One-dimensional velocity distribution

The one-dimensional velocity distribution function will be used in Section 10.1.2 to calculate the frequency of collisions between gas molecules and a container wall. This collision frequency is important, for example, in determining heterogeneous reaction rates, discussed in Chapter 11. It is derived via a change of variables, as above. Equating the translational energy expression 8.9 with the kinetic energy, we have... 

Thus, considering only the positive root for a moment, the one-dimensional velocity distribution function is... 

This is called the one-dimensional velocity distribution, since only the y direction is included. It can also be converted to a probability distribution P vy)dvy, which should be interpreted as the chance of finding any one molecule with a velocity between vy and... 

The ideal gas law has been used in many examples in earlier chapters, and some of the important physical properties of gases (the one-dimensional velocity distribution, average speed, and diffusion) were presented in Chapter 4. This chapter puts all of these results into a more comprehensive framework. For example, in Section 7.3 we work out how the diffusion constant scales with pressure and temperature, and we explore corrections to the ideal gas law. 

We went from the one-dimensional velocity distribution to the three-dimensional speed distribution by adding in an extra v2 factor to account for the added degeneracy in velocity space. The one-dimensional diffusion equation 7.36 has a form which is mathematically very similar to the one-dimensional velocity distribution... 

Chapter 7 covers the kinetic theory of gases. Diffusion and the one-dimensional velocity distribution were moved to Chapter 4 the ideal gas law is used throughout the book. This chapter covers more complex material. I have placed this material later in this edition, because any reasonable derivation of PV = nRT or the three-dimensional speed distribution really requires the students to understand a good deal of freshman physics. There is also significant coverage of dimensional analysis determining the correct functional form for the diffusion constant, for example. 

Equation (57) is the product of three one-dimensional velocity distributions,... 

Figure 12. The upper panels show one-dimensional velocity distributions of NO from the photodissociation of 2-chloro-2-nitrosopropane at 650 nm using the projection TOF method. The NO ions were produced by REMPI via the A2Z+ <- X2n transition using the branches indicated in the figure. The velocity distributions were taken with sD and ePR collinear and both cPR and the probe laser propagation direction perpendicular the axis of the detector, kD. The symbols O and indicate velocity distributions taken with eD parallel and perpendicular to kD, respectively. The lower panel shows the difference spectra of the velocity distributions, which are sensitive to the yS°(20) bipolar moment. The solid line shows a fit to the difference spectra using equations discussed in [42] and [170]. [Reprinted with permission from R. Uberna, R. D. Hinchliffe, and J. I. Cline, J. Chem. Phys., 105(22), 9847 (1996). Copyright 1996 American Institute of Physics.

The motion of activated complexes within the transition state may be analyzed in terms of classical or quantum mechanics. In terms of classical physics, motion along the reaction coordinate may be analyzed in terms of a one-dimensional velocity distribution function. In terms of quantum mechanics, motion along the reaction coordinate within the limits of the transition state corresponds to the traditional quantum mechanical problem involving a particle in a box. 

One-dimensional velocity distribution Specific conductivity, hard-sphere diameter for a collision, length parameter in Lennard-Jones potential, symmetry number of a molecule Intrinsic lifetime of a photoexcited state Azimuthal angular velocity in spherical polar coordinates, azimuthal angle in spherical polar coordinates, angle of deflection Quantum yield at wavelength A Fluorescence (phosphorescence) quantum efficiency... 

Assuming a thennal one-dimensional velocity (Maxwell-Boltzmaim) distribution with average velocity /2k iT/rr/tthe reaction rate is given by the equilibrium flux if (1) the flux from the product side is neglected and (2) the thennal equilibrium is retamed tliroughout the reaction ... 

The Doppler-selected TOF technique is one of the laser-based techniques for measuring state-specific DCSs.30 It combines two popular methods, the optical Doppler-shift and the ion TOF, in an orthogonal manner such that in conjunction with the slit restriction to the third dimension, the desired center-of-mass three-dimensional velocity distribution of the reaction product is directly mapped out. Using a commercial pulsed dye laser, a resolution of T% has been achieved. As demonstrated in this review, such a resolution is often sufficient to reveal state-resolved DCSs. 

