The Maxwell distribution

In a container of gas, the individual molecules are traveling in various directions with different speeds. We assume that the motions of the molecules are completely random. Then we set the following problem. What is the probability of finding a molecule with a speed between the values c and c + dc, regardless of the direction in which the molecule is traveling  

This problem can be broken down into simpler parts the solution of the problem is achieved by combining the solutions of the simpler problems. Let u, v, and w denote the components of velocity in the x, y, and z directions, respectively. Let dn be the number of molecules that have an x component of velocity with a value in the range between u and u + du. Then the probability of finding such a molecule is by definition dnJN, where N is the number of molecules in the container. If the interval width, du, is small, it is reasonable to expect that doubling the width will double the number of molecules in the interval. Thus dnJN is proportional to du. Also the probability dnJN will depend on the velocity component u. Thus we write 

At this point we must make clear why the function depends on u and not simply on u. Because of the random nature of molecular motion, the probability of finding a molecule with an x component in the range utou du must be the same as the probability of finding 

In writing Eq. (4.20) in this way, we assume implicitly that the probability dnJN is not in any way dependent on the values of the y or z components, v and w. This assumption is valid but will not be justified here. 

We now ask a more involved question What is the probability of finding a molecule that has simultaneously an x component in the range u to u du and a y component in the range i to i + dvlLot the number of molecules that satisfy this condition be dn y, then the probability of finding such a molecule is by definition dn jN, the product of the probabilities of finding molecules that satisfy the conditions separately. That is, dn jN = (dnjN)(dnjN), or 

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The velocity distribution of the electrons in a plasma is generally a complicated function whose exact shape is detennined by many factors. It is often assumed for reasons of convenience in calculations tliat such velocity distributions are Maxwellian and tliat tlie electrons are in tliennodynamical equilibrium. The Maxwell distribution is given by... 

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. 

We studied the effect of the mechanical stretching field on the conformations of the macromolecules in the melt. It is known that for a freely jointed chain the Maxwell distribution of end-to-end distances holds50). 

S FIGURE 4.27 The Maxwell distribution t again, but now the curves correspond to the speeds of a single substance inf molar fyg mass 50 g-mol 1) at different... 

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. 

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

Consider the Maxwell distribution of speeds found in Fig. 4.27. (a) From the graph, find the location that represents the most probable speed of the molecules at each temperature. 

The root mean square speed rrrm of gas molecules was derived in Section 4.10. Using the Maxwell distribution of speeds, we can also calculate the mean speed and most probable (mp) speed of a collection of molecules. The equations used to calculate these two quantities are i/mean = (8RT/-nM),a and... 

The velocity distribution in a neutron gas at equilibrium is subject to the laws of the kinetic theory of gases. The neutron velocities at equilibrium obey the Maxwell distribution... 

Doppler broadening arises from the random thermal motion of the atoms relative to the observer. The velocity V, of an atom in the line of sight will vary according to the Maxwell distribution, the atoms moving in all directions relative to the observer. The frequency will be displaced by... 

Exercise. Compute from (4.14) the jump moments, taking for F the Maxwell distribution. Show that (4.1) and (4.2) hold when V is small compared to the average speed of the gas molecules, and can therefore be used to describe equilibrium fluctuations if Mpm. 

The supply of activated molecules is thus maintained (1) by the Maxwell distribution, according to which a constant fraction of the total number are in the active state when there is no removal by chemical change, and (2) by the complete replacement of all such removals in the way postulated. The number of active molecules is thus always a constant fraction of the total concentration of reactant, and the reaction is thus kinetically unimolecular (see also p. 173). 

This expression for f(v) is now called the Maxwell distribution of speeds. AN is the number of molecules in the narrow range of speeds between v and v + Au, and N is the total number of molecules in the sample (Box 4.2). 

FIGURE 4.27 The range of molecular speeds for several gases, as given by the Maxwell distribution. All the curves correspond to the same temperature. The greater the molar mass, the lower the average speed and the narrower the spread of speeds. 

The molecules of all gases have a wide range of speeds. As the temperature increases, the root mean square speed and the range of speeds both increase. The details of the range of speeds are described by the Maxwell distribution, Eq. 28. 

Let s consider the fraction of molecules that collide with a kinetic energy equal to or greater than Emm. Because kinetic energy is proportional to the square of the speed, this fraction can be obtained from the Maxwell distribution of speeds (Section 4.13). As indicated for a specific reaction by the shaded area under the blue curve in Fig. 13.17, at room... 

These velocity components vtb in a given direction follow from the Maxwell distribution... 

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