Collisions of molecules

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. 

Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by colhsions between molecules. An approximate expression based on Brown, et al. J. Appl. Phys., 17, 802-813 [1946]) for the mean free path is... 

Conduction takes place at a solid, liquid, or vapor boundary through the collisions of molecules, without mass transfer taking place. The process of heat conduction is analogous to that of electrical conduction, and similar concepts and calculation methods apply. The thermal conductivity of matter is a physical property and is its ability to conduct heat. Thermal conduction is a function of both the temperature and the properties of the material. The system is often considered as being homogeneous, and the thermal conductivity is considered constant. Thermal conductivity, A, W m, is defined using Fourier s law. 

In the discussion so far, the fluid has been considered to be a continuum, and distances on the molecular scale have, in effect, been regarded as small compared with the dimensions of the containing vessel, and thus only a small proportion of the molecules collides directly with the walls. As the pressure of a gas is reduced, however, the mean free path may increase to such an extent that it becomes comparable with the dimensions of the vessel, and a significant proportion of the molecules may then collide direcdy with the walls rather than with other molecules. Similarly, if the linear dimensions of the system are reduced, as for instance when diffusion is occurring in the small pores of a catalyst particle (Section 10.7), the effects of collision with the walls of the pores may be important even at moderate pressures. Where the main resistance to diffusion arises from collisions of molecules with the walls, the process is referred to Knudsen diffusion, with a Knudsen diffusivily which is proportional to the product where I is a linear dimension of the containing vessel. 

Burshtein A. I., Storozhev A. V. The angular momentum relaxation due to multiparticle collisions of molecules with atoms, Chem. Phys. 164, 47-55 (1992). 

The study [39] shows that similar equation is valid for adsorption of NH- and NH2-radicaIs, too. There are a lot of experimental data lending support to the validity of the proposed two-phase scheme of free radical chemisorbtion on semiconductor oxides. It is worth noting that the stationary concentration of free radicals during the experiments conducted was around 10 to 10 particles per 1 cm of gas volume, i.e. the number of particle incident on 1 cm of adsorbent surface was only 10 per second. Regarding the number of collisions of molecules of initial substance, it was around 10 for experiments with acetone photolysis or pyrolysis provided that acetone vapour pressure was 0,1 to 0,01 Torr. Thus, adsorbed radicals easily interact at moderate temperatures not only with each other but also with molecules which reduces the stationary concentration of adsorbed radicals to an even greater extent. As we know now [45] this concentration is established due to the competition between the adsorption of radicals and their interaction with each other as well as with molecules of initial substance in the adsorbed layer (ketones, hydrazines, etc.). 

Collisions of molecules with the walls of the passage provide the resistance to diffusion when the mean free path, A, is appreciably greater than the diameter, d, of the passage. In experimental work the ratio, A/d, is taken as 10 or more to isolate the Knudsen effect. That investigator did experiments with small capillaries and deduced the equation... 

In our account here we neglect a third aspect of a spectral line, specifically its shape, beyond its characteristic frequency and strength. A natural line shape is almost impracticable to observe and would yield on analysis little or no additional information about intrinsic molecular properties. Another shape merely reflects components of molecular velocities in a direction parallel to the direction of propagation. Apart from these effects, further broadening of spectral lines due to finite durations, between collisions, of molecules in particular quantum states is attributed to interactions between colliding molecules rather than directly to... 

When a gas is cooled by 1 °C, from 0°C to —1 °C, it loses l4 3 of its pressure. Since pressure is due to movement of the gas molecules, Lord Kelvin realised that a gas would have no pressure at -273 °C that is, there would be no movement of the gas molecules -273 °C is absolute zero. Since the rate of a reaction depends upon the movement and hence collision of molecules, the relevant temperature scale for chemical (and therefore biochemical) reactions is one whose zero is -273 °C. It is known as the Kelvin or absolute scale. 

If we postulate that the rate-controlling mechanism involves the collision or interaction of a single molecule of A with a single molecule of B, then the number of collisions of molecules A with B is proportional to the rate of reaction. But at a given temperature the number of collisions is proportional to the concentration of reactants in the mixture hence, the rate of disappearance of A is given by... 

AU reactions occur by collisions between molecules or by collisions of molecules with surfaces. We will consider reactions at surfaces later, but here we consider the theory of homogeneous reactions. We wiU not attempt a quantitative or thorough description of reaction mechanisms but will only describe them in enough detail to be able to see how the engineer can control them. These collisions occur as sketched in Figure 4-12. 

If a mechanical degradation of a solution of two polymers is carried out by high speed stirring, the formation of a block copolymer is not probable as the scission of polymer molecules at low concentration is not caused mainly by intermolecular interaction, such as by collision of molecules and through entanglements, but by displacements due to hydrodynamic forces in velocity gradients. Nakamo and Minoura (98) did obtain reaction by stirring a benzene solution of polyethylene oxide and poly(methyl methacrylate). 

