Maxwell-Boltzmann flux

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

An easy way to find this correction factor is to look at the history of an exit trajectory. This history is followed by starting atx = xb trajectories with velocity sampled from a Maxwell-Boltzmann distribution in the outward direction—these represent the outgoing equilibrium flux, then inverting the velocity (y -> —v) so that the particle is heading into the well, and integrating the equations of motion... 

Time of flight measurements can yield useful information in a similar vein. The work of Comsa et al. (1980), for instance shows a shift of D desorption from Pd(100) from a Maxwell-Boltzmann, surface thermalised desorbing flux to one with fast Dj molecules emerging from the surface with a narrow distribution of energies after sulphur is deposited on the surface... 

To specify completely the neutron activity and to choose the proper cross sections for calculating the reaction rate constant, it is necessary to know the distribution of neutron concentration, or neutron flux, with respect to energy. In a thermal reactor the distribution of neutrons in thermal equilibrium with nuclei at an absolute temperature T is similar to the distribution of gas molecules in thermal equilibrium and can be approximated by the Maxwell-Boltzmann distribution... 

The reference speed u is arbitrarily chosen as 2200 m/s, which is the most probable speed for a Maxwell-Boltzmann distribution at temperature f= 293.2 K. The cross section a is now the specially defined effective cross section that, when multiplied by the 2200 m/s flux gives the proper reaction rate constant. 

The Westcott g and s factors can also be used to determine the effective thermal cross section a, such that when multiplied by the integrated Maxwell-Boltzmann thermal flux the proper reaction rate with a nuclide is obtained, as already defined by Eq. (2.55). From Eqs. (2.55) and (2.62), a is related to a by... 

Figure 3.16 is a cutaway view of this reactor. The reactor vessel is a cylinder 13 ft in diameter with an ellipsoidal bottom. The top of the vessel is closed with a flanged and bolted ellipsoidal head, which is removed for refueling. When in operation the reactor is filled with water at a pressure of 155 bar (15.5 MPa). The water enters the inlet nozzle at the left at a temperature of 282 C and leaves the outlet nozzle at the right at 317 C. The effective average temperature of the water is 301.6 C, which will be taken as the temperature of the Maxwell-Boltzmann component of the neutron flux. 

Table 3.13 gives effective cross sections for thermal neutrons and other nuclear properties of the materials in the core of this reactor. These effective cross sections have been calculated by the procedure recommended by Westcott, which has been outlined in Chap. 2, from data provided by Westcott [W3] and Critoph [Cl]. To obtain appropriate nuclear reaction rates, these effective cross sections are to be multiplied by the thermal-neutron flux, wmb. where Hub is density of neutrons in the Maxwell-Boltzmann part of the spectrum and C is the average speed of the Maxwell-Boltzmann neutrons. 

The so-called cold source neutrons emerge from a small volume ( 20 liters) of liquid deuterium maintained at around 25 K. Thermal neutrons are those moderated usually with heavy water D2O at around 330 K. A block of hot graphite at T 2000 K functions as a source of hot neutrons. The Maxwell-Boltzmann distributions for T = 25, 330, and 2000 K are illustrated in Figure 1.1. The flux, that is, the number of neutrons of velocity v that emerge from the moderator per second is proportional to v times f(v), and therefore in terms of the neutron flux that is available for scattering measurement the distribution is a little skewed in favor of higher v in comparison to that shown in Figure 1.1. 

Thus, if V is known, the reaction rate is proportional to the total neutron flux (f>. For example, for the commonly encountered Maxwell-Boltzmann distribution of thermal neutrons, v = Iv / fir, where Vp is the most probable neutron speed. 

A useful measure of the flux distortion is provided by the concept of the effective neutron temperature Tn- We define this temperature as that number which when used in (4.197) gives the best least-squares fit of a Maxwell-Boltzmann distribution to the computed flux (solid-line curve) in the range 0 < x < 35. The ratio of the effective neutron temperature to the moderator temperature Tn/Ts is indicated in the figure for the first two cases. The ratio has been omitted from the last case (A = 9, = 2 ) because the flux was so severely distorted from a... 

From the kinetic theory of gases, an expression for the net-mass flux at the interphase can be derived based on the works of Hertz [223] and Knudsen [224]. From a statistical consideration under the assumption of a Maxwell-Boltzmann distribution for the velocity of the gas molecules, the maximum condensation mass flux can be calculated. The evaporation mass flux has to equal the condensation mass flux at equilibrium. The resulting Hertz-Knudsen equation for calculating the area specific net-mass flux is given below ... 

In this equation, now represents the flux of a single incident particle. It is usually the case, however that the scatterer is not initially in one of its pure eigenstates. Instead, the target is normally in thermal equilibrium at some known temperature T. In this case, the scatterer is in a mixed quantum state and it is necessary to perform a summation over possible initial states, j), weighted according to the Maxwell-Boltzmann probability distribution. Pi = exp(- /fegT)/2y exp(- y/fegT), where 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 " ... 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

The Boltzmann equation is considered valid as long as the density of the gas is sufficiently low and the gas properties are sufficiently uniform in space. Although an exact solution is only achieved for a gas at equilibrium for which the Maxwell velocity distribution is supposed to be valid, one can still obtain approximate solutions for gases near equilibrium states. However, it is evident that the range of densities for which a formal mathematical theory of transport processes can be deduced from Boltzmann s equation is limited to dilute gases, since this relation is reflecting an asymptotic formulation valid in the limit of no coUisional transfer fluxes and restricted to binary collisions only. Hence, this theory cannot without ad hoc modifications be applied to dense gases and liquids. 

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