The Maxwell Distribution of Velocities

FIGURE 10-9 Schematic diagram of two gases separated by a moveable barrier. 

Boltzmann generalized Maxwell s theory and found that the probability p that a molecule will be found in a state with an energy E is given by Equations 10-15, 

Just remember two things. First, when considering molecular properties, we will find that energy is distributed according to this exponential form. Second, thermodynamic quantities can be calculated from these equations they are the essential link between the microscopic molecular world and the macroscopic world of thermodynamic quantities. But let s consider this in a simpler form, Boltzmann s original discovery and formulation of his holy grail —a mathematical description of entropy. We will start by reminding you of the character of spontaneous or irreversible processes. 

---

In the first place, we shall find that the four quantities Ty px0y py0y pz0 must be constant at all points of space, for equilibrium. By comparison with Eq. (2.4) of Chap. IV, the formula for the Maxwell distribution of velocities, we see that T must be identified with the temperature, which must not vary from point to point in thermal equilibrium. The quantities pxo, pyo, p 0 are the components of a vector representing the mean momentum of all the molecules. If they are zero, the distribution (2.15) agrees exactly with Eq. (2.4) of Chap. IV. If they are not zero, however, Eq. (2.15) represents the distribution of velocities in a gas with a certain velocity of mass motion, of components pxo/my pyQ/my pzo/m. The quantities px — pxo, etc., represent components of momentum relative to this momentum of mass motion, and the relative distribution of velocities is as... 

Before calculating V and V we shall derive a property of the above equation which simplifies calculations greatly and also can serve as a check. Evidently equation (1) is satisfied by the Maxwell distribution if 7 = 0. This together with the principle of detailed balance permits us to give equation (1) a symmetric form. If we denote the Maxwell distribution of velocity by M v)... 

Due to the necessity of producing a condensed volume, the content of this book has been devoted to a limited part of physics and physical chemistry that excludes more complicated systems. More sophisticated Formal Objects that are the distributed dipoles and systems with assemblies of these objects have not been studied further. These concepts enable us to handle systems with several energy levels that are of paramount importance in condensed matter physics and in many other fields. The Maxwell distribution of velocities in a gas is one of the important systems that are modeled with this Formal Object. 

Before proceeding to the Maxwell distribution of velocities it will be useful to derive equation (11-59). This was assumed without proof in the appendix to the last chapter. It is... 

Let us look at the physical conditions which make local equilibrium a valid assumption. First we must look at the concept of temperature. From statistical mechanics it can be seen that temperature is well defined when the velocity distribution is Maxwellian. According to the Maxwell distribution of velocities, the probability P y) that a molecule has a velocity v is given by... 

In the transformation from the microscopic world to the macroscopic one, we also need to eonsider the effect of molecular collision on the distribution of molecular velocity or energy in these systems. The majority of molecules will have a velocity close to the mean value for the molecules, but there are always some molecules with velocity much greater than and others with velocity much lesser than the mean velocity. The distribution of velocities of gas molecules was first described by Maxwell in 1860. The Maxwell distribution of velocities 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. 

The initial condition is, as usual, that the radical pair is separated by a distance r0, both radicals A and B initially have Maxwell distribution of velocities... 

See, however, the postulate of Maxwell [2] about the anisotropic distribution of velocities in gases having a stationary shearing motion, which he states without any attempt at proof ... 

Equation (2.4) is one form of the famous Maxwell distribution of velocities. 

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 ... 

A high-power microwave pulse of typical duration Tp = 1/isec is applied to a gas sample in the cavity at thermodynamic equilibrium and at a pressure of several millitorr or less. Our discussion is of course limited to the region aw k to as discussed earlier, although this restriction does not appear to be important experimentally. To help simplify some later results, we will for convenience also specify w= wc and aw aw(. that is, all polarization and emission processes are to be carried out well within the cavity bandwidth, with the cavity tuned to the carrier. Polarization of the gas is described by Equations 102 and 105. Using these and Equation 109, the field emitted from a static gas in the cavity is obtained.7 Integrating over a Maxwell distribution of velocities for the static gas, one obtains... 

The Maxwell-Boltzmann distribution is a result of remarkable generality it is independent of pressure and applies to any material, regardless of composition or phase. Figure i-4 shows this distribution for water at three temperatures. At the triple point, the solid, liquid, and vapor, aU have the same distribution of velocities. 

The table of thermal cross sections typically includes values for monoenergetic neutrons, 0.0253 eV (velocity of2,200 m s ) rather than cross sections averaged over the entire Maxwell distribution. In practice, however, thermal cross sections are measured or used, not for a single velocity, but for the entire Maxwell distribution of velocities present in a nuclear reactor. In the design of nuclear reactors, for example, neutron flux and cross sections appropriate for the entire Maxwell distribution are of course essential, whereas the monoenergetic neutron flux (o) and the corresponding cross sections are most useful in computations of production rates of radionuclides in nuclear reactors. 

Let the equilibrium in velocities be established much more rapidly than that in coordinates. Then, considering the space-time evolution of the system, one may suppose that the Maxwell distribution in velocities occurs at all times, so that ... 

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. 

A determination of the neutron temperature would therefore involve the measurement of the average kinetic energy of the neutrons, or their "root mean square" velocity (vj-ms = y ). These quantities, however, are practically not measureable. The root-mean-square velocity is related to the most probable velocity Vg of the Maxwell distribution of thermal neutrons, the latter velocity being a measureable quantity ... 

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... 

The initial velocities may also be chosen from a uniform distribution or from a simp Gaussian distribution. In either case the Maxwell-Boltzmann distribution of velocities usually rapidly achieved. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

If Restart is not checked then the velocities are randomly assigned in a way that leads to a Maxwell-Boltzmann distribution of velocities. That is, a random number generator assigns velocities according to a Gaussian probability distribution. The velocities are then scaled so that the total kinetic energy is exactly 12 kT where T is the specified starting temperature. After a short period of simulation the velocities evolve into a Maxwell-Boltzmann distribution. 

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... 

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

Often, we will be interested in how the velocities of molecules are distributed. Therefore we need to transform the Boltzmann distribution of energies into the Maxwell-Boltzmann distribution of velocities, thereby changing the variable from energy to velocity or, rather, momentum (not to be confused with pressure). If the energy levels are very close (as they are in the classic limit) we can replace the sum by an integral ... 

However, it was Maxwell in 1848 who showed that molecules have a distribution of velocities and that they do not travel in a direct line. One experimental method used to show this was that ammonia molecules are not detected in the time expected, as derived from their calculated velocity, but arrive much later. This arises l om the fact that the ammonia molecules tnterdiffuse among the air moixules by intermolecular collisions. The molecular velocity calculated for N-ls molecules from the work done by Joule in 1843 was 5.0 xl02 meters/sec. at room temperature. This implied that the odor of ammonia ought to be detected in 4 millisec at a distance of 2.0 meters from the source. Since Maxwell observed that it took several minutes, it was fuUy obvious that the molecules did not travel in a direct path. 

The frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

---