Maxwell’s distribution

About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again, in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature, which can be directly linked to the thermodynamic temperature, with the temperature of Maxwell s distribution, and with the perfect gas law. 

For the distribution of velocities in the gas, in any given plane we have, according to Maxwell s distribution law ... 

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,... 

Equation (7) is called Maxwell s distribution law after Clerk Maxwell (1831-79), Professor of Physics at Cambridge (England), who obtained it in 1860, before Boltzmann in 1871 obtained his wider distribution law, Maxwell derived his distribution lav/ from the conservation of energy together with the assumption that the motion is separable in three mutually orthogonal directions. The latter assumption was violently attacked by mathematicians, but we now iccognize that the assumption is both reasonable and true,... 

Based on the value that takes (0 =) and Fig. 1.16 we can see that the unsymmetrical distribution is in question. The x -distribution is derived from Maxwell s distribution of molecular velocities in gases [5],... 

Maxwell s Distribution of Velocities.—lieturning to our distribution law (1.5), let us first consider the case where there is no potential energy, so that the distribution is independent of position in space. Then the fraction of all molecules for which the momenta lie in dpx dpy dpt is... 

This is the more familiar form of Maxwell s distribution law. We give a... 

Eq. 3.4 is called the Maxwell s distribution of molecular velocities in one dimension. It is easy to derive the Maxwell s distribution of molecular velocities in three dimensions by multiplying the three one-dimension distributions with one another. Thus,... 

This result was obtained by Maxwell in 1860 and is called the Maxwell s distribution of molecular velocities. It is customary to writep(c)dcas dNIN, where /Vis the total number of gas molecules. The quantity dN/Nor p(c)dc gives the fraction of molecules with speeds between c and c + dc. The molecular mass m = Mn/NA where Mm is the molar mass and NA is the Avogadro number. Accordingly, Eq. 3.7 may also be written as... 

The Maxwell s distribution of molecular velocities is plotted in the following figure. 

In collision theory we postulate that not all collisions are reactive only those collisions with energy E or greater will react. Assume that the fraction of collisions which have enough energy to result in reaction is given by Maxwell s distribution function. 

Maxwell s distribution law states that the fraction of molecules of mass in a given volume having a velocity in a given direction between u and u- - du)s... 

We shall make the calculations for nitrogen at 1 atm and at 25 X. According to Maxwell s distribution law the fraction of molecules having a velocity component between u and +in a given direction is... 

The main aspect of kinetic theory is that the molecules in a gas are in a state of continuous random motion. The speed of molecules depends on the temperature, and can have a remge of values. Maxwell s distribution curve is perfect to analyze this fact. In the Maxwell s distribution curve, the relative number of molecules are plotted against the molecular speed on the x-axis. Take a look at the curve in Figure 6-2. 

Maxwell s Distribution of Speeds We will now return to our derivation of... 

The equation (12) is Maxwell distribution function for a gas and describes the ecjuilihrium state of distribution function f. The constants can find if replace equation (12) whit conservation laws obtains the common form of Maxwell s distribution function... 

A point of view very import, is the obtaining of equilibrium distribution function, it was mentioned before, of this depends obtaining apvpropiiately the Navier-Stokes equations. Maxwell s distribution that is employing like equilibrium distribution g for unitary particle mass and "D" dimensions is... 

The number of molecules n possessing an energy in excess of a given value of kinetic energy E is given by Maxwell s Distribution Law... 

Clausius made the simplification of imagining that all the molecules in a sample of gas possessed the same velocity. James Clerk Maxwell (1831-1879) realised that there was a wide distribution of molecular velocities, and in 1859 he applied statistical techniques to the problem. He showed how the various possible velocities are distributed between the molecules and he showed how an average velocity could be calculated. Later, in 1871, Ludwig Eduard Boltzmann (1844-1906) succeeded in generalising Maxwell s distribution law to cases where there are forces (such as gravity) acting on the molecules. 

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