Time between collisions

The average time between collisions is then v and in this time tlie particle will typically travel a distance X, the mean free path, where... 

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. 

It is helpful to distinguish three different types of problem to which Newton s laws of motion may be applied. In the simplest case, no force acts on each particle between collisions. From one collision to the next, the position of the particle thus changes by v,5f, where v, is the (constant) velocity and 6t is the time between collisions. In the second situation, the particle experiences a constant force between collisions. An example of this type of motion would be that of a charged particle moving in tr uniform electric field. In the third case, the force on the particle depends on its position relative to the other particles. Here the motion is often very difficult, if not impossible, to describe analytically, due to the coupled nature of the particles motions. 

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... 

If die average time between collisions is 2t dieii die relaxation time is dehned by... 

In impact theory the gas density is restricted by the criterion for the validity of the binary collision approximation. Roughly speaking, the time between collisions must be greater than their duration... 

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... 

Unfortunately, femtosecond laser pulses are not so readily predisposed to study collisions between atoms and molecules by the pump-probe approach. The reason is that, typically, the time between collisions in the gas phase is on the order of nanoseconds. So, with laser pulses of sub-lOOfs duration, there is only about one chance in ten thousand of an ultrashort laser pulse interacting with the colliding molecules at the instant when the transfer of atoms is taking place in other words, it is not possible to perform an accurate determination of the zero of time. 

As the system pressure is decreased at constant temperature, the time between collisions will increase, thereby providing greater opportunity for unimolecular decomposition to occur. Consequently, one expects the reaction rate expression to shift from first-order to second-order at low pressures. Experimental observations of this transition and other evidence support Linde-mann s theory. It provides a satisfactory qualitative interpretation of unimolecular reactions, but it is not completely satisfactory from a... 

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... 

If the average time between collisions is 2r then the relaxation time is defined by... 

At low cM, the rate-determining step is the second-order rate of activation by collision, since there is sufficient time between collisions that virtually every activated molecule reacts only the rate constant K appears in the rate law (equation 6.4-22). At high cM, the rate-determining step is the first-order disruption of A molecules, since both activation and deactivation are relatively rapid and at virtual equilibrium. Hence, we have the additional concept of a rapidly established equilibrium in which an elementary process and its reverse are assumed to be at equilibrium, enabling the introduction of an equilibrium constant to replace the ratio of two rate constants. 

Table 4.26 Values of average velocity u, molecular diameter a, mean free path X., and average time between collisions t for some gas molecules...

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

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 ... 

By lifetime we mean the average time that an excess carrier exists before annihilation by a carrier of the opposite sign. This is as opposed to relaxation time, the average time between collisions, or trapping time, the average time in a band before being trapped. We have used the same symbol, t, to represent both the lifetime and the relaxation time because this symbol... 

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... 

Under ordinary mass spcctrometric conditions only unimolecular reactions of excited ions occur, but at higher ionization chamber pressures bimolecular ion molecule reactions are observed in which both the parent ions and their unimolecular dissociation product ions are reactants. Since it requires a time of 10 5 sec. to analyze and collect the ions after their formation all of the ions in the complete mass spectrum of the parent molecule are possible reactants. However, in radiation chemistry we are concerned with the ion distribution at the time between molecular collisions which is much shorter than 10 5 sec. For example, in the gas phase at 1 atm. the time between collisions is 10 10 sec. and in considering the ion molecule reactions that can occur one must know the amount of unimolecular decomposition within that time. By utilizing the quasi-equilibrium theory of mass spectra6 it is possible to calculate the ion distribution at any time. This has been done for propane at a time of 10 10 sec.,24 and although the parent ion is increased by a factor of 2 the relative ratios of the other ions are about the same as in the mass spectrum observed in 10 r> sec. Thus for gas phase radiolysis the observed mass spectrum is a fair first approximation to the ion distribution. In... 

For the purpose of this work, we will call a complex of two or more interacting atoms/molecules a supermolecule. Supermolecules may exist for a short time only, e.g., the duration of a fly-by encounter ( 10-13 s). Alternatively, supermolecules may be bound by the weak van der Waals forces and thus exist for times of the order of the mean free time between collisions ( 10-10 s), or longer. In any case, it is clear that, in general, supermolecules possess a spectrum of their own, in excess of the sum of the spectra of the individual (non-interacting) molecules that make up the supermolecule. These spectra are the collision-induced spectra, the subject matter of this monograph. 

Time scales. For an understanding of spectral line shapes of induced absorption, at not too high gas densities, it is useful to distinguish three different times associated with collisions, namely the average time between collisions, the duration of a molecular fly-by and the duration of the spectroscopic interaction. 

A sufficiently rarefied gas, or a mixture of gases, consists of a number of neutral molecules of species 1 and 2 (which may or may not be the same). We may assume a distribution of velocities (measured in the laboratory frame), fi ( ) d3u, that may be modeled by a Maxwellian distribution function, with i = 1 or 2, as long as the duration of the average collision is short compared to the time between collisions. For binary collisions, one usually transforms from laboratory coordinates, Vj, to relative ( >12) and center-of-mass (1>cm) velocities,... 

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