Diatomic molecule, heat capacity

There is a well-known precedent for the difference in the results predicted by stiff and rigid models. The constant-volume heat capacity of a diatomic ideal gas is predicted to be jkT per molecule for a stiff classical model and jkT for a classical rigid rotor. A quanmm mechanical analysis of a diatomic gas yields the... 

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... 

Table III contains the bond energies of some gaseous diatomic molecules. The heat capacities (Cp) for such molecules all range from near 7 cal./mole-°K. at room temperature to 9 cal./mole-°K. at 3700°K.

Vapor pressure data, as reviewed by Brewer and Kane 37), can be represented by assuming the tetratomic molecule to be the gaseous species, with a heat of sublimation at 298 K. of 34,500 cal./mole, an entropy at 298 K. of 75.00 e. u., and a reasonable estimate of the heat capacity. According to this view, there is no appreciable concentration of the diatomic species in the vapor at saturation pressure below 1000 K., which sets a lower limit of about 48,000 cal./mole for the heat of sublimation at 298 K. for the diatomic gas. Comparison with the bond energies of P4 and Sb4 gives support to this value. 

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... 

For all diatomic molecules, with the exception of hydrogen below 300 K and of deuterium below 200 K, a considerable simplification is possible for temperatures above the very lowest. In the first place, the nuclear spin factor may be ignored for the present (see, however, 24j), since it is independent of temperature and makes no contribution to the heat capacity. The consequence of the nuclei being identical is then allowed for by introducing a s]rmmetry number a, giving the number of equivalent epatial orienta-turns that a tnolecule can occupy as a result of simple rotation. The value of F is 2 for symmetrical diatomic molecules, and for unsymmetrical molecules... 

The derived values of the heat of formation of AuSe(g) agree reasonably well. Since the entropy and heat capacity of a diatomic molecule calculated from estimated molecular parameters are expected to be quite accurate, the review adopts the mean of the two results and obtains ... 

This value is adopted since the heat capacity and entropy calculated from estimated molecular parameters can be expected to be quite accurate for a diatomic molecule. The additional uncertainty originating from the recalculation to the standard temperature is hence moderate in relation to the uncertainty in the reaction enthalpy. The result is included in Appendix E since it was calculated with non-TDB auxiliary data. 

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. 

The molar heat capacity of a diatomic molecule is 29.1 J/K mol. Assuming the atmosphere contains only nitrogen gas and there is no heat loss, calculate the total heat intake (in kilojoules) if the atmosphere warms up by 3°C during the next 50 years. Given that there are 1.8 X 10 ° moles of diatomic molecules present, how many kilograms of ice (at the North and South Poles) will this quantity of heat melt at 0°C (The molar heat of fusion of ice is 6.01 kJ/mol.)... 

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