Statistical thermodynamics Gibbs entropy function

Classical thennodynamics deals with the interconversion of energy in all its forms including mechanical, thermal and electrical. Helmholtz [1], Gibbs [2,3] and others defined state functions such as enthalpy, heat content and entropy to handle these relationships. State functions describe closed energy states/systems in which the energy conversions occur in equilibrium, reversible paths so that energy is conserved. These notions are more fully described below. State functions were described in Appendix 2A however, statistical thermodynamics derived state functions from statistical arguments based on molecular parameters rather than from basic definitions as summarized below. 

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

Figure 11. (a) The calculated partial molar entropy of oxygen (sQJ and (b) the calculated partial molar enthalpy of oxygen (fto2) as a function of 8 for La02Sr08Fe0 55Tio4503 s. Symbols are calculated by the Gibbs-Helmholtz equation. Lines correspond to the partial molar quantities calculated by statistical thermodynamics. 

Based upon experimentally observed spectroscopic data, statistical thermodynamic calculations provide thermodynamic data which would not be obtained readily from direct experimental measurements for the species and temperature of interest to rocket propulsion. If the results of the calculations are summarized in terms of specific heat as a function of temperature, the other required properties for a particular specie, for example, enthalpy, entropy, the Gibb s function, and equilibrium constant may be obtained in relation to an arbitrary reference state, usually a pressure of one atmosphere and a temperature of 298.15°K. Or alternately these quantities may be calculated directly. Significant inaccuracies in the thermochemical data are not associated generaUy with the results of such calculations for a particular species, but arise in establishing a valid basis for comparison of different species. 

In modem physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity... 

In view of the formal identity of the expressions for the Gibbs entropy of quantum and classical isobaric-isothermal ensembles, all the previously made formal connections between the thermodynamics and the statistical mechanics of closed, isothermal classical systems with variable volttme and fixed pressure also apply to closed, isothermal qrrantum systems with variable vol-mne and fixed pressttre. [See Eqs. (86)-(91b).] One need only replace the classical ensemble averages by qrrantrrm ensemble averages and reinterpret the classical isobaric-isothermal partition function as a qrrantum isobaric-isothermal partition function. 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

Temperature (T), pressure (P), partial specific Gibbs function (G), and entropy (S) for a cell in a nonequilibrium state depend upon the internal energy (U) and the mass units (m,) of the molecular species V exactly in the same manner as in an equilibrium situation. In other words each cell of a system in nonequilibrium is treated exactly in the same manner (statistically as well as thermodynamically) as if equilibrium exists in each of these cells. 

Figure 13 Thermodynamic parameters for the interaction between glucose oxidase and n-dodecyltrimethylammonium bromide (DTAB) as a function of the number of moles of DTAB bound per glucose oxidase molecule (v). AG- ( ), AH (A), and ASp (V) are the Gibbs energy, enthalpy, and entropy per mole of DTAB bound. The symbols and correspond to AG , and TASV corrected for statistical effects. (Data taken from Ref. 88.)...

The molecular properties, such as geometry, vibrational frequencies, and rotational constants, are needed to compute thermodynamic properties such as enthalpy, entropy, and Gibbs free energy through calculation of the partition functions of the substances using statistical mechanics methods. 

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 entropy theory is the result of a statistical mechanical calculation based on a quasi-lattice model. The configurational entropy (S, ) of a polymeric material was calculated as a function of temperature by a direct evaluation of the partition function (Gibbs and Di Marzio (1958). The results of this calculation are that, (1) there is a thermodynamically second order liquid to glass transformation at a temperature T2, and 2), the configurational entropy in the glass is zero i.e. for T > T2, => 0 as T... 

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