Isotropic molecular velocity distribution

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 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)). 

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... 

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. 

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. 

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... 

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