The molecular velocity distribution

In FPTRMS, transport of the reactive species of interest from the reactor to the detector can make a contribution to the observed time dependence such that the chemical kinetics becomes convoluted with mass transport rates. This will have to be accounted for in data analysis if reliable rate coefficients are to be obtained. If the physical rate processes are sufficiently fast they will make a negligible contribution to the kinetics. In this section we examine the above four factors to see when they influence the chemical kinetics. The first, third, and fourth items put an upper limit on the rate at which decays and growths can be reliably determined, and the second one sets a lower limit on the decay rate. 

For most FPTRMS designs the reactor to source distance is of the order of 1 cm, and measurable distortion can be expected when time constants of observed transients are smaller than about 1 ms. This was predicted by a 

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The thermodynamic equilibrium of the system was controlled by examining the molecular velocity distribution (MVD) and the first-neighbor distribution. In the case where only a first-neighbor interaction is present, the latter, as will be shown in Section IV, has an analytical expression given by... 

The molecular velocity distribution was monitored by averaging over the total number of time steps and all the particles. It was foimd to reproduce the Maxwell distribution relative to the average temperature of the system (the value of (v y/(v ) - was about 0.01). We wish to point out, however, that the thermalization in the one-dimensional case offered some difficulties, in contrast with the two-dimensional case, where it requires usually only a few hundred time steps. This occurred principally at low densities where the collisions between the particles are mainly binary collisions. In this case, with our starting conditions, the distribution was found to be... 

Therefore, the conceptual ideas of kinetic theory rely on the assumption that the mean flow, transport and thermodynamic properties of a collection of gas molecules can be obtained from the knowledge of their masses, number density, and a probabilistic velocity distribution function. The gas is thus described in terms of the distribution function which contains information of the spatial distributions of molecules, as well as about the molecular velocity distribution, in the system under consideration. 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann energy (or momentum or internal coordinate) distribution of reactants is postulated to persist during a reaction. In the collision theory, mainly due to Hinshelwood,7 the number of energetic, reaction producing collisions is calculated under the assumption that the molecular velocity distribution always remains Maxwellian. In the absolute... 

Taatjes, C.A. (2007) How Does the Molecular Velocity Distribution Affect Kinetics Measurements by Time-resolved Mass Spectrometry Int. J. Chem. Kinet. 39 565-570. 

In a series of impressive publications. Maxwell [95-98] provided most of the fundamental concepts constituting the statistical theory recognizing that the molecular motion has a random character. When the molecular motion is random, the absolute molecular velocity cannot be described deterministically in accordance with a physical law so a probabilistic (stochastic) model is required. Therefore, the conceptual ideas of kinetic theory rely on the assumption that the mean flow, transport and thermodynamic properties of a collection of gas molecules can be obtained from the knowledge of their masses, number density, and a probabilistic velocity distribution function. The gas is thus described in terms of the distribution function which contains information of the spatial distributions of molecules, as well as about the molecular velocity distribution, in the system under consideration. An important introductory result was the Maxwellian velocity distribution function heuristically derived for a gas at equilibrium. It is emphasized that a gas at thermodynamic equilibrium contains no macroscopic gradients, so that the fluid properties like velocity, temperature and density are uniform in space and time. When the gas is out of equilibrium non-uniform spatial distributions of the macroscopic quantities occur, thus additional phenomena arise as a result of the molecular motion. The random movement of molecules from one region to another tend to transport with them the macroscopic properties of the region from which they depart. Therefore, at their destination the molecules find themselves out of equilibrium with the properties of the region in which they arrive. At the continuous macroscopic level the net effect... 

However, it was Maxwell in 1848 who showed that molecules have a distribution of velocities and that they do not travel in a direct line. One experimental method used to show this was that ammonia molecules are not detected in the time expected, as derived from their calculated velocity, but arrive much later. This arises l om the fact that the ammonia molecules tnterdiffuse among the air moixules by intermolecular collisions. The molecular velocity calculated for N-ls molecules from the work done by Joule in 1843 was 5.0 xl02 meters/sec. at room temperature. This implied that the odor of ammonia ought to be detected in 4 millisec at a distance of 2.0 meters from the source. Since Maxwell observed that it took several minutes, it was fuUy obvious that the molecules did not travel in a direct path. 

Another method is to measure the disappearance rate of the excited parent molecules, that is, the intensity changes of the disk-like images at various delay times (therefore, at various photolysis laser positions) along the molecular beam. This is very useful when the dissociation rate is slow and the method described above cannot be applied. This measurement requires a small molecular beam velocity distribution and a large variable distance between the crossing points of the pump and probe laser beams with the molecular beam. The small velocity distribution can be obtained through adiabatic expansion, and the available distances between the pump and probe laser beams depend on the design of the chamber. For variable distances from 0 to 10 cm in our system and AV/V = 10% molecular beam velocity distribution, dissociation rates as slow as 3 x 103 s 1 under collisionless condition can be measured. 

Equations suitable for simulation of molecular weight distributions for any initial distribution and chosen values of G(scission) and G(crosslinking) have been developed and demonstrated. The molecular weight distributions may be obtained by GPC (with the limitation of changes in relative hydrodynamic volumes) and by sedimentation velocity in the ultracentrifuge. 

For the other extreme of the free molecular regime where Kn - oc, the particle radius is small compared to the mean free path. In this case, the thermal velocity distribution of the gas is not distorted by uptake at the surface. In effect, the gas molecules do not see the small particles. For this case, Fuchs and Sutugin (1970, 1971) show that for diffusion to a spherical particle of radius a... 

In porous media the flow of water and the transport of solutes is complex and three-dimensional on all scales (Fig. 25.1). A one-dimensional description needs an empirical correction that takes account of the three-dimensional structure of the flow. Due to the different length and irregular shape of the individual pore channels, the flow time between two (macroscopically separated) locations varies from one channel to another. As discussed for rivers (Section 24.2), this causes dispersion, the so-called interpore dispersion. In addition, the nonuniform velocity distribution within individual channels is responsible for intrapore dispersion. Finally, molecular diffusion along the direction of the main flow also contributes to the longitudinal dispersion/ diffusion process. For simplicity, transversal diffusion (as discussed for rivers) is not considered here. The discussion is limited to the one-dimensional linear case for which simple calculations without sophisticated computer programs are possible. 

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