Entropy statistical calculation

A second way of dealing with the relationship between aj and the experimental concentration requires the use of a statistical model. We assume that the system consists of Nj molecules of type 1 and N2 molecules of type 2. In addition, it is assumed that the molecules, while distinguishable, are identical to one another in size and interaction energy. That is, we can replace a molecule of type 1 in the mixture by one of type 2 and both AV and AH are zero for the process. Now we consider the placement of these molecules in the Nj + N2 = N sites of a three-dimensional lattice. The total number of arrangements of the N molecules is given by N , but since interchanging any of the I s or 2 s makes no difference, we divide by the number of ways of doing the latter—Ni and N2 , respectively—to obtain the total number of different ways the system can come about. This is called the thermodynamic probabilty 2 of the system, and we saw in Sec. 3.3 that 2 is the basis for the statistical calculation of entropy. For this specific model... 

Comparison and agreement with the calorimetric value verifies the assumption that So = 0. For example, we showed earlier that the entropy of ideal N2 gas at the normal boiling point as calculated by the Third Law procedure had a value of 152.8 0.4 J-K mol. The statistical calculation gives a value of 152.37 J K -mol-1, which is in agreement within experimental error. For PH3, the Third Law and statistical values at p 101.33 kPa and T— 185.41 K are 194.1 0.4 J K, mol 1 and 194.10 J-K 1-mol 1 respectively, an agreement that is fortuitously close. Similar comparisons have been made for a large number of compounds and agreement between the calorimetric (Third Law) and statistical value is obtained, all of which is verification of the Third Law. For example, Table 4.1 shows these comparisons for a number of substances. 

For most substances, the Third Law and statistical calculations of the entropy of the ideal gas are in agreement, but there are exceptions, some of which are summarized in Table 4.2. The difference results from residual entropy, So, left in the solid at 0 Kelvin because of disorder so that St - So calculated from Cp/TdT is less than the St calculated from statistical methods. In carbon monoxide the residual disorder results from a random arrangement of the CO molecules in the solid. Complete order in the solid can be represented schematically (in two-dimensions) by... 

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... 

Entropies can be calculated or estimated, and hence enthalpies can be derived from equilibrium measurements. Gaseous entropies are calculated by statistical mechanics using experimental or estimated molecular dimensions and fundamental frequencies (93). For solids, numerous methods based on additivity rules, or regularities in series of compounds, are available. Khriplovich and Paukov (140), for example, list 20 such relationships and were able to estimate entropies to about 1%. Empirical equations are also available for ion entropies (59). 

III. JThe Statistical Calculation of the Entropy of the Adsorbed Material... 

A good example of a reaction whose transition-state structure has been adduced on grounds of statistical mechanics is the dimerization of butadiene in the gas phase to give 3-vinylcyclohexene (Glasstone et al., 1941c). Entropies were calculated for a diradical transition state and a transition state involving a six-membered ring. The experimentally observed value of AS conformed to that expected for a diradical intermediate. 

Symbol Gibbs free energy is denoted eponymously by G, after Josiah Willard Gibbs, ca. 1873, who single-handedly created much of chemical thermodynamics. In the older literature F was sometimes used. Equation since G = H TS, the free energy of a molecule can be calculated from its enthalpy (above) and entropy at temperature 7 the entropy is calculated by standard statistical mechanics methods [130]. 

Botli tile classical and statistical equations [Eqs. (5.40) and (16.20)] yield absolute values of entropy. As is evidentfrom Table 16.3, good agreementbetween the statistical calculations and those based on calorimetric data is obtained. Results such as these provide impressive evidence for tile validity of statistical mechanics and quantum theory. In some instances results based on Eq. (16.20) are considered more reliable because of uncertainties in heat-capacity data or about tlie crystallinity of the substance near absolute zero. Absolute entropies provide much of tile data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 13. 

The thermal entropy of normal deuterium was found to be 33.90 E.u. mole . Normal deuterium consists of two parts of ortho- to one part of para-molecules at low temperatures the former occupy six and the latter nine closely spaced levels. The spin of each deuterium nucleus is 1 unit. Show that the practical standard entropy of deuterium gas at 25 C is 34.62 e.u. mole" (Add the entropy of mixing to the thermal entropy and subtract the nuclear spin contribution.) Compare the result with the value which would be obtained from statistical calculations, using moment of inertia, etc. in Chapter VI. 

