Boltzmann distribution, of vibrational states

Many relaxation studies have used the 2536 A Hg resonance line to excite benzene in systems with total gas pressure above about 10 torr. Although this line excites vibrational levels about 2000 cm above the zero-point level of the first excited singlet state (see Fig. 10), the fluorescence structure shows that radiative decay comes from a Boltzmann distribution of vibrational states. The proposition thus arises that all electronic relaxation in these high -pressure systems occurs after thermal equilibration of the vibrational levels. 

Just as above, we can derive expressions for any fluorescence lifetime for any number of pathways. In this chapter we limit our discussion to cases where the excited molecules have relaxed to their lowest excited-state vibrational level by internal conversion (ic) before pursuing any other de-excitation pathway (see the Perrin-Jablonski diagram in Fig. 1.4). This means we do not consider coherent effects whereby the molecule decays, or transfers energy, from a higher excited state, or from a non-Boltzmann distribution of vibrational levels, before coming to steady-state equilibrium in its ground electronic state (see Section 1.2.2). Internal conversion only takes a few picoseconds, or less [82-84, 106]. In the case of incoherent decay, the method of excitation does not play a role in the decay by any of the pathways from the excited state the excitation scheme is only peculiar to the method we choose to measure the fluorescence (Sections 1.7-1.11). 

The hydroxyl concentration profile for a stoichiometric CH -air flame is presented in Figure 8. Here the maximum mole fraction observed and the predicted mole fraction are equal to better than 10% accuracy. The abscissas of the theoretical and the experimental results were matched by setting the theoretically predicted temperature equal to the measured hydroxyl rotational temperature. At all positions in the flame the hydroxyl 2j[(v,=o) state exhibited a Boltzmann distribution of rotational states. This rotational temperature is equal to the N2 vibrational temperature to within the +100 K precision of the laser induced fluorescence and laser Raman scattering experiments. An example of this comparison is given in Figure 9. 

Aside from the poor comparison with experiment, the work of Montroll and Shuler is interesting in that they give an estimate for the effect on the dissociation rate of a nonequilibrium distribution of vibrational states. This is accomplished by assuming an initial Boltzmann distribution in two computations one, in which the distribution is artificially maintained throughout the dissociation and another, in which deviations from equilibrium are allowed. The results show that when E lkt > 10, the rates differ by <10%, and when E jkT = 5, the difference is about 20%. 

The cross section and rate constant expressions for an A + reaction, where both reactants are polyatomics, are the same as those above for an atom + diatom reaction, except for polyatomics there are more vibrational and rotational quantum numbers to consider. If the polyatomics are symmetric tops with rotational quantum numbers J and K, the state specific cross section becomes a, (urel, vA, JA, KA, vB, JB, KB), where vA represents the vibrational quantum numbers for A. If the polyatomic reactants have Boltzmann distributions of vibrational-rotational energies, the reactive cross section becomes a function of viel and T = TA = TB [i.e., ffr(vTCl T)] and is determined by summing over the quantum numbers as is done in Eq. 

A complete solution to this problem would require solving all of the coupled differential equations simultaneously and specifying all of the rate constants. Obviously this is not possible in general, but the question can be asked what properties of the system are necessary to explain the experimental observations. This question has been extensively examined with methods ranging from simplified models to large-scale computer solutions of the differential equations [16, 37, 38]. A basic conclusion reached is that during the course of the reaction a nonequilibrium distribution of vibrational states exists. This is because the higher vibrational states are more likely to dissociate and therefore become depleted (relative to a Boltzmann distribution) as the... 

The intensity distribution among rotational transitions in a vibration-rotation band is governed principally by the Boltzmann distribution of population among the initial states, giving... 

The frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

It is also possible to obtain excited neutral species by heating the molecules in a furnance. This method was employed to obtain a vibrationally excited N2 beam that was reacted with 0+ ions.127 Since the molecules undergo a large number of collisions with the walls of the furnace before escaping into the beam, a Boltzmann distribution of internal-energy states is established. With such an apparatus, the source temperature is measured by an optical pyrometer and is typically in the range 1000-3000° K. Several reactions of ions with excited neutrals are listed in Table III. 

