Entropy approximation and

Zhang, T-L., Ding, Y.-S. and Chou, K.-C. (2008) Prediction protein structural classes with pseudoamino acid composition approximate entropy and hydrophobidtypattern./. Theor. Biol., 250,186-193. 

Joshua S. Richman, J. Randall Morman, Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol Heart Cite Physiol 278, 2000,pp. H2039-H2049 Pincus S.M., Approximate entropy in cardiology, Herzschr Elektro-phys,2000, pp. 11 139-150... 

This is the result for monatomic fluids and is well approximated by a sum of tliree Lorentzians, as given by the first tliree temis on the right-hand side. The physics of these tliree Lorentzians can be understood by thinking about a local density fluctuation as made up of tliemiodynamically independent entropy and pressure fluctuations p = p s,p). The first temi is a consequence of the themial processes quantified by the entropy... 

Entropies and heat capacities can thus now be calculated using more elaborate models for the vibrational densities of states than the Einstein and Debye models discussed in Chapter 8. We emphasize that the results are only valid in the quasiharmonic approximation and can only be as good as the accuracy of the underlying force-field calculation of such properties can thus be a very sensitive test of interatomic potentials. 

These ideas have been employed to compute configurational entropy and hence test the AG relation via molecular simulation of several model systems.91-94,101,102 The approach used in those studies is conceptually simple. First, the total entropy of the fluid S is calculated by integration of standard thermodynamic relationships, for example, as discussed below. Then, the configurational contribution to the entropy Sc = S — Sv b, is approximated by subtracting from the total entropy an estimate for the vibrational contribution, Svib. 

AG is the free energy change of a process with AS and AV as the accompanying changes in entropy and volume. As a first approximation, one can assume AS and AV to be nearly constant within a limited p and T range, and use the integrated equation... 

This reaction, having equal number of mols of gas reactants and products, has a negligible change in entropy and thus a negligible heat effect if carried out reversibly at constant temperature. The maximum work available from a fuel cell under these circumstances would then be approximately the enthalpy change of the reaction, i.e., the heat of combustion of the... 

By convention, the standard Gibbs function for formation AfG of graphite is assigned the value of zero. On this basis, AfG gg of diamond is 2900 J mol Entropies and densities also are listed in Table 8.2. Assuming that the entropies and densities are approximately constant, determine the conditions of temperature and pressure under which the manufacture of diamonds from graphite would be thermodynamically and kinetically practical [2]. 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

I am suggesting that often the applicability of Barkley-Butler type plots, that is, the linear relationship between the entropy and enthalpy of activation in a series may come about because of there being a distribution of reaction paths. Small variations in the importance of low activation energy, low probability paths could then account for the data in Dr. Taube s table. By contrast, transition state theory in its approximate application, invariably leads to diagrams of energy vs. reaction path which, in spite of all protest, one reaction path, whatever it is, one transition state, and one energy. 

A rare exception to the dominance of enthalpic driving force is provided by the unstrained eight-membered ring (Me2SiO)4, which undergoes ROP to give poly(dimethylsiloxane). For this process, A//rop is approximately zero but the remarkable skeletal flexibility of the polysiloxane backbone makes the TAArop term favourable and entropy provides the driving force for ROP. Exceptional behaviour occurs in the extremely rare cases of monomers where the polymerisation is driven by entropy and A7/rop is unfavourable. In these cases, the formation of polymer is favoured at elevated temperatures. For example, cyclo-Sg will polymerise above 150 °C (where the favourable TAArop term dominates), but slowly depolymerises back to the cyclic Sg monomer at room temperature. ... 

The inter- and Intramolecular contributions to the entropy and energy of fusion are calculated for several linear aliphatic polyesters and polyamides assuming the fusion process consists of two Independent contributions the volume expansion (intermolecular contribution) and the increase in the conformational freedom of each polymer chain on melting (intramolecular contribution). The intramolecular entropy and energy contributions are obtained from the configurational partition function and Its temperature coefficient calculated for an isolated, unperturbed polymer chain using the RIS approximation. 

All the normal modes are present in the results of a semiempirical frequency calculation, as is the case for an ab initio or DFT calculation, and animation of these will usually give, approximately, the frequencies of these modes. A very extensive compilation of experimental, MNDO and AMI frequencies has been given by Healy and Holder, who conclude that the AMI error of 10% can be reduced to 6% by an empirical correction, and that entropies and heat capacities are accurately calculated from the frequencies [104], In this regard, Coolidge et al. conclude -surprisingly, in view of our results for the four molecules in Figs. 6.5-6.8 - from a study of 61 molecules that (apart from problems with ring- and heavy atom-stretch for AMI and S-H, P-H and O-H stretch for PM3) both AMI and PM3 should provide results that are close to experimental gas phase spectra [105]. 

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