Statistical thermodynamics internal rotation

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. 

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... 

So far, we have used the statistical approach to calculate the thermodynamic properties of an ideal gas. Translational, rotational, vibrational, and electronic contributions were included, along with internal rotations where applicable. 

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

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. 

Quantum chemistry applies quantum mechanics to problems in chemistry. The influence of quantum chemistry is evident in all branches of chemistry. Physical chemists use quantum mechanics to calculate (with the aid of statistical mechanics) thermodynamic properties (for example, entropy, heat capacity) of gases to interpret molecular spectra, thereby allowing experimental determination of molecular properties (for example, bond lengths and bond angles, dipole moments, barriers to internal rotation, energy differences between conformational isomers) to calculate molecular properties theoretically to calculate properties of transition states in chemical reactions, thereby allowing estimation of rate constants to understand intermolecular forces and to deal with bonding in solids. 

J. Chao, R. C. Wilhoit and B. J. Zwolinski, Ideal gas thermodynamic properties of ethane and propane , J. Phys. Chem. Ref. Data, 2, 427 (1973). Review and evaluation of structural parameters (including vibrational frequencies and internal rotation properties) tabulation of thermodynamic properties [C°, S°, H° — H°), (H° — H )/T, - G°-Hl)/T, AfG°,AfH°, logK ] for 0< T (K)< l500 calculated by statistical thermodynamic methods [rigid-rotor harmonic oscillator (RRHO) approximation]. 

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. 

Free internal rotation has been assumed in thermodynamic calculations of nitromethane, > toluene, aaa-trifluorotoluene, / -fluorotoluene, 2-picoline, 3-picoline, and but-2-yne. For all these molecules satisfactory agreement between calorimetric data and statistical calculations could be obtained only by assuming the internal rotation to be essentially unrestricted. 

Free Internal Rotation of Several Symmetric Tops. Extension of statistical thermodynamic calculations from internal rotation of one top to several tops is straightforward. The rotational partition function may be expressed by... 

Thermodynamic contributions from the internal rotation of several symmetric tops may be readily calculated by appropriate summation of terms in Table 4. Few reliable calculations, however, have been reported. Thermodynamic properties of propane and several methyl-substituted benzenes have been reported, for example, but subsequent more accurate work has shown the necessity for considering that the internal rotation may be restricted. " Although the subsequent calculations for m-xylene and p-xylene used 6-fold internal rotation barriers of 2.1 to 3.1 kJ mol", more recent statistical calculations for toluene employing the presence of free rotation suggest that internal rotation in the two xylenes may be effectively unrestricted. 

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

Other methods treat the molecules as classical objects, interacting through forces of various kinds and using the formalism of statistical thermodynamics to obtain the desired quantities as averages. The chief of these methods are Monte Carlo (MC), so called because it depends on (simulated) chance in the form of random nnmbers in evalnating average properties, and molecular dynamics (MD), which nses the laws of motion to explicitly represent the evolution in time of an assembly of molecnles. The desired properties are obtained as time averages. There are also varions hybrid methods, in which, for instance, the translation and rotation of molecnles are assnmed to behave classically bnt the internal vibrations are treated qnantnm mechanically or the solnte is treated quantnm mechanically but the solvent molecnles are treated classically, and methods in which solute molecules are treated as neither in vacno nor snrronnded by other molecnles, but as in a cavity in a continnons dielectric. 

Beginning in 1937, I had been very much interested in the thermodynamic properties of various hydrocarbon molecules and hence of those substances in the ideal gas state. This arose out of work with Kemp in 1936 on the entropy of ethane ( 1) which led to the determination of the potential barrier restricting internal rotation. With the concept of restricted internal rotation and some advances in the pertinent statistical mechanics it became possible to calculate rather accurately the entropies of various light hydrocarbons (2). Fred Rossini and I collaborated in bringing together his heat of formation data and my entropy and enthalpy values to provide a complete coverage of the thermodynamics of these hydrocarbons in the ideal gas state O). As an aside I cite the recent paper of Scott ( ) who presents the best current results on this topic. 

First, the literature should be searched for spectroscopic data, and if these are sufficient, the thermodynamic properties can be calculated by statistical mechanics formulas. McBride and Gordon s program (1967) is recommended for this purpose. The latest verson, PAC3, includes, among many possible calculation methods, an accurate calculation method for internal rotation contributions, which are important when organic species are involved, and a subroutine which automatically calculates the coefficients of the NASA polynomials. Wilhoit s extrapolation method was recently included in the code. 

The internal partition function for molecules having inversion may be factored, to a good approximation, into overall rotational and vibrational partition functions. Although inversion tunnelling results in a splitting of rotational energy levels, the statistical weights are such that the classical formulae for rotational contributions to thermodynamic functions may be used. The appropriate symmetry number depends on the procedure used to calculate the vibrational partition function. 

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