Maxwell velocity distribution function average

In actuality, molecular velocities are not all the same. At any time some molecules are moving much faster than the average while others are moving more slowly than the average. For a perfect gas the velocity distribution (in one dimension) is given by the Maxwell-Boltzmann distribution function,... 

This function should have the property Cvf—t) = Cv (t), and at r = 0 should agree with the average (V ) predicted by the Maxwell velocity distribution,... 

Simple collision theories neglect the internal quantum state dependence of a. The rate constant as a function of temperature T results as a thennal average over the Maxwell-Boltzmaim velocity distribution p Ef. 

Using Maxwell s distribution for particle velocity, Martin obtained the following expression for the average random particle velocity as a function of particle size and concentration,... 

The velocity correlation function can be determined from Eq. (5.9.17) by taking the dot product of V(0) with each term in the equation followed by averaging over a Maxwell distribution of initial velocities. Then... 

Maxwell found that he could represent the distribution of velocities statistically by a function, known as the Maxwellian distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was... 

The average velocity of a gas molecule is determined by the molecular weight and the absolute temperature of the gas. Air molecules, like many other molecules at room temperature, travel with velocities of about 500 m s"1 but there is a distribution of molecular velocities. This distribution of velocities is explained by assuming that the particles do not travel unimpeded but experience many collisions. The constant occurrence of such collisions produces the wide distribution of velocities. The quantitative treatment was carried out by Maxwell in 1859, and somewhat later by Boltzmann. The phenomenon of collisions leads to the concept of a free path, that is the distance traversed by a molecule between two successive collisions with other molecules of that gas. For a large number of molecules, this concept must be modified to a mean free path which is the average distance travelled by all molecules between collisions. For molecules of air at 25°C, the mean free path X at 1 mbar is 0.00625 cm. It is convenient therefore to use the following relation as a scaling function ... 

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