Maxwell-Boltzmann distribution curve

Most of the chemical reactions occur in the condensed phase or in the gas phase under conditions such that the number of intermolecular collisions during the reaction time is enormous. Internal energy is quickly distributed by these collisions over all the molecules according to the Maxwell-Boltzmann distribution curve. 

For the case n = 30, TF is shown as a function of r in Fig. 16. W(r)r dr, like the well-known Maxwell-Boltzmann distribution curve, is thus asymmetrical it also has greater breadth on account of the smaller number of statistically independent elements. The result is that we cannot regard the most probable value as the only one likely to be present, as we would in the ordinary statistical treatment of gases. We must, rather, consider neighboring values and attach much greater significance to fluctuation phenomena than in gas statistics. We shall return later to this point, when the occurrence of x-ray interferences in stretched rubber are discussed. 

The Maxwell-Boltzmann distribution curve is a probability density function. State the assumptions underlying the Maxwell-Boltzmann distribution curve and explain why it is a probability function. 

For the majority of the graphs in chemistry the area under the graph does not represent a useful physical quantity. However, the area under the curve is relevant to the Maxwell-Boltzmann distribution curve (Chapter 6), which is useful in accounting for the rate of reaction at different temperatures. It is a frequency distribution curve which shows the distribution of kinetic energies among reacting gas particles at a particular absolute temperature, T (in kelvins) (Figure 11.37). 

Figure 1.9 Molecular energies follow the Maxwell-Boltzmann distribution energy distribution of nitrogen molecules (as y) as a function of the kinetic energy, expressed as a molecular velocity (as x). Note the effect of raising the temperature, with the curve becoming flatter and the maximum shifting to a higher energy...

These two different concepts lead to different mathematical expressions which can be tested with the experimental data. The derivation is similar to that of equations (1-5) but with the inclusion of a term, calculated from the Maxwell-Boltzmann distribution, for the fraction of molecules in the activated state. With these formulas it can be shown that when the reciprocal of the velocity constant is plotted against the reciprocal of the initial pressure a straight line is produced, according to Theory I, but a curved line is produced if Theory II is correct. Moreover the extent of the curvature depends on the complexity of the molecule. It is found that simple molecules like nitrous oxide give astraight line, and more complicated molecules, like azomethane, give er curved line. ... 

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. 

If a probability distribution is symmetrical about its maximum, like the familiar bell curve, the most probable value and the average value are the same. The Maxwell-Boltzmann distribution is not symmetrical the area under the curve to the right of the maximum is somewhat larger than the area under the curve to the left of the maximum. (The next paragraphs use the mathematical form of the distribution to explain this fact.) Consequently, u will be larger than the most probable value of u. 

Fig. 3. A typical translational energy distribution of Cl atomic photofragments generated in the 193 nm photodissociation of solid CI2 molecules on the Si wafer. The smoothed curves are sum of two Maxwell-Boltzmann distributions with two different temperatures (Et = (3/2)kT).

Fig. 78 Time-of-arrival curves of the 28 amu fragment a irradiation with 308 nm and 260 mj cm 2 and b irradiation with 248 nm and 260 mj cnT2. The curves (1) and (2) represent Gaussian distribution, the curve (3) is fitted by a decaying Maxwell-Boltzmann distribution, while the curve (4) is the sum of the three fitted curves. REPRINTED WITH PERMISSION OF [Ref. 223], COPYRIGHT (2002) Elsevier Science...

A useful measure of the flux distortion is provided by the concept of the effective neutron temperature Tn- We define this temperature as that number which when used in (4.197) gives the best least-squares fit of a Maxwell-Boltzmann distribution to the computed flux (solid-line curve) in the range 0 < x < 35. The ratio of the effective neutron temperature to the moderator temperature Tn/Ts is indicated in the figure for the first two cases. The ratio has been omitted from the last case (A = 9, = 2 ) because the flux was so severely distorted from a... 

The total area under the curve is directly proportional to the total number of molecules and the area under any portion of the curve is directly proportional to the number of molecules with kinetic energies in that range. When the temperature of a gas, liquid or solution is increased, a number of changes occur in the shape of the Maxwell-Boltzmann distribution (see Figure 6.18) ... 

FIGURE 19.4 Distributions of speeds for various gases. Note how the curve for H2 is shifted to higher velocities at 500 K. These curves are collectively called Maxwell-Boltzmann distributions. 

The coefficient D, being proportional to a normalizing coefficient C of the Maxwell-Boltzmann energy distribution W = C exp(- / ), is determined by the parameters of the hat-curved model as... 

This equation relates the reaction rate (cross section) of a material at the temperature to its cross section at a lower temperature Tn- As in all the preceding analyses, it is assumed that the nuclei are distributed according to the Maxwell-Boltzmann relation (4.198). The above equation may be used then to compute the cross-section curves for a reactor operating at any temperature from the known cross-section data of the reactor material which have been determined at some temperature Tn. Note that U = Tn then Eq. (4.233) reduces to an identity [see also (4.225)]. In general, the indicated integration must be carried out in detail. There is one special case, however, which leads to an especially simple result. If the measured cross-section curve varies as l/v, which is the case for many absorbers in the low-energy range, then it is easily shown that the reaction rate is independent of the moderator temperature. For example, if we take aa v) = Co/y, where Co is some constant, then from (4.233)... 

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