The heat capacity and entropy were calculated in [94TOM/SUS] using statistical mechanics and experimental values of the molecular constants. The values obtained are... 

The heat capacity and entropy were calculated from statistical mechanics employing molecular parameters obtained by quantum mechanical calculations in [960HA/ZYW]... 

The heat capacity and entropy were calculated from statistical mechanics employing molecular parameters obtained by quantum mechanical calculations in [960HA/ZYW] to be c (Ge2Se2, g, 298.15 K) = (79.0 2.4) J-K -moP and S°(Ge2Se2, g, 298.15 K)... 

Two other approaches beside Bergson s have been used to derive torsion barriers for S—S bonds namely, from the torsional frequency of the bond as observed in Raman spectra, and from measured thermodynamic data as compared with statistically calculated ones. According to Scott and coworkers the torsional frequencies in dimethyl disulfide (204), diethyl disulfide (205), and disulfur dichloride (139) correspond to barrier heights of 9.5, 13.2, and 14.2 kcal/mole, respectively. On the other hand, agreement between calculated and observed entropy and heat capacity of dimethyl disulfide was obtained (146) by use of 6.8 kcal/mole for the effective barrier height. From observed and calculated heat capacities, Feh r and Schulze-Rettmer (76) arrived at a value of 2.7 kcal/mole for the barrier height in hydrogen disulfide. 

Going beyond statistical analyses, information encoded in molecular descriptor value distributions in databases of natural or synthetic compounds was analyzed in quantitative terms by application of an entropy-based information-theoretic approach. - Descriptor value distributions, represented in histograms, can be reduced to their information content using Shannon entropy (SE) calculations. Differences in information content between databases can be quantified using differential SE (DSE) analysis. " An extension of this approach, SE-DSE analysis," makes it possible to classify molecular descriptors according to their relative information content in diverse databases and to identify those descriptors that are most responsive to systematic differences between compound databases. Figure... 

Methylbut-l-ene.— Thermodynamic functions for all the pentenes were originally calculated by an incremental method from data for the lower mono-olefins. Later, experimental values of the entropy and heat capacity were used in a statistical calculation which exemplifies the treatment necessary for several molecules. 

Each arrangement is called a microstate. Hence, at a higher temperature there is a greater number of vibrational microstates. When the number of microstates increases, entropy increases. When the number of microstates decreases, entropy decreases. Entropy is a measure of the number of ways that energy can be shared out among molecules. Entropy values calculated in this way are called statistical entropies. 

Chapter 17 introduced some of the basic concepts that led to the development of a statistical approach to energy and entropy. This is statistical thermodynamics. By the end of the chapter, equations were applied to monatomic gases, and thermodynamic state functions—mostly entropy—were calculated whose values were very close to experimental values. Also, in some of the exercises you were asked to derive some simple expressions that were also derived from phenomenological thermodynamics. For example, we know from early chapters in this book that the equation AS = i In (V2/V1) is applicable for an isothermal change in volume of an ideal gas. We can also get this expression using the Sackur-Tetrode statistical thermodynamic expression for S. These correspondences are just two examples where phenomenological and statistical thermodynamics are consistent with each other. That is, they ultimately make the same predictions about the state functions of a system, and how they change with a process. 

Statistical thermodynamics can be used to calculate both the enthalpic and the entropic contributions to the free energy of mixing by means of statistical calculations. Attempts were first made to apply a theory appropriate to simple regular solutions, the Hildebrand model, to the case of macromolecular solutions. This model does not adequately account for the specific behavior of polymer solutions primarily because the entropy of mixing in a polymer solution is strongly affected by the connectivity of the polymer—that is, by the existence of covalent bonds between the repetitive units. 

The entropy is calculated from the number of conformations available to the molecules. This sounds simple but we will see that there are many difficulties which have to be solved first. In this chapter we restrict ourselves very much to fundamental theoretical problems based on molecular pictures, rather than on attempts to resolve them phenomenologically. This requires the generalization of Gibbsian statistical mechanics to cover the existence of permanent constraints, mathematical realization of chains, crosslinks and entanglements, and discussion of applications to real systems. 

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. 

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... 

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. 

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