Isotropic molecular velocity

In a molecular gas, collisions represent a mighty mechanism for energy redistribution between available degrees of freedom of the molecules which results in establishing an almost isotropic molecular velocity distribution irrespective of the... 

Using equation (14) and assuming isotropic orbital velocity distributions / (vi — v ) the corresponding expression to equation (2) for the orbital contribution to the molecular stopping was obtained as... 

In this section we shall be concerned with a molecular theory of the transport properties of gases. The molecules of a gas collide with each other frequently, and the velocity of a given molecule is usually changed by each collision that the molecule undergoes. However, when a one-component gas is in thermal and statistical equilibrium, there is a definite distribution of molecular velocities—the well-known Maxwellian distribution. Figure 1 shows how the molecular velocities are distributed in such a gas. This distribution is isotropic (the same in all directions) and can be characterized by a root-mean-square (rm speed u, which is given by... 

Consider isotropic molecular motion in a Cartesian coordinate system. If there are n molecules per unit volume, about one-third of them have velocities along the x-direction. Half of these, i.e., per unit volume, move in the (-l-x)-direction and the other half of them move in the (—x)-direction. Accordingly, one-sixth of the N molecules move in the (+r/)-direction, another one-sixth of the them move in the (—j/)-direction, another one-sixth of them in the (- -2)-direction, and finally the last one-sixth of them wiU move in the (—2)-direction. 

A number of mechanisms are known to contribute to spin-lattice relaxation in a molecule dipole-di-pole, spin-rotation, quadrupolar, scalar, and CSA, each associated with a corresponding relaxation rate. Analytical interest focuses only on the intemu-clear C-H dipole-dipole relaxation with a rate I IDD = 1/Tidd = (f//1.988)(l/Ti), which is proportional to the inverse sixth power of the carbon-proton distance rca, and depends on the rotational and internal molecular motion. For medium-sized molecules of nearly spherical shape (isotropic molecular reorientation) in solvents of low viscosity the condition of extreme narrowing holds, i.e. ct)cTc l for all Larmor frequencies ft)c, and therefore Ri=ANtc. The correlation time %c is a measure of the velocity of the molecule s rotational diffusion jumps. For N nearest protons within a distance rca the quantity A takes the form A = jrca... 

Equation (51) assumes that the eddy properties are isotropic. In addition, no effect of other gradients such as temperature or gravity upon the molecular transport is taken into account. The expression was written for a single component and it is necessary to solve a set of such expressions one for each component, simultaneously if the interrelation of the material transport upon the hydrodynamic velocity is to be taken into account. 

In the case of an orifice, Pr = 1. For other types of pipes and ducts Pr < 1. A significant part of assessing molecular flow in ducts involves the estimation of Pr. An initial assumption is that molecules arrive at the entrance plane of a duct with an isotropic velocity distribution. Conductance under molecular flow conditions is independent of the pressure but obviously the throughput is proportional to the Ap as stated by the definition of C (e.g Equation (2.6)). 

Let us now select a three-dimensional system of cartesian axes fixed with respect to the vessel. Our assumption of complete randomness of molecular motion tells us that we shall expect to find as many molecules moving with velocity components of a given range along the x axis as along the y and z axes. That is, the motion is isotropic. If we define three distribution functions P(y ), P vy), and P(vz) such that P Vx) dvx represents the fraction of all the N molecules which have x velocity components between Vx and Vx + dVx and the other two functions have similar relations with Vy and Va, the assumption of randomness tells us that the three functions are the same for the different components. The assumption of independent motion further tells us that the fraction of all molecules with... 

Despite our use of a capillary model to characterize a porous medium, most porous beds employed for chromatographic purposes are random and generally the medium is isotropic. In such media, the effective solute dispersivity still arises from the nonuniform pore velocity coupled with molecular diffusion... 

For flow with high Knudsen number, the number of molecules in a significant volume of gas decreases, and there could be insufficient number of molecular collisions to establish an equilibrium state. The velocity distribution function will deviate away from the Maxwellian distribution and is non-isotropic. The properties of the individual molecule then become increasingly prominent in the overall behavior of the gas as the Knudsen number increases. The implication of the larger Knudsen number is that the particulate nature of the gases needs to be included in the study. The continuum approximaticui used in the small Knudsen number flows becomes invalid. At the extreme end of the Knudsen number spectrum is when its value approaches infinity where the mean free path is so large or the dimension of the device is so small that intermolecular collision is not likely to occur in the device. This is called collisionless or free molecular flows. 

Essentially a hinge is composed of three layers caused by a high velocity of the pol)mner melt two highly oriented layers and one isotropic core layer. The mechanical behavior of the oriented layers is similar to that of hard elastic fibers which show a high stiffness and a high strain recovery. The higher the molecular orientation and the layer thickness, the better the hinge properties are. The layer thickness can be increased by low temperatures (melt and/or mold). 

The determination of accurate intermolecular potentials has been a key focus in the understanding of collision and half-collision dynamics, but has been exceedingly difficult to obtain in quantitative detail for even the simplest molecular systems. Traditional methods of obtaining empirical intermolecular potential information have been from analysis of nonideal gas behavior, second virial coefficients, viscosity data and other transport phenomena. However, these data sample highly averaged collisional interactions over relative orientations, velocities, impact parameters, initial and final state energies, etc. As a result intermolecular potential information from such methods is limited to estimates of the molecular size and stickiness, i.e., essentially the depth and position of the energy minimum for an isotropic well. 

The pressure of a gas is the force per unit area exerted due to molecular collisions with the walls of the container. Since force is momentum change per unit time, pressure is determined by computing the momentum transfer per unit time per unit area. Consider a wall at x = L if at each collision there is perfect reflection the x component of velocity changes from w to — m and the net momentum transfer is Imu. While perfect reflection is not a reasonable assumption we know that the gas has an isotropic velocity distribution and from (1.8), F(w, p, w) = F(—u, v, w). Thus, on the average, the fraction of molecules leaving the walls with an x component of velocity —M is the same as that striking the wall with x component of velocity u. The net effect is a mean momentum transfer of 2mu. 

Figure 2-13 Spectrum of velocity [E(n)] and temperature or concentration [r(n)] fluctuation wavenumbers (m ) in the equilibrium range of homogeneous isotropic turbulence for the case of large Sc or Pr (modified from Batchelor, 1959). In the Batchelor scale, k is either the thermal diffusivity (k/pCp) or the molecular diffusivity (Dab)-...

So far, our discussion of length scales has focused on the smallest scales of motion. At the larger scales of motion, Taylor (1921) considered the turbulent dispersion of fluid particles by homogeneous isotropic turbulence in the absence of molecular diffusion. In his model, each fluid particle leaving a point source in a uniform velocity field is expected to deviate from the linear mean path in a random manner, depending on the local nature of the turbulence. The RMS deviation of the particle paths is observed as a continued divergence, spread, or dispersion as the particles are carried downstream. This eddy motion occurs even... 

Birefringence - The light is transmitted with equal velocities in an isotropic substance. In anisotropic materials, molecular structure is such that the transmission vibration varies as a funetion of direction. A material is optically anisotropic when its refractive index depends on direetion. Birefringence is calculated as the dififerenee inreirae-tive index in two selected perpendieular direetions. 

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