Rotational motion, diatomic gases

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

The most direct application of particle-on-a-sphere result is to the rotational motion of diatomic molecules in a gas. As with vibrations (see Section 3.2), the real situation looks a little more complicated, but can be solved in a similar way. A molecule actually rotates about its centre of mass the coordinates 8 and can be used to define its direction in space. If we replace the mass in Schrodinger s equation by the reduced mass given by eqn 3.22, and let r be the bond length, then the moment of inertia is... 

It is known that the solution molecule interacts with the solute molecules via so called polarization or van der Waals forces. These forces are also active in the formation and binding of nobel-gas-diatom complexes. This kind of interaction is so weak that the vibrational and rotational motion of the diatomic in the van der Waals complex is similar to that of the free diatomic. A van der Waals system of the kind AB — X is thus considered to be a good candidate as model system for these studies of solute - solvent interaction. 

Another, more quantitative way to view the distinction among these phases of matter is provided by the time-dependence of translational and rotational motions. Lets briefly consider the rotational motion of a diatomic molecule as an example. As the molecule rotates freely in the gas phase, the path followed by one of the nuclei describes a circle around the center of mass. If the molecule collides with another, the orientation of the rotational motion is likely to change, which breaks the circular... 

In addition to translational and electronic motion, a diatomic gas has rotational and vibrational motion. To a good approximation the energy is a sum of four separate terms, as in Eq. (22.2-37) ... 

Infrared spectroscopy has broad appHcations for sensitive molecular speciation. Infrared frequencies depend on the masses of the atoms iavolved ia the various vibrational motions, and on the force constants and geometry of the bonds connecting them band shapes are determined by the rotational stmcture and hence by the molecular symmetry and moments of iaertia. The rovibrational spectmm of a gas thus provides direct molecular stmctural information, resulting ia very high specificity. The vibrational spectmm of any molecule is unique, except for those of optical isomers. Every molecule, except homonuclear diatomics such as O2, N2, and the halogens, has at least one vibrational absorption ia the iafrared. Several texts treat iafrared iastmmentation and techniques (22,36—38) and thek appHcations (39—42). 

For a diatomic or polyatomic ideal gas, Cy is greater than j R, because energy can be stored in rotational and vibrational motions of the molecules a greater amount of heat must be transferred to achieve a given temperature change. Even so, it is still true that... 

According to this a monatomic gas, with its three degrees of freedom for linear motion, must have the molecular heat R this is, of course, true, and is always rightly reckoned as one of the finest results of the kinetic theory. Diatomic rigid molecules have five, triatomic rigid molecules six degrees of freedom, since in the first case there are two, in the last case three, more degrees of freedom of rotation. As the above table shows, there occurs in many cases a remarkable approximation to the values... 

The reaction considered in Example 10.7 involves 3 moles of hydrogen gas on the reactant side and 3 moles of water vapor on the product side. Would you expect AS to be large or small for such a case We have assumed that AS depends on the relative numbers of molecules of gaseous reactants and products. On the basis of that assumption, AS should be near zero for the present reaction. However, AS is large and positive. Why The large value for AS results from the difference in the entropy values for hydrogen gas and water vapor. The reason for this difference can be traced to the difference in molecular structure. Because it is a nonlinear, triatomic molecule, H2O has more rotational and vibrational motions (see Fig. 10.12) than does the diatomic... 

Mobility in this region is dominated by short-time motion, typically < 2 ps. After that time, all correlation of molecular motion is lost due to frequent collisions with the cavity walls. The center-of-mass velocity autocorrelation function of the penetrant exhibits typical liquid-like behavior with a negative region due to velocity reversal when the penetrant hits the cavity wall [59]. This picture has recently been confirmed by Pant and Boyd [62] who monitored reversals in the penetrant s travelling direction when it hits the cavity walls. The details of the velocity autocorrelation function are not very sensitive to the force-field parameters used. On the other hand, the orientational correlation function of diatomic penetrants showed residuals of a gas-like behavior. Reorientation of the molecular axis does not have the signature of rotational diffusion, but rather shows some amount of free rotation with rotational correlation times of the order of a few tenths of a picosecond, although dependent in value on the Lennard-Jones radii of the penetrant s atoms. 

Many of the ideas that are essential to understanding polyatomic electronic spectra have already been developed in the three preceding chapters. As in diatomics, the Born-Oppenheimer separation between electronic and nuclear motions is a useful organizing principle for treating electronic transitions in polyatomics. Vibrational band intensities in polyatomic electronic spectra are frequently (but not always) governed by Franck-Condon factors in the vibrational modes. The rotational fine structure in gas-phase electronic transitions parallels that in polyatomic vibration-rotation spectra (Section 6.6), except that the rotational selection rules in symmetric and asymmetric tops now depend on the relative orientations of the electronic transition moment and the principal axes. Analyses of rotational contours in polyatomic band spectra thus provide valuable clues about the symmetry and assignment of the electronic states involved. 

We will apply this expression only to a monatomic gas. For a diatomic or polyatomic gas the thermal conductivity is more complicated, since it is not an adequate approximation to assume that rotational and vibrational motions are equilibrated at the upper and lower planes. 

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