Statistical thermodynamics rotations

Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

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... 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

Determination of the Rotational Barrier in Ethane by Vibrational Spectroscopy and Statistical Thermodynamics 166... 

Fig. 4.7 Scheme of statistical thermodynamic calculations of ideal-gas entropy for the compounds without internal rotation... 

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. 

Cantor, R.S. Dill, K.A. Statistical thermodynamic theory for the melting of n-alkanes from their rotator phases. Macromolecules 1985, 18, 1875. 

The measured value of Kp is compared with the value calculated using statistical thermodynamics and vibrational-rotational spectroscopic information of the type obtained in Exp. 37. 

To relate these thermodynamic quantities to molecular properties and interactions, we need to consider the statistical thermodynamics of ideal gases and ideal solutions. A detailed discussion is beyond the scope of this review. We note for completeness, however, that a full treatment of the free energy of solvation should include the changes in the rotational and vibrational partition functions for the solute as it passes from the gas phase into solution, AGjnt. ... 

In the association process some degrees of freedom of the reacting system change their nature (from translation and rotation to internal motions). Statistical thermodynamics suggests us the procedures to be used in gas phase calculations application to processes in solution requires a careful analysis. The additional internal motions are in general quite floppy, and their separation from rotational motions of the whole C is a delicate task. 

Data analysis options include the preparation of isoenergy contour maps (CMAPS), linear energy vs. rotation angle plots (LINPLOT) and tables of local energy minima, statistical thermodynamic probabilities, and entropy terms to enable the reporting of "free" energies in addition to "conformational" energies. 

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. 

Equation 6.16 is consistent in general form with previous treatments [24,119,123-126] of the effects of crosslinks on Tg, and with expectations based on statistical thermodynamic theories of the glass transition [13-17]. The new concept here is the use of Nrot as a normalization factor, to multiply the value of n. The product n-Nrot is the average number of rotational... 

The equilibrium constants Kf are not measurable and we must resort to statistical thermodynamics to estimate these values theoretically. The partition function (Q) is a quantity with no simple physical significance but it may be substituted for concentrations in the calculation of equilibrium constants (Eqns. 4 and 5) [5], (It is assumed that there is no isotopic substitution in B.) Partition functions may be expressed as the product of contributions to the total energy from translational, rotational and vibrational motion (Eqn. 6). 

The third-law entropies, S°, of the two gases can be calculated by using standard formulas of statistical thermodynamics.62 At room temperature, the entropy is due exclusively to the translational and rotational components. Owing to its lower mass and moment of inertia, the absolute entropy of H2 (31.2 cal/mol-deg) is 14.6 cal/mol-deg lower than that for N2. If Eq. (7.17) is reexamined, it is clear that if the total entropies of the complexes in solution exactly canceled, the predicted entropy change would be 14.6 cal/mol deg. This is reasonably close to the average values obtained for the three complexes 11 4 cal/mol. deg. 

From elementary statistical thermodynamics we know that the equilibrium constant can be written in terms of the partition functions of the individual molecules taking part in a reaction. These quantities represent the sum over all energy states in the system—translational, rotational, vibrational, and electronic. The probability that a molecule will be in a particular energy state, f ,-, is given by the Boltzmann law,... 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

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