This question can be answered by means of scattering experiments for molecules in fields [1-3] or accurate calculations of the scattering matrices for collisions of molecules in nonperturbative external fields. The purpose of this chapter is to describe the theory for such calculations and present selected results illustrating the effects of external electromagnetic fields on molecular collisions at low temperatures. 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

Most of the recent literature on molecular collisions in external fields [1-3, 9, 10, 15, 16, 18, 19, 21, 23, 25, 27-84, 92] is focused on collisions of molecules at low and ultralow temperatures. As mentioned in the introduction, it is at temperatures < 10 K that strong electromagnetic fields are expected to have a noticeable effect on the scattering properties of molecules. The coupled-channel calculations... 

Figure 8.6 Collisions of ultracold molecules in a quasi-2D geometry. Presented are the rates of the loss of molecules from an optical lattice trap occurring due to chemical reactions. The squares represent the reactions of molecules prepared in the same (translational and internal) quantum states the circles are for collisions of molecules in different translational states but the same internal states the triangles are for molecules in different internal states. Adapted with permission from Ref. [1].

All collisions of molecules are perfectly elastic, that is, there is no loss of internal energy upon collision. 

There are several theories to account for chemical kinetics. The simplest is the collision theory, which will be used in Chapter 4 to calculate the rate constants for the collision of molecules in solution. A more sophisticated theory, one that... 

The region of flow where collisions of molecules with the container walls are more frequent than intermolecular gaseous collisions was the subject of detailed study by Knudsen(8) early in the twentieth century. From geometrical considerations it may be shown(9) that, for the case of a capillary of circular cross-section and radius r, the proportionality factor is Snr3/3. This results in a Knudsen diffusion coefficient ... 

In macro- and mesoporous membrane layers the nature of the flow is determined by the relative magnitude of the mean free path X of the molecules and the pore size dp. When the mean free path of the gas molecules is much larger than the pore size, i.e. X dp, collisions of molecules with the pore walls are predominant and the mass transport takes place by the well-known selective Knudsen diffusion process. If the pore radius is much larger than the mean free path of the molecules and a pressure difference over the membrane exists the mass transport takes place by non-selective viscous flow. 

Rate of production of atoms 2 (Rate of collision of molecules with the surface)... 

Based on the molecular collision model, that describes successfully experimental data for a large number of bimolecular reactions, the rate of the reaction can be calculated as the number of collisions of molecules having energy higher than the required value E [7] ... 

The collision theory In a gas, molecules A are moving and colliding together with molecules B. The total number of collisions of molecules A with molecules B, ZAB, at a given time, At, is ... 

The Pirani114 gauge (1906 range 1 bar to 10-4 mbar) measures the voltage across a heated filament due to collisions of molecules with the filament, which decrease its temperature and thus its electrical resistance. 

It readily follows that the number of collisions of molecules of type 1 with those of type 2 would be given by... 

If the collisions of molecules produce a chemical reaction, the Boltzmann equation is modified in obtaining the equations of change these problems are addressed and analyzed in the context of quantum theory, reaction paths, saddle points, and chemical kinetics. Mass, momentum, and energy are conserved even in collisions, which produce a chemical reaction. 

The kinetic equations serve as a bridge between the microscopic domain and the behavior of macroscopic irreversible processes through the description of hydrodynamics in terms of intermolecular collisions. Hydrodynamics can specify a large number of nonequilibrium states by a small number of reproducible properties such as the mass, density, velocity, and energy density of a fluid conserved during the collision of molecules. Therefore, the hydrodynamic equations can describe a wide range of relaxation processes of nonequilibrium states to equilibrium state. We call such processes decay processes represented by phenomenological equations, such as Fourier s law of heat conduction. The decay rates are determined by the transport coefficients. Reliable transport coefficients provide microscopic and macroscopic information, and validate the results of molecular dynamics. 

Chemical reaction rate depends on the collisions of molecules, per second per unit volume. Since the number of collisions of a species is proportional to its concentration, the chemical reaction rate is proportional to the product of concentrations (mass action law). Thus, for a single homogeneous elementary chemical reaction... 

Hie chemical reaction rate is usually dependent on the molar concentrations of the reactants and not on their mass fractions, because it depends on the chance of collision of molecules. However, here the definition of in terms of mass fractions is preferred, because it can readily be incorporated into mass balances. A definition in terms of moles or molar concentrations might invite the use of mole balances instead of mass balances. Since, contrary to conservation of mass, there is no such thing as conservation of moles (because one molecule might divide into several molecules, or several might condense into one), the use of mole balances is strongly dissuaded. More information concerning the definition of conversion can be found elsewhere [2]. 

Elementary reactions are individual reaction steps that are caused by collisions of molecules. The collision can occur in a more or less homogeneous reaction medium or at the reaction sites on a catalyst surface. Only three elementary kinetic processes exist mono-, bi-, and trimolecular processes. Of these, trimolecular processes are rarely found, because the chance of three molecules colliding at the same time is very small. Each elementary reaction consists of an activation of the reactants, followed by a transition state and decomposition of the latter into reaction products ... 

Free-molecule or Knudsen flow (Kn 1). The molecules are transported within the porous structure without intermolecular collisions however, the molecules do collide with the pore walls. Collisions between molecules can be ignored compared to collisions of molecules with the walls of the porous medium or tube. 

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