Statistical thermodynamics heat capacity

The remaining question is how we got from G3MP2 (OK) = —117.672791 to G3MP2 Enthalpy = —117.667683. This is not a textbook of classical thermodynamics (see Klotz and Rosenberg, 2000) or statistical themiodynamics (see McQuarrie, 1997 or Maczek, 1998), so we shall use a few equations from these fields opportunistically, without explanation. The definition of heat capacity of an ideal gas... 

Statistical thermodynamics tells us that Cv is made up of four parts, translational, rotational, vibrational, and electronic. Generally, the last part is zero over the range 0 to 298 K and the first two parts sum to 5/2 R, where R is the gas constant. This leaves us only the vibrational part to worry about. The vibrational contr ibution to the heat capacity is... 

The treatment of heat capacity in physical chemistry provides an excellent and familiar example of the relationship between pure and statistical thermodynamics. Heat capacity is defined experimentally and is measured by determining the heat required to change the temperature of a sample in, say,... 

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... 

Statistical thermodynamics provides the relationships that we need in order to bridge this gap between the macro and the micro. Our most important application will involve the calculation of the thermodynamic properties of the ideal gas, but we will also apply the techniques to solids. The procedure will involve calculating U — Uo, the internal energy above zero Kelvin, from the energy of the individual molecules. Enthalpy differences and heat capacities are then easily calculated from the internal energy. Boltzmann s equation... 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

From classic thermodynamics alone, it is impossible to predict numeric values for heat capacities these quantities are determined experimentally from calorimetric measurements. With the aid of statistical thermodynamics, however, it is possible to calculate heat capacities from spectroscopic data instead of from direct calorimetric measurements. Even with spectroscopic information, however, it is convenient to replace the complex statistical thermodynamic equations that describe the dependence of heat capacity on temperature with empirical equations of simple form [15]. Many expressions have been used for the molar heat capacity, and they have been discussed in detail by Frenkel et al. [4]. We will use the expression... 

Figure 4.2. Variation of heat capacity with temperature as calculated from the equations of Frenkel et al. [4]. The differences observed between isotopic species and the way heat capacity depends on molecular size and structure can be described thermodynamically, but they must be explained by the methods of quantum-statistical thermodynamics. The right-hand scale is for H2 and D2 the left-hand scale is for the other compounds.

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 ... 

Exploiting the principles of statistical mechanics, atomistic simulations allow for the calculation of macroscopically measurable properties from microscopic interactions. Structural quantities (such as intra- and intermolecular distances) as well as thermodynamic quantities (such as heat capacities) can be obtained. If the statistical sampling is carried out using the technique of molecular dynamics, then dynamic quantities (such as transport coefficients) can be calculated. Since electronic properties are beyond the scope of the method, the atomistic simulation approach is primarily applicable to the thermodynamics half of the standard physical chemistry curriculum. 

It is noteworthy that Gibbs himself was acutely aware of the qualitative failures of 19th-century molecular theory (as revealed, for example, by erroneous classical predictions of heat capacities Sidebar 3.8). In the preface to his Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics (published in the last year of his life), Gibbs wrote ... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

The thermophysical properties necessary for the growth of tetrahedral bonded films could be estimated with a thermal statistical model. These properties include the thermodynamic sensible properties, such as chemical potential /t, Gibbs free energy G, enthalpy H, heat capacity Cp, and entropy S. Such a model could use statistical thermodynamic expressions allowing for translational, rotational, and vibrational motions of the atom. 

Below the statistical-thermodynamical calculation of configurational heat capacity of ordering twocomponent fullerite from fullerenes = C(, , [Pg.219]

The elaborated statistical-thermodynamical calculation makes it possible to elucidate the character of temperature dependence of configurational heat capacity... 

The enthalpies of formation and atomization, the heat capacities, and the entropies of the selenium rings Se (n = 5-12), as derived from mass spectrometric measurements and statistical thermodynamics (8-10,12, 13, 57), are given in Table VI. 

Other experimental and theoretical methods have been developed for the determination of the heat of sublimation of solid iodine these too are suitable for undergraduate laboratory experiments or variations on this experiment. Henderson and Robarts have employed a photometer incorporating a He-Ne gas laser, the beam from which (attenuated by a CUSO4 solution) has a wavelength of 632.8 nm, in a hot band near the long-wavelength toe of the absorption band shown in Fig. 3. Stafford has proposed a thermodynamic treatment in which a free-energy function ifef), related to entropy, is used in calculations based on the third law of thermodynamics. In this method either heat capacity data or spectroscopic data are used, and as in the present statistical mechanical treatment, the heat of sublimation can be obtained from a measurement of the vapor pressure at only one temperature. 

The aforementioned macroscopic physical constants of solvents have usually been determined experimentally. However, various attempts have been made to calculate bulk properties of Hquids from pure theory. By means of quantum chemical methods, it is possible to calculate some thermodynamic properties e.g. molar heat capacities and viscosities) of simple molecular Hquids without specific solvent/solvent interactions [207]. A quantitative structure-property relationship treatment of normal boiling points, using the so-called CODESS A technique i.e. comprehensive descriptors for structural and statistical analysis), leads to a four-parameter equation with physically significant molecular descriptors, allowing rather accurate predictions of the normal boiling points of structurally diverse organic liquids [208]. Based solely on the molecular structure of solvent molecules, a non-empirical solvent polarity index, called the first-order valence molecular connectivity index, has been proposed [137]. These purely calculated solvent polarity parameters correlate fairly well with some corresponding physical properties of the solvents [137]. 

This function may be calculated from molecular data by tlie methods of quantum statistical mechanics. It may also be obtained from experimentally determined heat capacity data, from which the required entropies are deduced using the third law of thermodynamics. The calculation may be put into several equivalent forms, and is discussed in the text-books. An account of the theory is given by Fowler and Guggenheim 7 its application by... 

Somewhat analogous considerations apply to the entropy of water vapor. The result derived from heat capacity measurements is again lower than the statistical value, and this can be accounted for by random orientation of the water molecules in the solid. The situation is complicated, however, by the distribution of hydrogen bonds in the ice crystal, and by other factors. In this instance, also, the crystal is not perfect, and so the entropy would not be zero at 0 K. The statistical value of the entropy is therefore the correct one to be used in thermodynamic calculations. 

In treating the various topics in this book the particular method employed has been determined in each case by considerations of simplicity, usefulness and logical development. In some instances the classical, historical approach has been preferred, but in others the discussion follows more modern lines. Whenever feasible the generalized procedures, involving reduced temperatures and pressures, which have been evolved in recent years chiefly by chemical engineers, are introduced. As regards statistical methods, the author feels that the time has come for them to take then-place as an essential part of chemical thermodynamics. Consequently, the applications of partition functions to the determination of heat capacities, entropies, free energies, equilibrium constants, etc., have been introduced into the text in the appropriate places where it is hoped their value will be appreciated. 

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