Collisional relaxation constants

The strong role of collisions in decohering a system is readily seen by considering fie density matrix pj of a system that has reached thermal equilibrium at tempera-rs T through collisional relaxation, that is, pf = Qexp(—Hs./kgT). Here Hs is the system Hamiltonian, Q is a normalization factor, and kB is the Boltzmann constant, fmsiderable insight is obtained if we cast the density matrix in the energy repre-... 

In writing rate equations only for it is tacitly assumed that translational relaxation is instantaneous on the time scale of all the other rate processes. Hence, a well-defined temperature, T, characterizes the translational degrees of freedom of the lasing molecules and all degrees of freedom of the nonlasing species. This heat bath temperature appears as a parameter in the collisional rate constants. It also enters the gain coefficients via the linewidth and in the case of rotational equilibrium mainly via the population inversion. Thus (1) and (2) should be supplemented by a rate equation for T. Additional kinetic equations describe the time dependence of the nonlasing species concentrations. 

Reaction (25) represents collisional relaxation of the excited oxygen atom. While halogenated hydrocarbons form too small a fraction of the available collision partners to be of atmospheric consequence for the relaxation channel, this pathway must be considered for laboratory experiments relying on 0( D) or 0( P) vs. time profiles to deduce the rate constant of reaction (24). In reaction (26), a sizable fraction of the 190 kJ mol excess energy of 0( D) is transferred to internal excitation of the HCFC, which then can dissociate to products. Between reactions (24) and (26) a range of products is possible, including OH + R, CIO -I- R, and 0( P) -I- HCl -f chlorofluoroalkene. 

Data for the reaction of 0( D) with HCFCs are scarce. We are unaware of any published product studies. Rate constants for reaction (24), provided primarily by two studies, are available for a variety of HCFCs, but only at 298 K. Table 4 collects the measured rate constants, giving values based on the removal of 0( D), which represents the sum of the rate constants for collisional relaxation and reaction, and giving rate constants for the removal of the HCFC. The rate constants reported by Green and Wayne [61] were measured relative to the reaction 0( D) -f N2O - products, using a rate constant of 2.2 x 10 °cm s for the latter reaction. These were subsequently corrected for the currently accepted value of 1.16 x 10 °cm s by Warren et al. [60]. The corrected values are listed in Table 4. 

A more carefully constructed approach to energy transfer explicitly accounts for the energy distribution of the chemically activated product (5,6,26,27), The initial distribution of energies upon production as well as the transient distributions formed by collisional relaxation of internal energy are used to calculate rate constants for imimolecular reaction. The formalism of Bunker (7,8,9) based on general RRKM theory is convenient for recoil chemical activation, since it explicitly accounts for both rotational and vibrational excitation in the product. In the cyclobutane model system reported, a stepladder approach to deactivation was incorporated, with the step size being a parameter determined by the best fit to the data (6). The overall processes considered are illustrated in Equation 4. 

Figure 6 shows a higher resolution spectrum of the fluorescence from another excited state, l Ag (N-31, Fi) to the v=0 level of b IIu. Because these are both good Hund s case(b) states, due to small spin-orbit coupling, large rotational constants, and high J, the spectrum exhibits a P,Q,R pattern. The weaker features are due to collisional relaxation in the upper state, as in Na2 (Figure 1). The Q(N=31) line in this fluorescence scan is broader than the instrumental resolution of about 0.8 cm due to predissociation by the a Ey" vibrational continuum. The R(30) and P(32) lines are sharp. However, the collisionally relaxed Q(30) and Q(32) lines are sharp and the R and P lines adjacent to R(30) and P(32) are broad. The reason for this selectivity in the predissociation of the b II rotational levels is explained by examining Figure 7.

The eigenvalues of this eigenvalue problem are found by diagonalizing the matrix M. The situation is more complicated if collisional relaxation is not fast compared to chemical reaction. In that case, the solution will yield more than one small negative eigenvalues and the overall reaction will proceed on a non-exponential time scale. In other words, if collisional relaxation interferes with unimolecular reaction, the reaction process cannot be described by a time-independent rate constant A uni-We now take a look at the corresponding chemically activated reaction,... 

This is a hint that the N+ fine structure energy may not be available for promoting the reaction. A careful trapping experiment corroborates this. In this experiment the N+ ions have been created by electron bombardment in a 350 K ion source. Under such conditions it is safe to assume that this leads to a statistical population of the three fine- structure states Pq, Pi, and P2, i.e. to the ratio 1 3 5. If the excited ions react faster, one would get two breaks in the slope of the reaction decay curve. In the experiment the number of N+ ions decreases with a single time constant over several orders of magnitude. Also the competition between collisional relaxation and reaction has been excluded. For more critical tests, experiments with state selectively prepared N+( P) reactants are needed or one has to find an independent method to prepare and test a low temperature fine structure ensemble (see Chapter 6). 

The ion-molecule reactions of collisionally relaxed NH2 with typical representatives of organic compounds in the gas phase at ambient temperature are compiled in Table 22. The anions were analyzed by mass spectrometry in early experiments and later by Fourier transform (ion cyclotron resonance) mass spectrometry. More recent investigations usually apply the flowing afterglow technique or its offspring, the selected-ion flow tube (SIFT) technique. These methods allow the identification of anions only the other products have to be deduced from the mass balance. Rate constants were determined by the flowing afterglow and the SIFT techniques. The products frequently form by proton abstraction which may be followed by elimination or by nucleophilic substitution. Reaction enthalpies and... 

Nj. The rate constant k = 0.3xi0 cm -molecule" s" for the reaction N2H N2 >N2H N2 was deduced presuming that the collisional relaxation of vibrationally excited N2H by N2 proceeds via a proton transfer reaction [13]. For isotope exchange reactions in the N2 N2H system, see p. 30. 

The time constant r, appearing in the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1/r = Pe — Pd, where Pe is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. The reciprocal of this quantity is the number of collisions Z required to de-excite a quantum of vibrational energy e = hv. This number can be explicitly calculated from Equation 4 since Z = 1/P, and it can be experimentally derived from the measured relaxation times. 

Consider a closed system characterized by a constant temperature T. The system is prepared in such a way that molecules in energy levels are distributed in departure from their equilibrium distribution. Transitions of molecules among energy levels take place by collisional excitation or deexcitation. The redistribution of molecular population is described by the rate equation or the Pauli master equation. The values for the microscopic transition probability kfj for transition from ith level toyth level are, in principle, calculable from quantum theory of collisions. Let the set of numbers vr be vibrational quantum numbers of the reactant molecule and vp be those of the product molecule. The collisional transitions or intermolecular relaxation processes will be described by ... 

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