Statistical mechanical calculation

Chesnut D A and Salsburg Z W 1963 Monte Carlo procedure for statistical mechanical calculation in a grand canonical ensemble of lattice systems J. Chem. Phys. 38 2861-75... 

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... 

Loring RA (1990) Statistical mechanical calculation of inhomogeneously broadened absorption line shapes in solution. J Phys Chem 94 513-515... 

Table 5 lists equilibrium data for a new hypothetical gas-phase cyclisation series, for which the required thermodynamic quantities are available from either direct calorimetric measurements or statistical mechanical calculations. Compounds whose tabulated data were obtained by means of methods involving group contributions were not considered. Calculations were carried out by using S%g8 values based on a 1 M standard state. These were obtained by subtracting 6.35 e.u. from tabulated S g-values, which are based on a 1 Atm standard state. Equilibrium constants and thermodynamic parameters for these hypothetical reactions are not meaningful as such. More significant are the EM-values, and the corresponding contributions from the enthalpy and entropy terms. 

To use equation 2.10 correctly, we need to know how the heat capacities vary in the experimental temperature range. However, these data are not always available. A perusal of the chemical literature (see appendix B) will show that information on the temperature dependence of heat capacities is much more abundant for gases than for liquids and solids and can be easily obtained from statistical mechanics calculations or from empirical methods [11]. For substances in condensed states, the lack of experimental values, even at a single temperature, is common. In such cases, either laboratory measurements, using techniques such as differential scanning calorimetry (chapter 12) or empirical estimates may be required. 

To apply this equation to calculate the difference between the standard enthalpies of formation of AB and AB+, we need to know the values of Af//°(e ,g), A, and A. The heat capacities of AB and AB+ are easily evaluated lfom statistical mechanics calculations, provided that their structures and vibrational frequencies are available. Usually, A 0. However, with regard to AfH°(e, g) and A, the apparently simple task of assigning their values has been a source of controversy involving two scientific communities the calorimetrists and the mass spectrometrists. 

Giauque, whose name has already been mentioned in connection with the discovery of the oxygen isotopes, calculated Third Law entropies with the use of the low temperature heat capacities that he measured he also applied statistical mechanics to calculate entropies for comparison with Third Law entropies. Very soon after the discovery of deuterium Urey made statistical mechanical calculations of isotope effects on equilibrium constants, in principle quite similar to the calculations described in Chapter IV. J. Kirkwood s development showing that quantum mechanical statistical mechanics goes over into classical statistical mechanics in the limit of high temperature dates to the 1930s. Kirkwood also developed the quantum corrections to the classical mechanical approximation. 

These techniques include methods that are highly empirical in nature, such as those based on various forms of additivity principles, as well as those that are based on the use of statistical mechanical calculations. Although the latter methods represent precise tools to determine S and Cp from molecular properties, they are of little utility for the prediction of AHj. 

Energy diagrams for bond rotation in 2,2,4,4-tetramethylpentane, 2,2,4,4,6,6-hexamethytheptane, and 2,2,4,4,6,6,8,8-octamethylnonane are generated in a completely a priori manner. A relatively simple conformational model gives a good representation of the conformations calculated, and permits a statistical mechanical calculation of the characteristic ratio. 

Conformational energies of various oligooxyethylene isomers are calculated by the empiricel force field method, and statistical mechanics calculations of the chain dimensions and the dipole moments are carried out. 

Later work by Porter and Hussain349 suggested minor revisions in the numerical constants obtained. The activation energy at low [NO] was found to be — 5.02 kcal/mole in the temperature range 300-583°K. More detailed statistical mechanical calculations of partition functions gave a value of AH — —11.45 kcal/mole. This result was dependent on the previously obtained value of AT9> 9 at 333°K. Thus, these more careful measurements failed to improve the agreement between experiment and theory. 

Statistical mechanical calculations show that the translational entropy of a molecule depends on / In Mx (plus some smaller terms), where R is the gas constant and Mr is the molecular weight. Suppose that a molecule Y undergoes... 

Fig.12. Computation by Monte Carlo methods of the first four order parameters of an ensemble of 1000 chromophores (of dipole moment 13 Debye) existing in a medium of uniform dielectric constant. At the beginning of the calculation, the chromophores are randomly ordered thus, ==O. During the first 400 Monte Carlo steps, an electric poling field (600 V/micron) is on but the chromophore number density (=10 7 molecules/cc) is so small that intermolecular electrostatic interactions are unimportant. The order parameters quickly evolve to well-known equilibrium values obtained analytically from statistical mechanics (black dots in figure also see text). During steps 400-800 the chromophore number density is increased to 5xl020 and intermolecular electrostatic interactions act to decrease order parameters consistent with the results of equilibrium statistical mechanical calculations discussed in the text. Although Monte Carlo and equilibrium statistical mechanical approaches described in the text are based on different approximations and mathematical methods, they lead to the same result (i.e., are in quantitative agreement)...

