Maxwell distribution of molecular

This factor is reminiscent of the radial distribution function for electron probability in an atom and the Maxwell distribution of molecular velocities in a gas, both of which pass through a maximum for similar reasons. 

J I I Describe the effect of molar mass and temperature on the Maxwell distribution of molecular speeds (Section 4.11). 

Maxwell distribution of molecular speeds The formula for calculating the percentage of molecules that move at any given speed in a gas at a specified temperature. 

In Eq. (20.32) we have the result that the expectation value of the square of the energy is equal to the square of the expectation value of the energy. This could not be correct if the energy were in some way distributed. The reader will recall that in dealing with the Maxwell distribution of molecular speeds we found that... 

The Maxwell distribution of molecular speeds permits the evaluation of such important quantities as the pressure p exerted by a dilute gas and the collision frequency Z in the gas under given conditions. The pressure is then given by... 

Gaseous reactants collide but most collisions do not yield a chemical reaction. The nonreactive collisions of molecules with one another and with the walls of the vessel cause the high energy molecules to lose some of their energy, but the Maxwell distribution of molecular velocities is rapidly restored and maintained. 

The ordinate [the ratio dn/dv, or the coefficient of dv in (lO)] is the Maxwell distribution of molecular (neutron) speeds the number of neutrons per unit volume with speeds between v and v + dv is represented on the graph by the area of the narrow vertical strip of width dv. 

A plot of the Maxwell distribution for the same gas at several different temperatures shows that the average speed increases as the temperature is raised (Fig 4.27). We knew that already (Section 4.9) but the curves also show that the spread of speeds widens as the temperature increases. At low temperatures, most molecules of a gas have speeds close to the average speed. At high temperatures, a high proportion have speeds widely different from their average speed. Because the kinetic energy of a molecule in a gas is proportional to the square of its speed, the distribution of molecular kinetic energies follows the same trends. 

A study of the Maxwell-Boltzmann distribution of molecular energies at temperatures T Kelvins and at (T + 10) K illustrates this. 

The Maxwell-Boltzmann distribution of molecular energies can be used to explain how a catalyst works at constant temperature. 

Chemical processes, in contrast, are processes that are not limited by rates of energy transfer. In thermal processes, chemical reactions occur under conditions in which the statistical distribution of molecular energies obey the Maxwell-Boltzmann form, i.e., the fraction of species that have an energy E or larger is proportional to e p(—E/RT). In other words, the rates of intermolecular collisions are rapid enough that all the species become thermalized with respect to the bulk gas mixture (Golden and Larson, 1984 Benson, 1976). 

Therefore the three-dimensional Maxwell-Boltzmann distribution of molecular speeds is... 

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],... 

This result was given in Eq. (2.28). The well-known Maxwell-Boltzmann distribution of molecular speeds, Eq. (2.27), is obtained after substitution of E = mv 2/2, dE = mvdv. 

We know that all the gas molecules do not travel with the same velocity. This is because the molecules are colliding with one another quite frequently and so their velocities keep on changing. Maxwell worked out the distribution of molecules between different possible velocities by using probability considerations. According to him, the distribution of molecular velocities is given by,... 

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. 

Solution The Maxwell equation for distribution of molecular velocities may be put as... 

FIG U R E 9.14 The Maxwell-Boltzmann distribution of molecular speeds in nitrogen at three temperatures. The peak in each curve gives the most probable speed, u p, which is slightly smaller than the root-mean-square speed, Urms The average speed Uav (obtained simply by adding the speeds and dividing by the number of molecules in the sample) lies in between. All three measures give comparable estimates of typical molecular speeds and show how these speeds increase with temperature. 

Use the Maxwell-Boltzmann distribution of molecular speeds to calculate root-mean-square, most probable, and average speeds of molecules in a gas (Section 9.5, Problems 41-44). 

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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