Maxwell-Boltzmann speed

The function f M) was predicted theoretically by Maxwell and Boltzmann about 60 years before it was first measured. It is called the Maxwell-Boltzmann speed distribution for a gas of molecules of mass m at temperature T and it has the following form ... 

An alternative interpretation of the Maxwell-Boltzmann speed distribution is helpful in statistical analysis of the experiment. Experimentally, the probability that a molecule selected from the gas will have speed in the range Au is defined as the fraction AN/N discussed earlier. Because AN/N is equal to f u) Au, we interpret this product as the probability predicted from theory that any molecule selected from the gas will have speed between u and u + Au. In this way we think of the Maxwell-Boltzmann speed distribntion f(u) as a probability distribution. It is necessary to restrict Au to very small ranges compared with u to make sure the probability distribution is a continuous function of u. An elementary introdnction to probability distributions and their applications is given in Appendix C.6. We suggest you review that material now. 

FIGURE 9.15 Mathematical form of the Maxwell-Boltzmann speed distribution. The factor cuts off the distribution at small values of u, whereas the exponential factor causes it to die off at large values of u. The competition between these effects causes the distribution to achieve its maximum value at intermediate values of u. 

The following diagram shows the Maxwell-Boltzmann speed distributions for gas at two different ten5)eratures T, and T2. Calculate the value of Ti-... 

The zeroth moment of a distribution is 1, the first moment is < i>, the second moment is < P>, etc. The higher moments of a distribution hence compute successively higher averages of the distributions of the independent variable for example, in classical statistical thermodynamics the mean square velocity is the second moment of the Maxwell-Boltzmann speed distribution for an ideal gas, and is directly related to average kinetic energy < KE > = m < v >/2, and hence to temperature [= 3k TI2 for a monatomic gas]. 

Figure 5.3 Maxwell-Boltzmann speed distribution for oxygen at various temperatures.

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. 

For rotational levels of i , < 400cm , the slow component of the TOF data was fitted by a Maxwellian speed distribution with a mean kinetic energy < KE >/2k 350 50 K (dividing by 2 to achieve Maxwell-Boltzmann equivalent temperatures). The peak surface temperature induced by the laser pulse in these experiments was calculated to be 300-320 K. The good agreement between the mean kinetic energy determined for this desorption... 

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. 

Find a formula for the most probable molecular speed, cmp. Sketch the Maxwell-Boltzmann velocity distribution and show the relative positions of (c), cmp, and cms on your sketch. 

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). 

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