Fig. 15. Comparison of experimental (for the CLD-OMet chromophore in PMMA) and theoretical (equilibrium statistical mechanical calculations described in the text) data. Experimental data are denoted by solid diamonds. The solid line theoretical curve was computed without adjustable parameters. Quantitative agreement can be obtained by adjusting parameters (chromophore dipole moment, molecular polarizability, shape, and host dielectric constant) within reasonable limits. The theoretical curve can be broken down into two parts. The purely electronic part of the electrostatic interaction is shown by the dashed line. The steric effect of nuclear repulsive interactions is shown by the dotted line...

Most experimental investigations consist in measurements of the isotherms, the heats of adsorption, or both. Therefore, we shall direct our attention to these two properties. From the point of view of a statistical mechanical calculation, the most convenient definition of the amount adsorbed, Na> is ... 

The value of the molar absorption coefficient s at each temperature can be found from Table 2 by interpolation. Then p may be calculated with Eq. (43). In the calculations of this experiment, considerable care must be taken with units. It is desirable to obtain p in pascals for the statistical mechanical calculations accordingly e should be converted into units of m moP, d should be in m, and R should be in units of J mop. ... 

The values of p and T can now be used for the statistical mechanical calculations. In order to calculate the rotational characteristic temperature t with Eq. (20), use the literature value for the rotational constant Bo = 0.037315 cm [or calculate Bo from the internuclear distance in the molecule, rg = 0.2667 nm, with Eqs. (17) to (19)]. From the literature value of the molecular vibrational frequency in the gas phase, Tg = 213.3 cm , calculate the vibrational characteristic temperature vu, with Eq. (22). From the phonon dispersion data in Table 1, calculate the 12 vibrational characteristic temperatures , -. 

Statistical mechanical calculations [3] have shown that the entropy change is given by... 

S. Engstrdm, Thesis On the Interpretation of Spectra of Quadrupolar Nuclei-Quantum Chemical and Statistical Mechanical Calculations, RhD dissertation, Lund University, 1980. 

It was found that chemisorption equilibrium is rapidly attained in most reacting systems through rapid desorption and readsorption. With a few exceptions, chemisorbed molecules can be regarded as immobile since statistical-mechanical calculations of the chemisorption equilibrium agree well with the experiment if two-dimensional translations and rotations of the chemisorbed molecules are assumed to be nonexistent. The chemisorbed state of di- or triatomic molecules can be molecular or atomic, depending on the nature of the adsorbent. For example, the carbon dioxide molecule is chemisorbed with complete dissociation into its three atoms on metallic surfaces, while on oxidic catalysts it is chemisorbed with only partial dissociation. 

The thermal factions for Sg(g) were calculated ( ) by adjusting the low valued vibrational frequencies such that 8(298.15 K) and a H (298.15 K) values could be found which reproduced the partial pressure of the octamer. The entropy of 82(g) is well-known (3), being based on reliable data and sound statistical mechanical calculations. 

A novel approach is reported for the accurate evaluation of pore size distributions for mesoporous and microporous silicas from nitrogen adsorption data. The model used is a hybrid combination of statistical mechanical calculations and experimental observations for macroporous silicas and for MCM-41 ordered mesoporous silicas, which are regarded as the best model mesoporous solids currently available. Thus, an accurate reference isotherm has been developed from extensive experimental observations and surface heterogeneity analysis by density functional theory the critical pore filling pressures have been determined as a function of the pore size from adsorption isotherms on MCM-41 materials well characterized by independent X-ray techniques and finally, the important variation of the pore fluid density with pressure and pore size has been accounted for by density functional theory calculations. The pore size distribution for an unknown sample is extracted from its experimental nitrogen isotherm by inversion of the integral equation of adsorption using the hybrid models as the kernel matrix. The approach reported in the current study opens new opportunities in characterization of mesoporous and microporous-mesoporous materials. 

Phase transitions in statistical mechanical calculations arise only in the thermodynamic limit, in which the volume of the system and the number of particles go to infinity with fixed density. Only in this limit the free energy, or any thermodynamic quantity, is a singular function of the temperature or external fields. However, real experimental systems are finite and certainly exhibit phase transitions marked by apparently singular thermodynamic quantities. Finite-size scaling (FSS), which was formulated by Fisher [22] in 1971 and further developed by a number of authors (see Refs. 23-25 and references therein), has been used in order to extrapolate the information available from a finite system to the thermodynamic limit. Finite-size scaling in classical statistical mechanics has been reviewed in a number of excellent review chapters [22-24] and is not the subject of this review chapter. 

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