We analyze the transport of a chemical (concentration C) along a fluid flow with mean velocity vx (x designates the direction of the flow). Examples are transport in a river (Chapter 24) or in an aquifer (Chapter 25). Imagine that at time t = 0, per cross-sectional area the total amount M (dimension ML-2) of a chemical is added to the flow at location x = 0. Then, the one-dimensional concentration distribution, C(x, t = 0) can be described by the 8-function (see Eq. 18-15) ... 

Figure 15 shows the CI2 bond distance for two different trajectories for which the initial conditions of the Ari25Cl2 cluster are the same, i.e. the configuration and the center of mass velocity of the clusters at the beginning of each trajectory (before the collision with the surface) are identical. The only differences between the two trajectories are the velocities (randomly chosen from a one-dimensional thermal distribution at 30 K) of the hard cubes that mimic the surface. Despite the rather low temperature of the surface, one of the trajectories results in the dissociation of the diatomic molecule while the other one ends with a vibrationally excited reactant molecule. The effect of the hard cube velocity on the energy of the atom scattering from the surface is negligible but the history of a single trajectory is extremely sensitive to the details of the collisions with the surface, as shown in Fig. 15. This is a characteristic of so called chaotic systems. In...

Assuming a thermal one-dimensional velocity (Maxwell-Boltzmann) distribution with average velocity... 

It follows that the normal distribution of the one-dimensional velocity, can be written as ... 

Consider a one-dimensional random walk, with a probability p of moving to the right and probability q = 1 — p of moving to the left. If p = g = 1/2, the distribution has mean p = 0 and spreads in time with a standard deviation a = sJijA. In general, though, p = (p — g)t and a = y pgt. In particular, as p moves away from the center value 1/2, the center of mass of the system Itself moves with velocity P = p — q. 

Below we consider a quasi-one-dimensional model of flow and heat transfer in a heated capillary, with hydrodynamic, thermal and capillarity effects. We estimate the influence of heat transfer on steady-state laminar flow in a heated capillary, on the shape of the interface surface and the velocity and temperature distribution along the capillary axis. 

The present model takes into account how capillary, friction and gravity forces affect the flow development. The parameters which influence the flow mechanism are evaluated. In the frame of the quasi-one-dimensional model the theoretical description of the phenomena is based on the assumption of uniform parameter distribution over the cross-section of the liquid and vapor flows. With this approximation, the mass, thermal and momentum equations for the average parameters are used. These equations allow one to determine the velocity, pressure and temperature distributions along the capillary axis, the shape of the interface surface for various geometrical and regime parameters, as well as the influence of physical properties of the liquid and vapor, micro-channel size, initial temperature of the cooling liquid, wall heat flux and gravity on the flow and heat transfer characteristics. 

Significant simplification of the governing equations may be achieved by using a quasi-one-dimensional model for the flow. Assume that (1) the ratio of meniscus depth to its radius is sufficiently small, (2) the velocity, temperature and pressure distributions in the cross-section are close to uniform, and (3) all parameters depend on the longitudinal coordinate. Differentiating Eqs. (8.32-8.35) and (8.37) we reduce the problem to the following dimensionless equations ... 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

Below the system of quasi-one-dimensional equations considered in the previous chapter used to determine the position of meniscus in a heated micro-channel and estimate the effect of capillary, inertia and gravity forces on the velocity, temperature and pressure distributions within domains are filled with pure liquid or vapor. The possible regimes of flow corresponding to steady or unsteady motion of the liquid determine the physical properties of fluid and intensity of heat transfer. 

The quasi-one-dimensional model described in the previous chapter is applied to the study of steady and unsteady flow regimes in heated micro-channels, as well as the boundary of steady flow domains. The effect of a number of dimensionless parameters on the velocity, temperature and pressure distributions within the domains of liquid vapor has been studied. The experimental investigation of the flow in a heated micro-channel is carried out. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

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