Diatomic molecules vibrational 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. 

This is an extremely simple expression for a partition function. Table 18.1 lists a few 6y values for some diatomic molecules. At temperatures well above 6y for each gas-phase molecule, the vibrational partition function is given simply by equation 18.18. At temperatures near 0y or lower, the more complete expression in equation 18.17 must be used. (But be careful For some of the molecules listed, the stable phase is not the gas phase at T < 0 )... 

Make a plot of Cy as a function of temperature for an ideal gas composed of diatomic molecules vibrating as a harmonic oscillator according to the energy level expression of E (kj mohi) = 23.0 (n + 1/2). 

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 vibrational temperature, defined for a diatomic harmonic oscillator by the temperature in Equation (5.22), is considerably higher because of the low efficiency of vibrational cooling. A vibrational temperature of about 100 K is typical although, in a polyatomic molecule, it depends very much on the nature of the vibration. 

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. 

As an exercise we leave the reader to show that the average energy of a diatomic molecule in the gas phase at temperatures where only the vibrational ground state is populated equals Sk T/l. What is it at high temperatures ... 

To help visualize this process, let us consider a diatomic molecule with the energy curves shown in Figure 2.5(a). In this example the ground and excited states have the same equilibrium intemuclear distance ra. Since in solution at room temperature almost all the molecules will be in the lowest vibrational level of the ground state vt° (subscripts refer to the electronic... 

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. 

As previously mentioned, for most diatomic molecules at room temperature, the population of excited vibrational levels is negligible. We therefore first consider transitions for which the initial vibrational level is u = 0. The selection rules (4.108) allow the transitions v = 0- l, 0— 2,... 

Exercise 28-4 Explain qualitatively how temperature could have an effect on the appearance of the absorption spectrum of a diatomic molecule A—B with energy levels such as are shown in Figure 28-1, knowing that most molecules usually are in their lowest vibrational state at room temperature. 

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. 

Step 1 is the unimolecular decomposition of a diatomic molecule, and step —1 is a recombination of two atoms. This recombination is a highly exothermic reaction, and this energy will reside in the newly formed bond, which is thus highly excited. If this energy is not removed in the time of one vibration, the molecule will split up on vibration and recombination will not occur. If a collision occurs before the first vibration, then energy transfer takes place and the bond will be stabilized. The third body fulfils this function. This is more effective at high pressures and low temperatures. 

---