Entropy statistical thermodynamics

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

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. 

Thus from an adsorption isotherm and its temperature variation, one can calculate either the differential or the integral entropy of adsorption as a function of surface coverage. The former probably has the greater direct physical meaning, but the latter is the quantity usually first obtained in a statistical thermodynamic adsorption model. 

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. 

By the standard methods of statistical thermodynamics it is possible to derive for certain entropy changes general formulas that cannot be derived from the zeroth, first, and second laws of classical thermodynamics. In particular one can obtain formulae for entropy changes in highly di.sperse systems, for those in very cold systems, and for those associated, with the mixing ofvery similar substances. 

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

The entropy of formation is calculated from 5° values obtained from Third Law measurements (Chapter 4) or calculated from statistical thermodynamics (Chapter 10). The combination of AfS with Af// gives AfG. For example, for the reaction at 298.15 K... 

The freely-jointed chain considered previously has no internal restraint, and hence, its internal energy is zero regardless of its present configuration. The entropy (S) is not constant, however, since the number of available configurations decreases with the chain end separation distance. The variation which follows from chain length change by a small amount (dr) at constant temperature (T) is given by the Boltzmann rule of statistical thermodynamics ... 

We can show that the thermodynamic and statistical entropies are equivalent by examining the isothermal expansion of an ideal gas. We have seen that the thermodynamic entropy of an ideal gas increases when it expands isothermally (Eq. 3). If we suppose that the number of microstates available to a single molecule is proportional to the volume available to it, we can write W = constant X V. For N molecules, the number of microstates is proportional to the Nth power of the volume ... 

We have shown that the statistical and thermodynamic entropies lead to the same conclusions. We can expect their more general properties to be the same, too ... 

Doubling the number of molecules increases the number of microstates from W to W2, and so the entropy changes from k In W to k In W2, or 2k In W. Therefore, the statistical entropy, like the thermodynamic entropy, is an extensive property. 

The equations used to calculate changes in the statistical entropy and the thermodynamic entropy lead to the same result. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

Data for a large number of organic compounds can be found in E. S. Domalski, W. H. Evans, and E. D. Hearing, Heat capacities and entropies in the condensed phase, J. Phys. Chem. Ref. Data, Supplement No. 1, 13 (1984). It is impossible to predict values of heat capacities for solids by purely thermodynamic reasoning. However, the problem of the solid state has received much consideration in statistical thermodynamics, and several important expressions for the heat capacity have been derived. For our purposes, it will be sufficient to consider only the Debye equation and, in particular, its limiting form at very low temperamres ... 

A satisfactory explanation for this discrepancy was not available until the development of statistical thermodynamics with its methods of calculating entropies from spectroscopic data and the discovery of the existence of ortho- and parahydrogen. It then was found that the major portion of the deviation observed between Equations (11.24) and (11.25) is from the failure to obtain a tme equilibrium between these two forms of H2 molecules (which differ in their nuclear spins) during thermal measurements at very low temperatures (Fig. 11.4). If true equilibrium were established at all times, more parahydrogen would be formed as the temperature is lowered, and at 0 K, all the hydrogen molecules would be in the... 

With the development of statistical thermodynamics and the calculations of the entropies of many substances from spectroscopic data, several other substances in addition to hydrogen have been found to have values of molar entropies that disagree with those calculated from thermal data alone [13] (Table 11.1). The discrepancies can be accounted for on the assumption that even near absolute zero not all molecules are in the same state and that tme equilibrium has not been attained. For CO, COCI2, N2O, NO, and CIO3F, the close similarity in the sizes of the atoms makes different... 

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

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... 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

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... 

The maximum entropy method (MEM) is an information-theory-based technique that was first developed in the field of radioastronomy to enhance the information obtained from noisy data (Gull and Daniell 1978). The theory is based on the same equations that are the foundation of statistical thermodynamics. Both the statistical entropy and the information entropy deal with the most probable distribution. In the case of statistical thermodynamics, this is the distribution of the particles over position and momentum space ( phase space ), while in the case of information theory, the distribution of numerical quantities over the ensemble of pixels is considered. 

As in statistical thermodynamics, the entropy is defined as In P. Since the numerator is constant, the entropy is, apart from a constant, equal to... 

Further understanding of the kinetic of template polymerization needs consideration of the process entropy. Applying a well known lattice model, it is easy to see that entropy changes, AS, in free polymerization and the template polymerization, differs considerably. According to the principles of statistical thermodynamics, the entropy of mixing is given by the equation ... 

Equation (1.44) states that the structural energy increases associated with the creation of defects are offset by entropy increases. The entropy is the number of ways the defects (both interstitials and vacancies) can be arranged within the perfect lattice, and it can be approximated using statistical thermodynamics as... 

Entropy will be represented by the letter S. Entropy is a measure of randomness or disorder in a system and has SI units of J/K. Recall that the Second Law of Thermodynamics states that the entropy change of all processes must be positive. We will see that the origins of entropy are best described from statistical thermodynamics, but for now let us concentrate on how we can use entropy to describe real material systems. 

The entropy of mixing can be thought of as a measure of the increase in the nnmber of spatial confignrations that become available to the system as a resnlt of the mixing process, ASconf, which can be shown with statistical thermodynamic argnments to be... 

For solutions of a polymer in a solvent, it can be shown, again through statistical thermodynamic arguments, that the entropy of mixing is given by... 

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