Mean time between collisions

Spectral lines are fiirther broadened by collisions. To a first approximation, collisions can be drought of as just reducing the lifetime of the excited state. For example, collisions of molecules will connnonly change the rotational state. That will reduce the lifetime of a given state. Even if die state is not changed, the collision will cause a phase shift in the light wave being absorbed or emitted and that will have a similar effect. The line shapes of collisionally broadened lines are similar to the natural line shape of equation (B1.1.20) with a lifetime related to the mean time between collisions. The details will depend on the nature of the intemrolecular forces. We will not pursue the subject fiirther here. 

When collisions occur between gas phase atoms or molecules there is an exchange of energy, which leads effectively to a broadening of energy levels. If t is the mean time between collisions and each collision results in a transition between two states there is a line broadening Av of the transition, where... 

The 16 ns natural lifetime of excited Na is much shorter than the 140 /rs mean time between collisions, thus the fine broadening due to collision-induced... 

There may be several reasons for the difference between gas phase and matrix photochemistry, and we outline one possible explanation. Even at 355 nm (XeF laser), a uv photon has more energy (equivalent to 335 kJ mol-1) than is needed to break one M—CO bond (89,90). In a matrix, the isolated Fe(CO)5 molecule is in intimate contact with the matrix material, and any excess energy can be rapidly lost to the matrix. In the gas phase, collisions are the principal pathway for loss of this excess energy. Under the conditions used in the gas phase photolysis, the mean time between collisions was relatively long and the excess energy could not... 

Mean Free Path. The mean free path of a gas molecule l and the mean time between collisions t are given by... 

To consider gas molecules as isolated from interactions with their neighbors is often a useless approximation. When the gas has finite pressure, the molecules do in fact collide. When natural and collision broadening effects are combined, the line shape that results is also a lorentzian, but with a modified half-width at half maximum (HWHM). Twice the reciprocal of the mean time between collisions must be added to the sum of the natural lifetime reciprocals to obtain the new half-width. We may summarize by writing the probability per unit frequency of a transition at a frequency v for the combined natural and collision broadening of spectral lines for a gas under pressure ... 

Equations (A9) show that the electron exhibits the expected cyclotron motion in the presence of the magnetic field. However, collisions must also be taken into account. Let N(t) be the number of particles that have not experienced a collision for time t (after some arbitrary beginning time, t = 0). Then it is reasonable to assume that the rate of decrease of N(t) will be given by dN oc —Ndt = —Ndt/1. The solution of this equation is N(t) = N0 exp (—t/ ), where N0 is the total number of particles. It can easily be shown that x is simply the mean time between collisions. The probability of having not experienced a collision in time t is, of course, N(t)/N0 = exp(—t/ ). The... 

For rough estimates, the collisional cross section may be assumed to be velocity-independent, Qn(vn) = Qo = constant (hard-sphere approximation), so that the mean time between collisions becomes... 

Later studies showed the same phenomena in deuterium and deuterium-rare gas mixtures [335, 338, 305], and also in nitrogen and nitrogen-helium mixtures [336] in nitrogen-argon mixtures the feature is, however, not well developed. The intercollisional dip (as the feature is now commonly called) in the rototranslational spectra was identified many years later see Fig. 3.5 and related discussions. The phenomenon was explained by van Kranendonk [404] as a many-body process, in terms of the correlations of induced dipoles in consecutive collisions. In other words, at low densities, the dipole autocorrelation function has a significant negative tail of a characteristic decay time equal to the mean time between collisions see the theoretical developments in Chapter 5 for details. 

The virial expansion of the time correlation functions is possible for times smaller than the mean time x between collisions. Accordingly, the spectral profiles may be expanded in powers of density, for angular frequencies much greater than the reciprocal mean time between collisions, co 1/r. Since at low density the mean time between collisions is inversely proportional to density, lower densities permit a meaningful virial expansion for a greater portion of the spectral profiles. 

It has been argued that, in the low-density limit, intercollisional interference results from correlations of the dipole moments induced in subsequent collisions (van Kranendonk 1980 Lewis 1980). Consequently, intercollisional interference takes place in times of the order of the mean time between collisions, x. According to what was just stated, intercollisional interference cannot be described in terms of a virial expansion. Nevertheless, in the low-density limit, one may argue that intercollisional interference may be modeled as a sequence of two two-body collisions in this approximation, any irreducible three-body contribution vanishes. 

Fig. 5.3. The dipole autocorrelation function, long-time behavior (schematic). Xd and xc are times of the order of the mean duration of a collision and the mean time between collisions, respectively.

Long-time behavior of correlation functions. The dipoles induced in successive collisions are correlated as Fig. 3.4 on p. 70 suggests. As a consequence, the dipole autocorrelation function has a negative tail of a duration comparable to the mean time between collisions, Fig. 5.3. Furthermore, the area under the negative tail is of similar order of magnitude as the area under the positive (or intracollisional) part of C(r). If the neg-... 

The mean life of the 1Pl state is very short, about 10"9 sec. At 1 atmosphere pressure the mean time between collisions of a given molecule is about 10 10 sec so that appreciable emission by 1Pl mercury atoms can occur even in the presence of strongly quenching gases at pressures of several millimeters. 

Mean Free Path and Mean Time Between Collisions... 

The ideal gas relation was derived under the assumption that each molecule travels undisturbed from wall to wall, which is certainly not true at common pressures and temperatures. To see this, we need to get some estimate of the mean free path k (the mean distance a molecule travels before it undergoes a collision), and the mean time between collisions. 

We also sometimes evaluate the mean time between collisions r, which is the... 

Suppose the density of a gas is kept constant, but the temperature is doubled. Predict what would happen to (a) the mean free path A. (b) the mean time between collisions r (c) the diffusion constant D. If you can, use your intuition about the physical process, rather than substitution into equations derived in this chapter. 

Partially filled bands of collective-electron states support metallic conductivity. The electrical conductivity is defined as the ratio of current density J = nev to electric field strength, E, where n is the number of carriers of charge e per unit volume and v is their average velocity. Since the average force on a charged particle is eE = m v/r, where r is the mean time between collisions and m is the effective mass, it follows that... 

Now the time between collisions is a random quantity. Sometimes the collisions may occur in rapid succession at others, there may be fairly long intervals. It is possible, however, to talk of a mean time between collisions, t. In Section 4.2.5, it was shown that the number of collisions (steps) is proportional to the time. If Acollisions occur in a time t, then the average time between collisions is t/N. Hence,... 

The velocity component is given by the ratio of /, the mean distance between collisions (i.e., the mean free path), to t, the mean time between collisions,... 

The parameter x now invites consideration. In the gas phase, x is the mean time between collisions. What is the significance of t in an ionic liquid ... 

A key feature of the DSMC technique in comparison to continuum methods is its relatively high computational expense. To allow the decoupling between molecular motion and intermolecular collisions to occur in a physically accurate way, the time-step used in the DSMC technique must be smaller than the mean time between collisions. Similarly, the size of the cells employed in the DSMC computational grid must be of the order of the local mean free path everywhere in the flow domain. These physical restrictions on the size of the numerical parameters results in the time steps and cell sizes employed in DSMC calculations being usually significantly smaller than those employed in continuum computations. For this reason, significant work has been performed in the optimization of the DSMC technique for different types of computer hardware. Examples of specific implementations are described in Refs. 23-26. 

Now the mean time between collisions diverges that is. 

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