 79 implies, that the phenomenological entropy production does agree with the statistical expression of the entropy production, which is given as the brace symbol. This was first pointed out by the author. ... 

We can now shed some additional light on the relation between statistical entropy and thermodynamic entropy, and also on the identification of the parameter fi with 1 /k T. Using the definition of the statistical entropy in Eq. (26.1-1) and replacing In(f ) by its largest term, In(Wmp), we obtain... 

In popular imagination, and also in various scientific commentaries, entropy is equated to disorder. The words order and disorder have no definite meaning in equilibrium thermodynamics, since no assumptions are made about the units that make up the whole. We will return to this point when we come to statistical thermodynamics. 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

The equilibrium state, which is denoted x, is by definition both the most likely state, p(x E) > p(x E), and the state of maximum constrained entropy, iS,(T (x /ij > iS 0(x j. This is the statistical mechanical justification for much of the import of the Second Law of Equilibrium Thermodynamics. The unconstrained entropy, as a sum of positive terms, is strictly greater than the maximal constrained entropy, which is the largest term, S HE) >. S(1 (x j. However, in the thermodynamic limit when fluctuations are relatively negligible, these may be equated with relatively little error, S HE) . S(1 (x j. 

The term entropy, which literally means a change within, was first used in 1851 by Rudolf Clausius, one of the formulators of the second law of thermodynamics. A rigorous quantitative definition of entropy involves statistical and probability considerations. However, its nature can be illustrated qualitatively by three simple examples, each demonstrating one aspect of entropy. The key descriptors of entropy are randomness and disorder, manifested in different ways. 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

The Entropy and Irreversible Processes.—Unlike the internal energy and the first law of thermodynamics, the entropy and the second law are relatively unfamiliar. Like them, however, their best interpretation comes from the atomic point of view, as carried out in statistical mechanics. For this reason, we shall start with a qualitative description of the nature of the entropy, rather than with quantitative definitions and methods of measurement. 

Of the three quantities (temperature, energy, and entropy) that appear in the laws of thermodynamics, it seems on the surface that only energy has a clear definition, which arises from mechanics. In our study of thermodynamics a number of additional quantities will be introduced. Some of these quantities (for example, pressure, volume, and mass) may be defined from anon-statistical (non-thermodynamic) perspective. Others (for example Gibbs free energy and chemical potential) will require invoking a statistical view of matter, in terms of atoms and molecules, to define them. Our goals here are to see clearly how all of these quantities are defined thermodynamically and to make use of relationships between these quantities in understanding how biochemical systems behave. 

Similarly, if one is interested in a macroscopic thermodynamic state (i.e., a subset of microstates that corresponds to a macroscopically observable system with bxed mass, volume, and energy), then the corresponding entropy for the thermodynamic state is computed from the number of microstates compatible with the particular macrostate. All of the basic formulae of macroscopic thermodynamics can be obtained from Boltzmann s definition of entropy and a few basic postulates regarding the statistical behavior of ensembles of large numbers of particles. Most notably for our purposes, it is postulated that the probability of a thermodynamic state of a closed isolated system is proportional to 2, the number of associated microstates. As a consequence, closed isolated systems move naturally from thermodynamic states of lower 2 to higher 2. In fact for systems composed of many particles, the likelihood of 2 ever decreasing with time is vanishingly small and the second law of thermodynamics is immediately apparent. 

In modem physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity... 

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

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