Maxwell-Boltzmann distribution coefficient

This rate coefficient can be averaged in a fifth step over a translational energy distribution P (E ) appropriate for the bulk experiment. In principle, any distribution P (E ) as applicable in tire experiment can be introduced at this point. If this distribution is a thennal Maxwell-Boltzmann distribution one obtains a partially state-selected themial rate coefficient... 

The velocity distribution/(v) depends on the conditions of the experiment. In cell and trap experiments it is usually a Maxwell-Boltzmann distribution at some well defined temperature, but /(v) in atomic beam experiments, arising from optical excitation velocity selection, deviates radically from the nonnal thennal distribution [471. The actual signal count rate, relates to the rate coefficient through... 

The rate constant is measured in units of moles dnr3 sec /(moles dnr3)", where n = a + b. Time may also be in minutes or hours. It should be noted that in case where the reaction is slow enough, the thermal equilibrium will be maintained due to constant collisions between the molecules and k remains constant at a given temperature. However, if the reaction is very fast the tail part of the Maxwell-Boltzmann distribution will be depleted so rapidly that thermal equilibrium will not be re-established. In such cases rate constant will not truly be constant and it should be called a rate coefficient. 

The Boltzmann rate coefficient, k T), at temperature T is then obtained by averaging this expression over the Maxwell-Boltzmann distribution of relative velocities,1 or relative energies. In terms of E,... 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

O or equivalently the drag coefficient becomes very large. (The noninertial limit typically corresponds to // 10 s.) In this limit the Maxwell-Boltzmann distribution for the angular velocities has set in so that orientation and angular velocity variables are decoupled from each other as far as the time behavior of the particle orientations is concerned. Thus on setting / = 0 in Eqs. (5.23)-(5.25), we have... 

Figure 2.2). (Such data are commonly collected in IMS smdies to determine K more precisely and verify that the measured K equals Klf)) needed to derive H using Equation 1.10.) A lower measurement accuracy obviously increases the apparent (E/AOc- In Ihs result, the high-field behavior is seen in differential IMS at much lower E/N than in conventional IMS (3.2.4). The distortion of Maxwell-Boltzmann distribution also causes ion heating by Equation 1.27. Hence a negligible deviation of K from K(relative measurement accuracy) is achieved when ATh < yT, where y is a coefficient dependent on x. Using Equation 1.27, that can be expressed as... 

The relations connecting the rate coefficients of forward and backward collisional processes follow from the microscopic detailed balance relations for reactive collisions (8)-(9) after averaging them with the Maxwell-Boltzmann distribution over the velocity and rotational energy. Thus for the rate coefficients of forward and backward reactions we obtain... 

The rate coefficient k T) of a bimolecular chemical reaction in the gas phase at a given temperature T results from the thermal average of a very large number of bimolecular reactive collisions. These collisions involve reagents in a variety of quantum states, the populations of which follow Boltzmann s law at this temperature, with a Maxwell-Boltzmann distribution of centre of mass kinetic energies Et (hereafter KEcm)- Ignoring any dependence of the reaction cross section (a) on the internal states of the reagents, the relationship between the rate coefficient and the reaction cross section is... 

The activation free energy AA can be used to compute the TST approximation of the rate constant = Ce, where C is the preexponential factor. Because not every trajectory that reaches the transition state ends up as products, the actual rate is reduced by a factor k (the transmission coefficient) as described earlier. The transmission coefficient can be calculated using the reactive flux correlation function method. " " " Starting from an equilibrated ensemble of the solute molecules constrained to the transition state ( = 0), random velocities in the direction of the reaction coordinate are assigned from a flux-weighted Maxwell-Boltzmann distribution, and the constraint is released. The value of the reaction coordinate is followed dynamically until the solvent-induced recrossings of the transition state cease (in less than 0.1 ps). The normalized flux correlation function can be calculated using " ... 

In plasma processes, the ionization rate coefficients are important quantities which are determined by using our calculated partial and total ionization cross sections and Maxwell-Boltzmann distribution of temperature/energy [26, 26] as follows ... 

The coefficient D, being proportional to a normalizing coefficient C of the Maxwell-Boltzmann energy distribution W = C exp(- / ), is determined by the parameters of the hat-curved model as... 

For a system having a Maxwell-Boltzmann energy distribution, current classical theories of ion-molecule interactions predict a collision or capture rate coefficient given by... 

The main assumptions of the model are following 1) particles are spherical 2) each particle has both the directed and chaotic components of die absolute velocity 2) a chaotic particle motion is carried out according to the Boltzmann-Maxwell law (the particle medium is considered as an ideal gas having its own pressure, density etc.) 3) a chaotic particle velocity drop is caused by both a viscous particle-gas friction force and inelastic particle-particle collisions (coefficient of energy losses due to inelastic collisions has to be rather low because Bolzmann-Maxwell velocity distribution is valid only for elastic particles and can be employed only for small non-elasticity) 4) particles do not get fragmented 5) a heat exchange between gas and particles is neglected. 

In relation to the applications, in particularly to plasma processes, ionization rate coefficients are rather more desirable than ionization cross sections. "We have evaluated a set of ionization rate coefficients as a function of electron temperature in the units of energy for the individual cations produced in electron collision with the SiH4 molecule. The calculations are made using the calculated ionization cross sections and Maxwell-Boltzmann energy distribution, and the results are presented in Figure 7 along with Table 2. 

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