Internal energy rotational contribution

Information about the structure of a molecule can frequently be obtained from observations of its absorption spectrum. The positions of the absorption bands due to any molecule depend upon its atomic and electronic configuration. To a first approximation, the internal energy E oi a, molecule can be regarded as composed of additive contributions from the electronic motions within the molecule (Et), the vibrational motions of the constituent atoms relative to one another E ), and the rotational motion of the molecule as a whole (Ef) ... 

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 A4.6 gives the internal rotation contributions to the heat capacity, enthalpy and Gibbs free energy as a function of the rotational barrier V. It is convenient to tabulate the contributions in terms of VjRTagainst 1/rf, where f is the partition function for free rotation [see equation (10.141)]. For details of the calculation, see Section 10.7c.

The contribution of translational and rotational motion to the internal energy can be estimated from the temperature. 

A linear molecule, such as any diatomic molecule, carbon dioxide, and ethyne (acetylene, HC=CH), can rotate about two axes perpendicular to the line of atoms, and so it has two rotational modes of motion. Its average rotational energy is therefore 2 X jkT = kT, and the contribution to the molar internal energy is NA times this value ... 

A nonlinear molecule, such as water, methane, or benzene, can rotate about any of three perpendicular axes, and so it has three rotational modes of motion. The average rotational energy of such a molecule is therefore 3 X jkT = ]kT. The contribution of rotation to the molar internal energy of a gas of nonlinear molecules is therefore... 

Internal energy is stored as molecular kinetic and potential energy. The equipartition theorem can be used to estimate the translational and rotational contributions to the internal energy of an ideal gas. 

The molar heat capacities of gases composed of molecules (as distinct from atoms) are Higher than those of monatomic gases because the molecules can store energy as rotational kinetic energy as well as translational kinetic energy. We saw in Section 6.7 that the rotational motion of linear molecules contributes another RT to the molar internal energy ... 

Thus, a molecule may exist in many states of different energy. The internal energy in a certain state may be considered to be made up of contributions from rotational energy, E ou vibrational energy, E vib and electronic energy, E i as described by equation (5.2) ... 

The thermal internal energy function calculated at 298.15 K [E — 0] is also listed in Table 8.1. The translational and rotational contributions are found using Eqs. 8.80 and 8.82, respectively. The vibrational contributions (Eq. 8.84) are much less, as expected. Mode 2 makes a significant contribution to the total internal energy at this temperature. Vibrational modes 5 and 6 also make smaller, but nonnegligible, contributions. The electronic contribution was calculated directly from Eq. 8.76. Through application of Eq. 8.118, the total enthalpy is [H — Ho] - 11146.71 J/mole. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

The falloff curve of the quantum yield is explained by the contribution of the internal (mostly rotational) energy to supplement the incident photon energy (550,810), (see Section 1-4.3 for details). Above 4358 A a small ( 0.01) but significant yield of NO was observed, which is attributed to reactions of electronically excited N02 (NO ) by Jones and Bayes (550). 

Recent advances in experimental techniques, particularly photoionization methods, have made it relatively easy to prepare reactant ions in well-defined states of internal excitation (electronic, vibrational, and even rotational). This has made possible extensive studies of the effects of internal energy on the cross sections of ion-neutral interactions, which have contributed significantly to our understanding of the general areas of reaction kinetics and dynamics. Other important theoretical implications derive from investigations of the role of internally excited states in ion-neutral processes, such as the effect of electronically excited states in nonadiabatic transitions between two potential-energy surfaces for the simplest ion-molecule interaction, H+(H2,H)H2+, which has been discussed by Preston and Tully.2 This role has no counterpart in analogous neutral-neutral interactions. 

Rotational energy contributes to the internal energy of a diatomic molecule, and classically any rotational speed is possible. We will return to rotational properties in Chapter 8, when we discuss quantum mechanics, which imposes restrictions on the rotational energy we will find that transitions between allowed rotational states let us measure bond lengths or cook food in microwave ovens. 

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