Speed Maxwell-Boltzmann distribution

One may also show that MPC dynamics satisfies an H theorem and that any initial velocity distribution will relax to the Maxwell-Boltzmann distribution [11]. Figure 2 shows simulation results for the velocity distribution function that confirm this result. In the simulation, the particles were initially uniformly distributed in the volume and had the same speed v = 1 but different random directions. After a relatively short transient the distribution function adopts the Maxwell-Boltzmann form shown in the figure. 

Consider, as an example, the calculation of the mean-square speed of an ensemble of molecules which obey the Maxwell-Boltzmann distribution law. This quantity is given by... 

Figure 10 A graphical illustration of the Maxwell-Boltzmann distribution laws. Normalized speed---is vlvp, and normalized energy - is EUcT.

The distribution function (24) for an ideal gas, shown in figure 6 is known as the Maxwell-Boltzmann distribution and is specified more commonly [118] in terms of molecular speed, as... 

D) Whether you can answer this question depends on whether you are acquainted with what is known as the Maxwell-Boltzmann distribution. This distribution describes the way that molecular speeds or energies are shared among the molecules of a gas. If you missed this question, examine the following figure and refer to your textbook for a complete description of the Maxwell-Boltzmann distribution. 

Distribution of Speeds in Gas— Maxwell-Boltzmann Distribution (Rae D6jur)... 

Ohm s law assumes that the drift speed of electrons in an electric field, vd = peEy is small compared to their average speed, in a Maxwell-Boltzmann distribution. At high electric fields, E > 10 kV/cm, vd no longer increases with electric field and approaches a limiting saturation speed, determined primarily by optical phonon emission. Figure 3 shows the variation of drift speed with electric field for electrons and holes in various semiconductors. 

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

The most probable speed (u ) in the Maxwell-Boltzmann distribution is found by setting the derivative of Eq. 10.27 with respect to v to zero, and solving for v = v ... 

Maxwell-Boltzmann distribution of speeds for H2 at 100, 400, and 900 K. Note that ump increases linearly with Tl/2. 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

Due to the simple product form of the Maxwell-Boltzmann distribution, the derivations given above are easily generalized to the expression for the relative velocity in three dimensions. Since the integrand in Eq. (2.18) (besides the Maxwell-Boltzmann distribution) depends only on the relative speed, we can simplify the expression in Eq. (2.18) further by integrating over the orientation of the relative velocity. This is done by introducing polar coordinates for the relative velocity. The full three-dimensional probability distribution for the relative speed is... 

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. 

In Section 5.2, we will derive the three-dimensional Maxwell-Boltzmann distribution n(v)dv of molecular speeds between v and v + dv in the gas phase ... 

PROBLEM 4.20.2. Show for a Maxwell-Boltzmann distribution of Eq. (4.20.1) that the most probable molecular speed vmp is given by Eq. (4.20.6). 

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. 

A probability distribution enables us to calculate the average of the values obtained in several repetitions of the experiment it describes. The procedure is described in Appendix C.6. For the Maxwell-Boltzmann distribution, this calculation gives the average speed u, which is... 

There are several possible ways to characterize a non-symmetrical probability distribution by a single number. The three different speeds discussed above serve this purpose for the Maxwell-Boltzmann distribution. Because the distribution is non-symmetrical, they are close to each other but are not equal. They stand in the ratio ... 

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). 

To obtain Eqs (1.203) and (1.206) we need to assume that P vanishes asx - 00 faster than Physically this must be so because a particle that starts at x = 0 cannot reach beyond some finite distance at any finite time if only because its speed cannot exceed the speed of light. Of course, the diffusion equation does not know the restrictions imposed by the Einstein relativity theory (similarly, the Maxwell-Boltzmann distribution assigns finite probabilities to find particles with speeds that exceed the speed of light). The real mathematical reason why P has to vanish faster than jg that in... 

Thus, although the individual molecular motions are chaotically unpredictable, their average behavior is entirely predictable and satisfies a particular probability distribution (the Maxwell-Boltzmann distribution). Quantities, such as the temperature that appear in Boyle s Law are measures of the average speed of the molecules. If we reran the tape of the history of our gas, we would find essentially the same average behavior, in accord with Boyle s Law, even though the individual trajectories of the molecules would be quite different. 

The reference speed u is arbitrarily chosen as 2200 m/s, which is the most probable speed for a Maxwell-Boltzmann distribution at temperature f= 293.2 K. The cross section a is now the specially defined effective cross section that, when multiplied by the 2200 m/s flux gives the proper reaction rate constant. 

Thus, if V is known, the reaction rate is proportional to the total neutron flux (f>. For example, for the commonly encountered Maxwell-Boltzmann distribution of thermal neutrons, v = Iv / fir, where Vp is the most probable neutron speed. 

Two types of distribution functions have been discussed so far the spatial distribution of molecules in a gravitational field, the Boltzmann distribution, and the speed distribution in a gas (the Maxwell distribution). These can be written in a combined form, the Maxwell-Boltzmann distribution law. 

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