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. 

Plots of molecular speed or energy versus the number of molecules with that speed or energy are often called Boltzmann distribution curves. The area under the curve reflects the total number of molecules in the sample. 

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). 

Boltzmann, L. 18. 19 Boltzmann constant 337 Boltzmann distribution law 514-23 bubble-pressure curve in vapor + liquid phase equilibrium 406... 

Fig. 5.2 Radial distribution curves, Pv

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...

At room temperature, most of the molecules are in the lowest vibrational level of the ground state (according to the Boltzmann distribution see Chapter 3, Box 3.1). In addition to the pure electronic transition called the 0-0 transition, there are several vibronic transitions whose intensities depend on the relative position and shape of the potential energy curves (Figure 2.4). 

Fig. 6-46. Differential capacity observed and computed for an n-type semiconductor electrode of zinc oxide (conductivity 0. 59 S cm in an aqueous solution of 1 M KCl at pH 8.5 as a function of electrode potential solid curve s calculated capacity on Fermi distribution fimction dashed curve = calculated capacity on Boltzmann distribution function. [From Dewald, I960.]...

Little is known about the fluorescence of the chla spectral forms. It was recently suggested, on the basis of gaussian curve analysis combined with band calculations, that each of the spectral forms of PSII antenna has a separate emission, with Stokes shifts between 2nm and 3nm [133]. These values are much smaller than those for chla in non-polar solvents (6-8 nm). This is due to the narrow band widths of the spectral forms, as the shift is determined by the absorption band width for thermally relaxed excited states [157]. The fluorescence rate constants are expected to be rather similar for the different forms as their gaussian band widths are similar [71], It is thought that the fluorescence yields are also probably rather similar as the emission of the sj tral forms is closely approximated by a Boltzmann distribution at room temperature for both LHCII and total PSII antenna [71, 133]. 

If it is assumed that the Curtin-Hammett principle applies, one need only to compare the energies of the minima on the solid and dashed curves to be able to predict the structure of the major product. These curves also allow a direct comparison of Cram s, Cornforth s, Karabatsos s and Felkin s model for 1,2 asymmetric induction. Both Figures show the Felkin transition states lying close to the minima. The Corn-forth transition states (Fig. 3) are more than 4 kcal/mol higher and should contribute little to the formation of the final products assuming a Boltzmann distribution for the transition states, less than one molecule, out of a thousand, goes through them. Similarly, Fig. 4 shows the Cram and Karabatsos transition states to lie more than 2.7 kcal/mol above the Felkin transition states, which means that they account for less than 1% of the total yield. 

Max Planck, utilizing his quantum theory postulates and modifications of the Boltzmann statistical procedure, established the theoretical formula for the spectral distribution curves of a black body ... 

Fig. 9. This Larmor spectrum was measured in the analysis trap by resonant excitation (at 104 GHz) of the transition between the two spin states (spin up and down) of the bound electron. The asymmetric line shape of the resonance curve is due to the strong magnetic inhomogeneity in the analysis trap in combination with the thermal Boltzmann distribution of the ion s axial oscillation amplitude...

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. ... 

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