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... 

Butadiene, the simplest conjugated diene, has been the subject of intensive theoretical and experimental studies to understand its physical and chemical properties. The conjugation of the double bonds makes it 15 kJ/mole (3.6 kcal/mol) (13) more thermodynamically stable than a molecule with two isolated single bonds. The r-trans isomer, often called the trans form, is more stable than the s-cis form at room temperature. Although there is a 20 kJ/mole (4.8 kcal/mol) rotational barrier (14,15), rapid equiUbrium allows reactions to take place with either the s-cis or r-trans form (16,17). 

Experimental information about tire energy levels of molecules is obtained from spectroscopic studies, in the infra-red for the rotational states and in the ultra-violet for die vibrational and most of the dissociation energies. Some thermodynamic data are also obtained for the dissociation energies using mass spectroscopy. 

Electrocyclic reactions of 1,3,5-trienes lead to 1,3-cyclohexadienes. These ring closures also exhibit a high degree of stereospecificity. The ring closure is normally the favored reaction in this case, because the cyclic compound, which has six a bonds and two IT bonds, is thermodynamically more stable than the triene, which has five a and three ir bonds. The stereospecificity is illustrated with octatrienes 3 and 4. ,Z, -2,4,6-Octatriene (3) cyclizes only to cw-5,6-dimethyl-l,3-cyclohexadiene, whereas the , Z,Z-2,4,6-octa-triene (4) leads exclusively to the trans cyclohexadiene isomer. A point of particular importance regarding the stereochemistry of this reaction is that the groups at the termini of the triene system rotate in the opposite sense during the cyclization process. This mode... 

The influence of barriers on thermodynamic properties must have importance in determining the rates of various chemical reactions. It seems certain that the activated complex for many reactions will involve the possibility of restricted rotation and that the thermodynamic properties of the complex will therefore be in part determined by the magnitude of the barriers. Whereas at the moment there is no direct way of determining such barriers, any general principles obtained for stable molecules should ultimately be applicable to the activated state. One might then hope to be able to estimate the barriers and the reaction rates a priori. 

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... 

Tables 10.1, 10.2, and 10.3e summarize moments of inertia (rotational constants), fundamental vibrational frequencies (vibrational constants), and differences in energy between electronic energy levels for a number of common molecules or atoms/The values given in these tables can be used to calculate the rotational, vibrational, and electronic energy levels. They will be useful as we calculate the thermodynamic properties of the ideal gas.

The thermodynamic functions of primary interest in chemistry are Cp.m, Sm, and Gm-Ho.m- The translational, rotational, and vibrational contributions are summarized in Table 10.4.u We will not attempt to derive all the equations in this table but will do enough to show how it is done. 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

We are interested in understanding how this relative motion of two parts of the molecule contributes to its thermodynamic properties. It turns out that the thermodynamic treatment of this internal rotation depends upon the relative... 

As an example of the application of this procedure, Pitzer3 has calculated the thermodynamic properties for dimethylcadmium, including free rotation. At 298.15 K, he obtained the following result for the entropy ... 

Figure 10.10 Internal rotation contribution to the heat capacity of CH3-CCI3 as a function of temperature. Reprinted from K. S. Pitzer. Thermodynamics, McGraw-Hill, Inc., New York, 1995, p. 374. Reproduced with permission of the McGraw-Hill Companies.

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. 

R. E. Pennington and K. A. Kobe. Contributions of Vibrational Anharmonicity and Rotation-Vibration Interaction to Thermodynamic Functions". J. Client. Plus., 22. 1442-1447 (1954). 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

Table A4.5 summarizes the equations for calculating anharmonicity and nonrigid rotator corrections for diatomic molecules. These corrections are to be added to the thermodynamic properties calculated from the equations given in Table A4.1 (which assume harmonic oscillator and rigid rotator approximations).

For diatomic molecules, Bo is the rotational constant to use with equation (10.125), while BQ applies to equation (10.124). They are related by Bo — Bc — ja. The moment of inertia /o/(kg-m2) is related to Z o/(cm ) through the relationship /o = A/(8 x 10 27r2 b< ) with h and c expressed in SI units. For polyatomic molecules, /a, /b, and Iq are the moments of inertia to use with Table A4.1 where the rigid rotator approximation is assumed. For diatomic molecules, /o is used with Table A4.1 to calculate the thermodynamic properties assuming the rigid rotator approximation. The anharmonicity and nonrigid rotator corrections are added to this value. 

The following equations are used to calculate the anharmonicity and nonrigid rotator corrections to the thermodynamic properties of diatomic molecules. 

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