Collisions, kinetic theory

The mean free path, A is another fundamental scale necessary for describing the dynamical properties of gases. Mean free path is defined as the average distance traveled by the molecule between two successive collisions. Kinetic theory of gases establishes the following two expressions for the mean free path of gases. 

Here, a molecule of C is formed only when a collision between molecules of A and B occurs. The rate of reaction r. (that is, rate of appearance of species C) depends on this collision frequency. Using the kinetic theory of gases, the reaction rate is proportional to the product of the concentration of the reactants and to the square root of the absolute temperature ... 

There is a restriction on this simple model for the C0-N02 reaction. According to the kinetic theory of gases, for a reaction mixture at 700 K and concentrations of 0.10 M, every CO molecule should collide with about 109 N02 molecules in one second. If every collision were effective, the reaction should be over in a fraction of a second. In reality, this does not happen under these conditions, the half-life is about 10 s. This implies that not every CO-N02 collision leads to reaction. 

Z, the collision frequency, which gives the number of molecular collisions occurring in unit time at unit concentrations of reactants. This quantity can be calculated quite accurately from the kinetic theory of gases, but we will not describe that calculation. 

The pressure behavior shown in Figure 4-3 is readily explained in terms of the kinetic theory of gases. There is so much space between the molecules that each behaves independently, contributing its share to the total pressure through its occasional collisions with the container walls. The water molecules in the third bulb are seldom close to each other or to molecules provided by the air. Consequently, they contribute to the pressure exactly the same amount they do in the second bulb—the pressure they would exert if the air were not present. The 0.0011 mole of water vapor contributes 20 mm of pressure whether the air is there or not. The 0.0050 mole of air contributes 93 mm of pressure whether the water vapor is there or not. Together, the two partial pressures, 20 mm and 93 mm, determine the measured total pressure. 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

When mass transfer rates are very high, limitations may be placed on the rate at which a component may be transferred, by virtue of the limited frequency with which the molecules collide with the surface. For a gas, the collision rate can be calculated from the kinetic theory and allowance must then be made for the fact that only a fraction of these molecules may be absorbed, with the rest being reflected. Thus, when even a pure gas is brought suddenly into contact with a fresh solvent, the initial mass transfer rate may be controlled by the rate at which gas molecules can reach the surface, although the resistance to transfer rapidly builds up in the liquid phase to a level where this effect can be neglected. The point is well illustrated in Example 10.4. 

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

The kinetic theory—also called the collision theory—... 

According to Trautz and Lewis, who gave the first treatment of reaction rates in terms of the kinetic theory of collisions in 1916-1918, the rate of collisions (not yet reaction) between the spheres A and B is jtd u... 

As a first approximation, we can introduce wall collisions by simply introducing separate effectiveness parameters for molecule-molecule and molecule-wall collisions, in which fV3il can be described through kinetic theory as... 

The chance of a collision will obviously depend upon the number of gas molecules per unit volume or, alternatively, upon the pressure. The chance of a collision will also depend upon the size of the gas molecules. For example, the chance of two basketballs thrown toward one another undergoing a collision is much greater than the chance of having a similar collision between two golf balls. An expression for the mean free path in terms of pressure and molecular diameter may be derived from kinetic theory. We give only the result, which may be expressed as... 

The principles of kinetic theory may be used to arrive at an expression for the number of collisions whose relative kinetic energy along the line of centers is greater than ec. The result is the following expression for the number of such collisions per unit volume per unit time. 

The kinetic theory of radon progeny attachment to aerosol particles assumes that unattached atoms and aerosol particles undergo random collisions with the gas molecules and with each other. The attachment coefficient, 3(d), is proportional to the mean relative velocities between progeny atoms and particles and to the collision cross section (Raabe, 1968a) ... 

Contamination from the gas phase will be proportional to the number of collisions a surface undergoes. Hence, from kinetic theory, the number of collisions per unit time and unit area, Zc, is given by ... 

In the kinetic theory, the gas molecules are represented by hard spheres colliding elastically with each other and with the container walls. Details of this theory are given, for example, in ref. [1], An important parameter that can be calculated by this model is A, the mean free path of a molecule between collisions. The mean free path A of molecules is ... 

In summary, polymeric flocculants generally increase peri-kinetic flocculation rates compared with perikinetic coagulation rates. This is not necessarily true for orthokinetic flocculation, and experimental results in the literature are seemingly in conflict. Collision rate theory predicts that the polymer adsorption step may become rate limiting in orthokinetic flocculation. The present study was designed to elucidate the relationship between polymer adsorption rates and particle flocculation rates under orthokinetic conditions. 

The BET surface area equation is based on Langmuir s kinetic theory of monolayer gas adsorption on surfaces [6], Langmuir theorized that the collision... 

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