Statistical thermodynamics enthalpy

TINAG/BRI] Nagarajan, G., Brinkley, D. C., Statistical thermodynamics Enthalpy, free energy, entropy, and heat capacity of some hexafluorides of octahedral symmetry, Z Naturforsck, 26A, (1971), 1658-1666. Cited on page 164. 

To simplify notation for these two terms let 2f0[G3(MP2)] s E0 and G3MP2 Enthalpy = //29g. The thermal correction to the enthalpy (TCH), converting energy at 0 K to enthalpy at 298, (H29% -E0 = -78.430772-(—78.4347736) = 0.0040016 h) is a composite of two classical statistical thermodynamic enthalpy changes for translation and rotation, and a quantum harmonic oscillator term for the vibrational energy. 

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... 

There is also no significant influence of statistic thermodynamical calculations on the reaction parameters. That can be seen in the Tables 3 and 4. In Table 4 the calculated reaction enthalpies and free reaction enthalpies are faced with experimental values estimated by means of thermochemical methods. 

Quantum-chemical calculations now can provide values of enthalpies of formation with a precision and accuracy comparable with thermochemical values and those calculated from statistical thermodynamics. The basis for these calculations is beyond the scope of this text, but it is interesting to observe some values calculated in this way for comparison with other values in Tables 4.3-4.5. The data in Table 4.6 were obtained by a method called Gaussian-3 (G3) [5]. 

The partition function is the central feature of statistical thermodynamics. From the partition function the various thermodynamic variables such as entropy, enthalpy, and free energy may be evaluated. It is also possible, in principle, to deduce the equation of state for a system from the partition function. 

It is useful to be able to express the pressure in terms of the partition function. (The result will be used subsequently in the statistical thermodynamic expression derived for the enthalpy.)... 

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. 

Table 7.3 The C-C bond energy of ethane by HF, MP2(fc), and DFT (B3LYP, M06, and TPSS) calculations, at 0 and 298 K. The basis set is 6-31G. Standard, tabulated bond energies are for dissociation at 298 K. Bond energy = 2(CH3 radical enthalpy) - (CH3CH3 enthalpy). For the radical the unrestricted method (UHF etc.) was used. For the 0 K dissociation enthalpy, the HF and MP2 calculations use energies corrected for ZPE, with the ZPE itself corrected by a factor of 0.9135 (HF) or 0.9670 (MP2) [77]. The 0 K dissociation enthalpy for the DFT calculations is uncorrected for ZPE, and the 298 K dissociation enthalpy is from standard statistical thermodynamics methods [79]. The experimental C-C energy of ethane has been reported as 90.1 0.1 kcal mol-1, i.e. 377 0.4 kJ mol-1 [80]. Calculations are by the author...

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. 

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. 

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. 

The probabilistic model of macromolecular association introduced in the previous section, for the case of large a and n B, may be recast into the formal language in terms of statistical thermodynamics. Recall from Chapter 1 that the chemical potential of a species has two terms, a structural energy (enthalpy) term and a concentration/entropy term ... 

Experimental determination of excess molar quantities such as excess molar enthalpy and excess molar volume is very important for the discussion of solution properties of binary liquids. Recently, calculation of these thermodynamic quantities becomes possible by computer simulation of molecular dynamics (MD) and Monte Carlo (MC) methods. On the other hand, the integral equation theory has played an essential role in the statistical thermodynamics of solution. The simulation and the integral equation theory may be complementary but the integral equation theory has the great advantage over simulation that it is computationally easier to handle and it permits us to estimate the differential thermodynamic quantities. 

The thermodynamic functions of the solubilities of many gases in molecular liquids at room temperature have been tabulated [22]. The enthalpy of sorption is negative (exothermic) if the sorption energy exceeds the energy needed to make a hole of molecular size in the polymer or molecular liquid, and positive (endothermic) otherwise. In rough empirical correlations [21], S and AHS are usually related to the boiling temperature, critical temperature, or Lennard-Jones 6-12 potential energy parameter of the gas molecule. AHS can also be modeled atomistically [23-25], and by statistical thermodynamic equation-of-state theories (Section 3.E and Ref. [26]). 

Jain, R. K., Simha, R., Balik, C. M Statistical thermodynamics of fluid mixtures-enthalpies and equations of state, Indian Journal of Pure Applied Physics, 22(ll),pp. 651-657(1984). 

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. 

Statistical thermodynamics make it possible to demonstrate that the Fermi energy Ep is equal to the partial derivative of the free enthalpy G ... 

Statistical thermodynamics provides exact formulas for the calculation of the fundamental quantities of heat capacities, enthalpies and entropies, provided that certain assumptions are valid. Among these assumptions are ... 

The early theories of phase separation are of the mean-field, cell, or cell-hole lattice (statistical thermodynamics) type. The theory takes into account the configurational entropic and enthalpic contributions, but since these are weak, the effects on miscibility are not as predictable as that for other PO blends. Nevertheless, the enthalpy as a difference of the solubility parameters well correlate with the experimental data being independent on SCBD and SCB if SCB < 5/100 C. This observation is unexpected, since the miscibility was reportedly controlled by entropy, e.g., chain stiffening led to phase separation. The newer theoretical models attempt incorporating the model macromolecular chain stmctures using either mathematical modeling via MC, molecular... 

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