Collision models frequency dependences

However, these parameters are temperature and Larmor frequency dependent. Such a model can be conveniently parameterized in terms of collision frequency (/), so that the rate of decorrelation is given by... 

An understanding of reaction rates can be explained by adopting a collision model for chemical reactions. The collision theory assumes chemical reactions are a result of molecules colliding, and the rate of the reaction is dictated by several characteristics of these collisions. An important factor that affects the reaction rate is the frequency of collisions. The reaction rate is directly dependent on the number of collisions that take place, but several other important factors also dictate the speed of a chemical reaction. 

Frequency Dependences Pertinent to Different Collision Models... 

In Figs. 45 and 46 we see the frequency the dependences typical for resonance-absorption. The isothermal line (at Ft) is characterized by the loss peak, whose intensity is evidently larger and bandwidth is narrower than those pertinent to two other lines. The Lorentz line (at FB) is characterized by (i) a smallest maximum loss and (ii) an absence of the loss shoulder at lower frequencies. This shoulder appears (at Ft and Fq) due to the denominator in Eqs. (370) and (372), pertinent to the self-consistent collision models. Since the plateau-like loss curve is indeed a feature characteristic for water in the... 

Collision-induced vibrational excitation and relaxation by the bath molecules are the fundamental processes that characterize dissociation and recombination at low bath densities. The close relationship between the frequency-dep>endent friction and vibrational relaxation is discussed in Section V A. The frequency-dependent collisional friction of Section III C is used to estimate the average energy transfer jjer collision, and this is compared with the results from one-dimensional simulations for the Morse potential in Section V B. A comparison with molecular dynamics simulations of iodine in thermal equilibrium with a bath of argon atoms is carried out in Section V C. The nonequilibrium situation of a diatomic poised near the dissociation limit is studied in Section VD where comparisons of the stochastic model with molecular dynamics simulations of bromine in argon are made. The role of solvent packing and hydrodynamic contributions to vibrational relaxation are also studied in this section. 

O. The production of organic hydrocarbons in this model then depends on carbon ion insertion reactions. A number of these have been measured in the laboratory and have been shown to proceed at essentially collision frequency. The production of C3 hydrocarbons starting with C+ is delineated in the set of reactions 17.7-17.22 listed above. 

In general, gap depends on the volume fractions of each particle type and on the particle diameters. However, it can also depend on other moments of the velocity distribution function. For example, if the mean particle velocities Uq. and Vp are very different, one could expect that the collision frequency would be higher on the upstream side of the slower particle type. The unit vector Xi2 denotes the relative positions of the particle centers at collision. If we then consider the direction relative to the mean velocity difference, (Uq, - U ) xi2, we can model the dependence of the pair correlation function on the mean velocity difference as °... 

As it has appeared in recent years that many hmdamental aspects of elementary chemical reactions in solution can be understood on the basis of the dependence of reaction rate coefficients on solvent density [2, 3, 4 and 5], increasing attention is paid to reaction kinetics in the gas-to-liquid transition range and supercritical fluids under varying pressure. In this way, the essential differences between the regime of binary collisions in the low-pressure gas phase and tliat of a dense enviromnent with typical many-body interactions become apparent. An extremely useful approach in this respect is the investigation of rate coefficients, reaction yields and concentration-time profiles of some typical model reactions over as wide a pressure range as possible, which pemiits the continuous and well controlled variation of the physical properties of the solvent. Among these the most important are density, polarity and viscosity in a contimiiim description or collision frequency. 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

Figure 3.5.2 shows the results obtained using M-5 and TS-500 samples with S/V values of 3.03 x 107 and 3.28 x 107 m 1, respectively, and porosities of 0.936 and 0.938, respectively. Note the significant deviation of the relaxation behavior from that ofbulk CF4 gas (dotted lines in Figure 3.5.2). The experimental data were first fitted to the model described above, assuming an increase in collision frequency due purely to the inclusion of gas-wall collisions, assuming normal bulk gas density. However, this model merely shifts the T) versus pressure curve to the left, whereas the data also have a steeper slope than bulk gas data. This pressure dependence can be empirically accounted for in the model via the inclusion of an additional fit parameter. Two possible physical mechanisms can explain the necessity of this parameter.

Let us now deduce the factors that control the rate of conversion of B and C to D and E by imagining the transformation process is portrayed well by what is known as a collision rate model. (Strictly speaking, the collision rate model applies to gas phase reactions here we use it to describe interactions in solution where we are not specifying the roles played by the solvent molecules.) First, in order to be able to react, the molecules B and C have to encounter each other and collide. Hence, the rate of reaction depends on the frequency of encounters of B and C, which is proportional to the product of their concentrations. The rate is also related to how fast B and C move in the aqueous solution. Next, the rate is proportional to the probability that B and C meet with the right orientation to be able to react, which we may refer to as the orientation probability . Third, only a fraction of collisions have a sufficient amount of energy (greater then or equal to Ea) to break the relevant bonds in B and C... 

This section considers the cross section for reactive collisions ar. Bimolecular reactions will be treated explicitly. The rate (frequency) of collisions depends on the collision cross section. The larger the cross section, the more often molecules run into one another. In a similar way the reactive cross section determines how often molecules run into one another and react. This section introduces the simple line-of-centers model for scaling of the reactive cross section with energy. 

The difference between the two results is in the pre-exponential term. In the quantum mechanical treatments either vet= vaKe or vet = 0.5(tu + reyl depending on the model adopted. In the Marcus equation the pre-exponential term is ZKee p( w lRT) and the time dependence is introduced through the collision frequency, Z. In any case, in the non-adiabatic limit, ve < v , and kobs is given by equation (41). In the adiabatic limit, ve > vn and kobs is given by equation (42), with vn in the range 10n —1013 s-1. 

In the low-frequency range (with x spectral function L(z) depends weakly on frequency x. Then Eq. (32) comes to the Debye-relaxation spectrum given by Eq. (33). Its main characteristics, such as the dielectric-loss maximum Xd and its frequency xD, are given by Eq. (34). A connection between these quantities and the model parameters becomes clear in an example of a very small collision frequency y. In this case, relations (34) come to... 

Function (2.2) can be considered as an empirical model used to best fit the experimental concentration-time data. In practice, laws different from (2.2) are also encountered, especially when dependence on the concentration is considered however, a simple theory based on the kinetic theory of gases can only explain the simplest of these empirical rate laws. The general idea of this theory is that reaction occurs as a consequence of a collision between adequately energized molecules of reactants. The frequency of collision of two molecules can explain simple reaction... 

Pressure-dependent rate constants for the syn-anti conformational process in larger alkyl nitrites provide a further test of the ability of RRKM theory to successfully model the kinetics of the internal rotation process in these molecules. Solution of the Lindemann mechanism shows that at the pressure where the rate constant is one-half of its limiting high-pressure value, Pm, the frequency of deactivating collisions is comparable to , the average rate that critically... 

---