For transitions in absorption the Boltzmann distribution of molecules over the vibrational levels of the ground state implies that a majority of bands observed will emanate from the lowest, zero-point level transitions from higher levels will be weakened in proportion to the Boltzmann factor exp [ — Jiv/lcT], Hot bands arising from excited vibrational levels can be identified by studying the effect of temperature on the relative intensities. At ordinary temperatures the Boltzmann factor decreases approximately tenfold for each 500 cm-1 of vibrational... 

According to quantum mechanics, only those transitions involving Ad = 1 are allowed for a harmonic oscillator. If the vibration is anhar-monic, however, transitions involving Au = 2, 3,. .. (overtones) are also weakly allowed by selection rules. Among many Au = 1 transitions, that of u = 0 <-> 1 (fundamental) appears most strongly both in IR and Raman spectra. This is expected from the Maxwell-Boltzmann distribution law, which states that the population ratio of the u = 1 and u = 0 states is given by... 

Fig. 2.4-2 shows the Boltzmann distribution of excited vibrational states with energies of hu = hcD (with F = 0. .. 4000 cm ) of an ensemble of molecules at temperatures of 10, 30, 100, 300, 1000, 3000 and 10 000 K, respectively. Thus, the ratios of the intensities of the Stokes and anti-Stokes Raman lines of the same frequency shift make it possible to determine the temperature of a sample (see discussion of Eq. 2.4-10),... 

The vibrational temperature of a molecule prepared in a supersonic jet can be estimated from the observed populations of its vibrational levels, assuming a Boltzmann distribution. The vibrational frequency of HgBr is 5.58 X 10 s , and the ratio of the number of molecules in the n = 1 state to the number in the n = 0 state is 0.127. Estimate the vibrational temperature under these conditions. 

Shannon [20] assumed that atoms or molecules at the reaction interface can have vibrational, and possibly also torsional and rotational, degrees of freedom with a Boltzmann distribution of energies. He also assumed that the rate determining step is decomposition and not any prior or subsequent process. Using transition-state terminology, the rate coefficient, k for the reaction ... 

To obtain hyperpolarizabilities of calibrational quality, a number of standards must be met. The wavefunctions used must be of the highest quality and include electronic correlation. The frequency dependence of the property must be taken into account from the start and not be simply treated as an ad hoc add-on quantity. Zero-point vibrational averaging coupled with consideration of the Maxwell-Boltzmann distribution of populations amongst the rotational states must also be included. The effects of the electric fields (static and dynamic) on nuclear motion must likewise be brought into play (the results given in this section include these effects, but exactly how will be left until Section 3.2.). All this is obviously a tall order and can (and has) only been achieved for the simplest of species He, H2, and D2. Comparison with dilute gas-phase dc-SHG experiments on H2 and D2 (with the helium theoretical values as the standard) shows the challenge to have been met. 

These values were, in turn, averaged over the manifold of rovibrational wavefunctions belonging to the vibrational ground state. Finally, a mean value was found on the assumption that there is a Maxwell-Boltzmann distribution of molecules in the rotational levels (for this the temperature 295K was assumed). 

Direct determination of non-Boltzmann distributions of the vibrational levels of the ground N2(X1S ) state has recently been performed by a new diagnostic technique, coherent anti-Stokes Raman spectroscopy21 with results consistent with calculated distributions. Kinetic data on N2 dissociation amenable to a comparison with theoretical predictions are scanty. Data from Ref.223 can however be quoted and are reported in Fig. 26. The dashed line has been calculated on the assumption that dissociation takes place via predissociated electronic states excited by direct electron impact. Observed dissociation rates are higher and a much better agreement has been claimed with dissociation rates calculated on the basis of a pure vibrational mechanism223 22b. ... 

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