Heat capacity polyatomic

In addition to the intermolecular potential, there is an intramolecular portion of the Helmholtz free energy. Cheetah uses a polyatomic model to account for this portion including electronic, vibrational, and rotational states. Such a model can be expressed conveniently in terms of the heat of formation, standard entropy, and constant-pressure heat capacity of each species. 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

From a molecular viewpoint, we know that heat capacity is closely connected to internal modes of molecular vibration. According to the classical equipartition theorem (Sidebar 3.8), a nonlinear polyatomic molecule of Aat atoms has ftmodes = 3Aat — 6 independent internal modes of vibration, each of which would contribute equally to heat capacity... 

For a monatomic gas, where the heat capacity involves only translational energy, V is independent of sound oscillation frequency (except at ultra-high frequencies, where a classical visco-thermal dispersion sets in). For a relaxing polyatomic gas this is no longer so. At sound frequencies, where the period of the oscillation becomes comparable with the relaxation time for one of the forms of internal energy, the internal temperature lags behind the translational temperature throughout the compression-rarefaction cycle, and the effective values of CT and V in equation (3) become frequency dependent. This phenomenon occurs at medium ultrasonic frequencies, and is known as ultrasonic dispersion. It is accompanied by... 

Alternatively, when process (3) is slower than (4) or (5), but faster than (1) or (2), A will again relax by the route (3) followed by (4) or (5), but now (3) will be rate determining. This will give a linear variation of 1// A with x. B will relax independently, and more rapidly, via (4) and (3), with linear dependence of 1// B on x. There will thus be a double relaxation phenomenon with two relaxation times, PA involving only the vibrational heat capacity of A, and / B only that of B, both showing linear concentration dependence. This mechanism is analogous to the relaxation behaviour discussed in Section 3.1 for pure polyatomic gases, which show double dispersion because vibration-vibration transfer between modes is slower than vibration-translation transfer from the lowest mode. 

Heat capacities of polyatomic molecules can be explained by the same arguments. As discussed in Chapter 3, bond-stretching vibrational frequencies can be over 100 THz. At room temperature k T heat capacity (which explains why most diatomics give cv % 5R/2, the heat capacity from translation and rotation alone). Polyatomic molecules typically have some very low-frequency vibrations, which do contribute to the heat capacity at room temperature, and some high-frequency vibrations which do not. 

When actual data are not available, a useful approximate rule for ordinary temperatures and pressures, is to take y as 1.67 for monatomic gases, 1.40 for diatomic gases, 1.30 for simple polyatomic gases, such as water, carbon dioxide, ammonia and methane. It may be noted that the heat capacity ratio for hydrogen gas increases at low temperatures toward the vfdue for a monatomic gas. This matter will be explained in Chapter VI. 

Van t Hoff and Kekule were not alone in recognizing the inherently dynamic character of molecules—relatively few of their contemporaries were "naive ball-and-stick guy[s]." The problem was how to use that awareness to solve chemical problems. On the one hand, chemists were not able to derive very much of practical value from treating molecules as dynamic objects for example, the kinetic theory of gases wasn t even capable of predicting correct heat capacity ratios for polyatomic molecules (Sackur, 1917, pp. 154-166). On the other hand, a large body of previously confusing experimental data readily made sense if one treated molecules as more or less rigid objects. 

Whereas monatomic molecules can only possess translational thermal energy, two additional kinds of motions become possible in polyatomic molecules. A linear molecule has an axis that defines two perpendicular directions in which rotations can occur each represents an additional degree of freedom, so the two together contribute a total of 1/2 R to the heat capacity. For a non-linear molecule, rotations are possible along all three directions of space, so these molecules have a rotational heat capacity of 3/2 R. Finally, the individual atoms within a molecule can move relative to each other, producing a vibrational motion. A molecule consisting of N atoms can vibrate in 3N-6 different ways or modes1. For mechanical reasons that we cannot go into here, each vibrational mode contributes R (rather than 1/2 R) to the total heat capacity. 

When a monatomic gas absorbs heat, all of the energy ends up in translational motion, and thus goes to increase its temperature. In a polyatomic gas, by contrast, the absorbed energy is partitioned among the other kinds of motions since only the translational motions contribute to the temperature, the temperature rise is smaller, and thus the heat capacity is larger. 

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