Entropy statistical mechanics

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by... 

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. 

Nearly ten years ago, Tsallis proposed a possible generalization of Gibbs-Boltzmann statistical mechanics. [1] He built his intriguing theory on a reexpression of the Gibbs-Shannon entropy S = —k Jp r) np r)dr written... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

The Boltzmann distribution is fundamental to statistical mechanics. The Boltzmann distribution is derived by maximising the entropy of the system (in accordance with the second law of thermodynamics) subject to the constraints on the system. Let us consider a system containing N particles (atoms or molecules) such that the energy levels of the... 

The thermal conductivity of soHd iodine between 24.4 and 42.9°C has been found to remain practically constant at 0.004581 J/(cm-s-K) (33). Using the heat capacity data, the standard entropy of soHd iodine at 25°C has been evaluated as 116.81 J/ (mol-K), and that of the gaseous iodine at 25°C as 62.25 J/(mol-K), which compares satisfactorily with the 61.81 value calculated by statistical mechanics (34,35). 

Chemistry can be divided (somewhat arbitrarily) into the study of structures, equilibria, and rates. Chemical structure is ultimately described by the methods of quantum mechanics equilibrium phenomena are studied by statistical mechanics and thermodynamics and the study of rates constitutes the subject of kinetics. Kinetics can be subdivided into physical kinetics, dealing with physical phenomena such as diffusion and viscosity, and chemical kinetics, which deals with the rates of chemical reactions (including both covalent and noncovalent bond changes). Students of thermodynamics learn that quantities such as changes in enthalpy and entropy depend only upon the initial and hnal states of a system consequently thermodynamics cannot yield any information about intervening states of the system. It is precisely these intermediate states that constitute the subject matter of chemical kinetics. A thorough study of any chemical reaction must therefore include structural, equilibrium, and kinetic investigations. 

The Gibbs free energy is given in terms of the enthalpy and entropy, G — H — TS. The enthalpy and entropy for a macroscopic ensemble of particles may be calculated from properties of the individual molecules by means of statistical mechanics. 

Also called the uncertainty or - because of its formal similarity to the entropy function used in statistical mechanics - Shannon entropy. 

Treatment of Solutions by Statistical Mechanics. Since the vapor pressure is directly connected with the free energy, in the thermodynamic treatment the free energy is discussed first, and the entropy is derived from it. In the treatment by statistical mechanics, however, the entropy is discussed first, and the free energy is derived from it. Let us first consider an element that consists of a single isotope. When the particles share a certain total energy E, we are interested in the number of recog-... 

Although these potential barriers are only of the order of a few thousand calories in most circumstances, there are a number of properties which are markedly influenced by them. Thus the heat capacity, entropy, and equilibrium constants contain an appreciable contribution from the hindered rotation. Since statistical mechanics combined with molecular structural data has provided such a highly successful method of calculating heat capacities and entropies for simpler molecules, it is natural to try to extend the method to molecules containing the possibility of hindered rotation. Much effort has been expended in this direction, with the result that a wide class of molecules can be dealt with, provided that the height of the potential barrier is known from empirical sources. A great many molecules of considerable industrial importance are included in this category, notably the simpler hydrocarbons. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

The name entropy is used here because of the similarity of Eq. (4-6) to the definition of entropy in statistical mechanics. We shall show later that H(U) is the average number of binary digits per source letter required to represent the source output. 

To close this chapter we emphasize that Hie statistical mechanical definition of macroscopic parameters such as temperature and entropy are well designed to describe isentropic equilibrium systems, but are not immediately applicable to the discussion of transport processes where irreversible entropy increase is an essential feature. A macroscopic system through which heat is flowing does not possess a single tempera-... 

Use the Third Law to calculate the standard entropy, S°nV of quinoline (g) p — 0.101325 MPa) at T= 298,15 K. (You may assume that the effects of pressure on all of the condensed phases are negligible, and that the vapor may be treated as an ideal gas at a pressure of 0.0112 kPa, the vapor pressure of quinoline at 298.15 K.) (c) Statistical mechanical calculations have been performed on this molecule and yield a value for 5 of quinoline gas at 298.15 K of 344 J K l mol 1. Assuming an uncertainty of about 1 j K 1-mol 1 for both your calculation in part (b) and the statistical calculation, discuss the agreement of the calorimetric value with the statistical... 

Chapter 9, on entropy and molecular rotation in crystals and liquids, is concerned mostly with statistical mechanics rather than quantum mechanics, but the two appear together in SP 74. Chapter 9 contains one of Pauling s most celebrated papers, SP 73, in which he explains the experimentally measured zero-point entropy of ice as due to water-molecule orientation disorder in the tetrahedrally H-bonded ice structure with asymmetric hydrogen bonds (in which the bonding proton is not at the center of the bond). This concept has proven fully valid, and the disorder phenomenon is now known to affect greatly the physical properties of ice via the... 

In the foregoing article we have applied the methods of statistical mechanics to a determination of the entropy of crystals and supercooled glasses, and have reached the following conclusions. 

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

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. 

This nonequilibrium Second Law provides a basis for a theory for nonequilibrium thermodynamics. The physical identification of the second entropy in terms of molecular configurations allows the development of the nonequilibrium probability distribution, which in turn is the centerpiece for nonequilibrium statistical mechanics. The two theories span the very large and the very small. The aim of this chapter is to present a coherent and self-contained account of these theories, which have been developed by the author and presented in a series of papers [1-7]. The theory up to the fifth paper has been reviewed previously [8], and the present chapter consolidates some of this material and adds the more recent developments. 

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. 

P. Attard, Thermodynamics and Statistical Mechanics Equilibrium by Entropy Maximisation, Academic Press, London, 2002. 

The work of Ludwig Boltzmann (1844-1906) in Vienna led to a better understanding, and to an extension, of the concept of entropy. On the basis of statistical mechanics, which he developed, the term entropy experienced an atomic interpretation. Boltzmann was able to show the connections between thermodynamics and the phenomenon of order and chance events he used the term entropy as a measure... 

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