Diatomic molecules rotational temperature

We see from (4.104) that, although the vibrational quantum number is not changing, the frequency of a pure-rotational transition depends on the vibrational quantum number of the molecule undergoing the transition. (Recall that vibration changes the effective moment of inertia, and thus affects the rotational energies.) For a collection of diatomic molecules at temperature T, the relative populations of the energy levels are given by the Boltzmann distribution law the ratio of the number of molecules with vibrational quantum number v to the number with vibrational quantum number zero is... 

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. 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

The rotational temperature is defined as the temperature that describes the Boltzmann population distribution among rotational levels. For example, for a diatomic molecule, this is the temperature in Equation (5.15). Since collisions are not so efficient in producing rotational cooling as for translational cooling, rotational temperatures are rather higher, typically about 10 K. 

Iodine vapor is characterized by the familiar violet color and by its unusually high specific gravity, approximately nine times that of air. The vapor is made up of diatomic molecules at low temperatures at moderately elevated temperatures, dissociation becomes appreciable. The concentration of monoatomic molecules, for example, is 1.4% at 600°C and 101.3 kPa (1 atm) total pressure. Iodine is fluorescent at low pressures and rotates the plane of polarized light when placed in a magnetic field. It is also thermoluminescent, emitting visible light when heated at 500°C or higher. 

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

If, now, we continue warming the substance sufficiently, we will reach a point at which the kinetic energies in vibration, rotation, and translation become comparable to chemical bond energies. Then molecules begin to disintegrate. This is the reason that only the very simplest molecules—diatomic molecules—are found in the Sun. There the temperature is so high (6000°K at the surface) that more complex molecules cannot survive. 

Upon absorption of light of an appropriate wavelength, a diatomic molecule can undergo an electronic transition, along with simultaneous vibrational and rotational transitions. In this case, there is no restriction on Au. That is, the selection rule Av = +1 valid for purely vibrational and vibrational-rotational transitions no longer applies thus numerous vibrational transitions can occur. If the molecule is at room temperature, it will normally be in its lower state, v" = 0 hence transitions corresponding to v" = 0 to v = 0,... 

R. W. Patch. Theory of pressure induced vibrational and rotational absorption of diatomic molecules at high temperatures. J.Q.S.R.T. 11 1311, 1971. 

For any homonudear diatomic molecule whose nuclei have nonzero spin, it should, in principle, be possible to isolate a modified form with only even-numbered rotational levels populated. (For /=0, half the rotational levels do not exist.) However, only for H2 and D2 has this been achieved. The small moments of inertia of these light molecules give a relatively large spacing between 7 = 0 and 7= 1 rotational levels, so that we can get nearly all the molecules into the 7 = 0 level at a temperature above the freezing point of the substance. 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

IR spectra of the fundamental vibrational band of small gaseous diatomic molecules, such as CO and NO, contain a large number of absorption lines that correspond to these vibrational-rotational energy transitions. Since many different rotational levels can be populated at ambient temperature, many different transitions at different energies may occur (Fig. 1). Vibrational-rotational lines are evident only in gas-phase spectra collected at sufficiently high resolution. These lines are not resolved in condensed-phase spectra because of frequent collisions between molecules hence, condensed-phase spectra are characterized by broad absorption bands occurring at the vibrational transition energies. 

The mean amplitude of molecular vibration can be calculated from the vibrational frequency [74] and vice versa. For the frustrated rotation of an upright diatomic molecule adsorbed on heavy substrate atoms and a vibrational mode which is doubly degenerate, the mean square amplitude at equilibrium temperature Ts is given by [68]... 

Table IX-2.—Characteristic Temperature for Rotation, Diatomic Molecules...

It is important to note in Figure 1 that both curves show a decrease with temperature, and it should be possible to fit B smoothly onto A by multiplying by a suitable scale factor, possibly as shown by the dashed line. To explain the data shown in Figure 1 the temperature dependence of fs/fg is needed. The rotational partition function for a diatomic molecule that is free is... 

Figure 3.25 Rotational collision number for several diatomic molecules at room temperature as a function of AEr (see text), according to Bauer and Kosche.

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