The Statistical Interpretation of Entropy

Because entropy is such an important state function, it is natural to seek a description of its meaning on the microscopic level. 

Entropy is sometimes said to he a measure of disorder. According to this idea, the entropy increases whenever a closed system becomes more disordered on a microscopic scale. This description of entropy as a measure of disorder is highly misleading. It does not explain why entropy is increased by reversible heating at constant volume or pressure, or why it increases during the reversible isothermal expansion of an ideal gas. Nor does it seem to agree with the freezing of a supercooled hquid or the formation of crystalline solute in a supersaturated solution these processes can take place spontaneously in an isolated system, yet are accompanied by an apparent decrease of disorder. 

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

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. 

The fundamental assumption of statistical mechanics is that accessible microstates of equal energy are equally probable, so that the system while in an equilibrium state spends an equal fraction of its time in each such microstate. The statistical entropy of the equilibrium state then turns out to be given by the equation 

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Next, we review findings of educational research about the main areas of physical chemistry. Most of the work done was in the areas of basic thermodynamics and electrochemistry, and some work on quantum chemistry. Other areas, such as chemical kinetics, statistical thermodynamics, and spectroscopy, have not so far received attention (although the statistical interpretation of entropy is treated in studies on the concepts of thermodynamics). Because many of the basics of physical chemistry are included in first-year general and inorganic courses (and some even in senior high school), many of the investigations have been carried out at these levels. 

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... 

Th is edition has been extensively revised in the light of recent contributions to the literature. Many new references have been added the re-writing of certain passages, especially of those concerning the statistical interpretation of entropy and the present understanding of order-disorder transitions, also reflects charges of emphasis. 

For an ideal chain there is no interaction among monomers, so that dU=0. Hence dG=-TdS, and using the statistical interpretation of entropy in terms of probability... 

This is a law about the equilibrium state, when macroscopic change has ceased it is the state, according to the law, of maximum entropy. It is not really a law about nonequilibrium per se, not in any quantitative sense, although the law does introduce the notion of a nonequilibrium state constrained with respect to structure. By implication, entropy is perfectly well defined in such a nonequilibrium macrostate (otherwise, how could it increase ), and this constrained entropy is less than the equilibrium entropy. Entropy itself is left undefined by the Second Law, and it was only later that Boltzmann provided the physical interpretation of entropy as the number of molecular configurations in a macrostate. This gave birth to his probability distribution and hence to equilibrium statistical mechanics. 

MSN.74. I. Prigogine, The statistical interpretation of nonequilibrium entropy, Acta Phys. Austriaca, Suppl. X, 401 50 (1973). 

If we attempt to interpret the observations with regard to residual entropy in Frame 16, section 16.4 and those features that are not entirely in accord with the Third Law, we see that equation (17.1) represents a statistical interpretation of entropy which gives a reasonable account of these departures from the Third Law as well as giving an entirely consistent account of the Third Law itself. 

A comparison of expression (54) with the statistical interpretation of the adsorbate entropy provides additional insight to the significance of the different terms in this equation. The total partition function adsorbed molecules where M > N is (5, 8)... 

The third law of thermodynamics lacks the generality of the other laws, since it applies only to a special class of substances, namely pure, crystalline substances, and not to all substances. In spite of this restriction the third law is extremely useful. The reasons for exceptions to the law can be better understood after we have discussed the statistical interpretation of the entropy the entire matter of exceptions to the third law will be deferred until then. 

As the number of probable states available to a system increases, the uncertainty as to which state the system occupies increases and the entropy defined in terms of probability increases. A statistical interpretation of entropy is related to the uncertainty of knowledge about the state of the system. 

As regards entropy, one way of dealing with these difficulties is simply to postulate its existence, rather than seeking to prove it. However this method seems to me not sufficiently satisfying for the student. Far better, in my view, to put forward the classical arguments as well as they can be put, and to develop simtdtaneously the statistical interpretation of the second law, so as to create a linkage of thermodynamics with the rest of physics and chemistry. 

We end with a brief exposition concerning the statistical interpretation of the entropy function. For a brief review of the fundamental concepts, see Chapter 10 and consult texts on statistical thermodynamics for an in-depth exposition. 

It is often claimed, with some justification, that statistical theory explains the basis of the second law and provides a mechanistic interpretation of thermodynamic quantities. For example, the Boltzmann expression for entropy is S = ks In W, where W is the number of ways the total energy is distributed among the molecules. Thus entropy in statistical theory is connected to the availability of microstates for the system. In classical theory, on the other hand, there are no pictures associated with entropy. Hence the molecular interpretation of entropy changes depends on statistical theory. 

For those who are familiar with the statistical mechanical interpretation of entropy, which asserts that at 0 K substances are nonnally restricted to a single quantum state, and hence have zero entropy, it should be pointed out that the conventional thennodynamic zero of entropy is not quite that, since most elements and compounds are mixtures of isotopic species that in principle should separate at 0 K, but of course do not. The thennodynamic entropies reported in tables ignore the entropy of isotopic mixing, and m some cases ignore other complications as well, e.g. ortho- and para-hydrogen. 

As we have seen, the third law of thermodynamics is closely tied to a statistical view of entropy. It is hard to discuss its implications from the exclusively macroscopic view of classical themiodynamics, but the problems become almost trivial when the molecular view of statistical themiodynamics is introduced. Guggenlieim (1949) has noted that the usefiihiess of a molecular view is not unique to the situation of substances at low temperatures, that there are other limiting situations where molecular ideas are helpfid in interpreting general experimental results ... 

Boltzmann, following Clausius, considered entropy to be defined only to an arbitrary constant, and related the difference in entropy between two states of a system to their relative probability. An enormous advance was made by Planck who proposed to determine the absolute entropy as a quantity, which, for every realizable system, must always be positive (third law of thermodynamics). He related this absolute entropy, not to the probability of a system, but to the total number of its possibilities. This view of Planck has been the basis of all recent efforts to find the statistical basis of thermodynamics, and while these have led to many differences of opinion, and of interpretation, we believe it is now possible to derive the second law of thermodynamics in an exact form and to obtain... 

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 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 ... 

The above realization of the abstract mesoscopic equilibrium thermodynamics is called a Canonical-Ensemble Statistical Mechanics. We shall now briefly present also another realization, called a Microcanonical-Ensemble Statistical Mechanics since it offers a useful physical interpretation of entropy. 

If we restrict ourselves to time intervals which are experimentally realizable, the statistical interpretation is equivalent to the requirement that the entropy of an isolated gas quantum should always increase.153 Beyond that, however, one can in fact defend two opposite points of view. 

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. 

In the molecular statistical analysis, Boltzmann defined the entropy S in any thermodynamic state as S = In fl, where O is the number of microstates available to the system in that same thermodynamic state. This equation is used for qualitative interpretations of entropy changes. It shows that any process that increases D will increase S, and any process that decreases O will decrease S. 

Provide a statistical interpretation of the change in entropy that occurs when a gas undergoes a volume change (Section 13.2, Problems 3-10). 

The interpretation of entropy as a measure of knowledge of an observer cannot be avoided in any of the statistical theories of thermodynamics advanced to date because in all these theories objective reality is represented by pure states only, and such states have no entropy. 

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