Partition function and thermodynamic properties

Since this is not a textbook for thermodynamics, only a short derivation will be given, with the aim to obtain quick and useful results. Toward the end of the chapter, the problems in the derivation will be pointed out and an ensemble theory is mentioned. Here, we use simplified versions of the microcanonical ensemble. 

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Structural characteristics of adsorbed chains, such as the mean distance of chain ends from the surface and the fraction of segments in each layer, derive from P(i,s). The partition function and thermodynamic properties depend on the eigenvalues of W (Flory, 1969). In the limit n - oo, calculation of thermodynamic functions simplifies because one eigenvalue, denoted by A, dominates the free energy per segment,... 

Drellishak, K. S. Partition functions and thermodynamic properties of high temperature gases, Ph. D. Northwestern Univ. 1%3... 

The partition functions or thermodynamic properties of A and B are usually well known and to estimate a value of k all that is required is a knowledge, or estimate, of, or of the thermodynamic properties of the... 

Again, therefore, all thermodynamic properties of a system in quantum statistics can be derived from a knowledge of the partition function, and since this is the trace of an operator, we can choose any convenient representation in which to compute it. The most fruitful application of this method is probably to the theory of imperfect gases, and is well covered in the standard reference works.23... 

Relationship Between The Partition Function And The Thermodynamic Properties... 

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

With the development of Equation 5.12 relating the partition function and the macroscopic properties, all of the macroscopic thermodynamic properties may be derived from Equation 5.7. For example, differentiating In E with respect to the absolute activity (A.) of./, provides the total number of guest molecules J over all the cavities i... 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

It is thus clear that there is a network of relations between the functions determining the folllowing partitions octanol - water (Pow), water - biota (BCF), water - generic organic carbon in sediment (Koc), water - atmosphere (Hc) — and between each of them and thermodynamic properties such as aqueous solubility (S), vapor pressure (P), and melting point (Tm). 

A major drawback of MD and MC techniques is that they calculate average properties. The free energy and entropy functions cannot be expressed as simple averages of functions of the state point y. They are directly connected to the logarithm of the partition function, and our methods do not give us the partition function itself. Nonetheless, calculating free energies is important, especially when we wish to determine the relative thermodynamic stability of different phases. How can we approach this problem ... 

The molecular field is thus related to the partition function and so it is possible to generate a self-consistent value of the molecular field,. Thermodynamic properties can then be calculated from the partition function. For example, Marcelja calculated the pressure as a function of the area per polar head group for surface monolayers at a variety of temperatures. His results showed good qualitative agreement with experimental results for such systems. 

Calculation of Thermodynamic Functions from Molecular Properties The calculation methods for thermodynamic functions (entropy S, heat capacities Cp and Cy, enthalpy H, and therefore Gibbs free energy G) for polyatomic systems from molecular and spectroscopic data with statistical methods through calculation of partition functions and its derivative toward temperature are well established and described in reference books such as Herzberg s Molecular Spectra and Molecular Structure [59] or in the earlier work from Mayer and Mayer [7], who showed, probably for the first time in a comprehensive way, that all basic thermochemical properties can be calculated from the partition function Q and the Avagadro s number N. The calculation details are well described by Irikura [60] and are summarized here. Emphasis will be placed on calculations of internal rotations. 

The most fundamental starting point for any theoretical approach is the quantum mechanical partition function PF), and the fundamental connection between the partition function and the corresponding thermodynamic potential. Once we have a PF, either exact or approximate, we can derive all the thermodynamic quantities by using standard relationships. Statistical mechanics is a general and very powerful tool to connect between microscopic properties of atoms and molecules, such as mass, dipole moment, polarizability, and intermolecular interaction energy, on the one hand, and macroscopic properties of the bulk matter, such as the energy, entropy, heat capacity, and compressibility, on the other. 

The statistical approach to adsorption, which was developed largely by Fowler and Guggenheim and Hill, depends on representing the adsorbed species in terms of a simplified physical model for which the appropriate expression for the partition function may be derived. The thermodynamic properties are then obtained using the established relationships between the partition functions and the classical thermodynamic properties. A brief summary of some of the more important relationships is given in Appendix A. 

The partition function, which is simply the sum of all Boltzmann factors for each allowable state of the system, provides the bridge between classical and statistical thermodynamics. To derive the relationships between the partition function and the classical thermodynamic properties we consider a closed system and a canonical ensemble. 

In most cases, for polyatomic molecules the nuclear partition function is again neglected, because it usually has a very small effect on the overall thermodynamic properties of polyatomic molecules. (Indeed, the only reason why we had to consider it for diatomic molecules is because it imposes an obvious, measurable effect on various observations, like spectra and thermodynamic properties to be considered in section 18.8.) In the high-temperature limit, a linear polyatomic molecule has the same rotational partition function as a homonuclear diatomic molecule ... 

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