Diatomic molecule, heat capacity vibrational

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. 

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. 

Vibrational Partition Function/ The thermodynamic quantities for an ideal gas can usually be expressed as a sum of translational, rotational, and vibrational contributions (see Exp. 3). We shall consider here the heat capacity at constant volume. At room temperature and above, the translational and rotational contributions to are constants that are independent of temperature. For HCl and DCl (diatomic and thus linear molecules), the molar quantities are... 

ISe. Classical Calculation of Heat Capacities.— For a diatomic molecule two types of rotation are possible, as seen above, contributing RT per mole to the energy. Since there are two atoms in the molecule, i.e., n is 2, there is only one mode of vibration, and the vibrational energy should be RT per mole. If the diatomic molecules rotate, but the atoms do not vibrate, the total energy content E will be the sum of the translational and rotational energies, i.e., RT + RT = RT, per mole hence,... 

SO that the heat capacity at constant volume should be about 5 cal. deg. mole The same result would be obtained if there is vibration of the atoms in the diatomic molecule, but no rotational motion. If, however, the molecule rotates and the atoms also vibrate, the energy content should be RT + RT + RT i.e., RTj per mole then... 

K the value of RT is approximately 2.5 kJ mol"1,) However Aevib is usually much greater than kT and under these circumstances vibrations do not contribute significantly to heat capacity (Fig. 9.7). Thus for nitrogen at room temperature the rotational contribution is approximately iiTand the vibrational contribution almost zero. Thus Cv 5/2)1 and y w 7/5 = 1.40 as opposed to the classical prediction of 1,29. For iodine the vibrational spacings are closer (Table 9.1) and we would predict y 1.29 in accord with the classical value. If the temperature is varied the heat capacity of a diatomic or polyatomic gas may show steps as the contributions from rotations and vibrations rise as the energy separations become comparable to kT, The positions of the steps depend on the moments of inertia and the vibrational frequencies of the molecules. 

There is another amazing aspect in the discovery of argon beyond its total chemical inertness. Rayleigh and Ramsay reported measurements of the speed of sound in argon that indicated that the ratio of its heat capacity at constant pressure to that at constant volume (Cp/Cy) was too high for a diatomic molecule. The only other similar observation was for monatomic mercury (vapor) whose atomic weight was known since it forms compounds. At constant volume, heat added to a diatomic molecule such as N2 goes into both movement of the molecule (translation) as well as vibration of the bond. In a monatomic substance there is no bond vibration and, thus, less capacity to absorb heat. 

Qualitatively sketch the heat capacity of an ideal gas of diatomic molecules as a function of temperature. Indicate the characteristic temperatures (in terms of vibrational frequency, moment of inertia, and so on) where various degrees of freedom begin to contribute. 

For a diatomic molecule, the vibrational contribution to the heat capacity can be calculated using statistical mechanics, assuming that the vibrations are well modeled by a harmonic oscillator ... 

The quantity , = hcv/k is called the vibrational temperature, wha-e h is Planck s constant, c is the speed of light, k is Boltzmann s constant, and v is the vibrational frequency in wave numbers (cm ). The vibrational frequencies for F2 and I2 are 919 and 214 cm, respectively. Use the formula above, together with the equipartition values for the translational and rotational contributions, to calculate the constant-pressure heat capacity for these two diatomic molecules. Compare your results with the expraimental values given in Table 7.4. 

If we sum the three contributions to calculate the value for the heat capacity at constant volume, as the characteristic temperatures of rotation are often lower than the Einstein temperature (see Table 7.3), the variation in the heat capacity, for example of a diatomic molecule, with temperature takes the form of the curve in Figure 7.12(c). At low temperatures, the only contribution is that of translation, given by 3R/2. Then if the temperature increases, the contribution of rotation, is added according to the curve in Figure 7.12(a) until the limiting value of this contribution is reached, then the vibrational contribution is involved until the molecule dissociates which makes the heat capacity become double that of the translational contribution of monoatomic molecules. The limiting value of the vibrational contribution is sometimes never reached. This explains why the values calculated in Table 7.2 are too low if we do not take into account the vibration and too high in the opposite case. 

The temperature of the system determines the validity of each of the approximations made that impact the value of the heat capacity of diatomic molecules. At high temperatures, vibrations are important and the heat capacity is best described by Eq. 7.50. For lower temperatures, the rigid rotor approximation becomes valid and Eq. 7.36 captures the heat capacity. The temperature thresholds depend on the molecule type. In Fig. 7.4, the heat capacity of a diatomic molecule is shown as a function of temperature. 

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