Maxwell-Boltzmann distribution of the

FIGURE 4.3 The Maxwell-Boltzmann distribution of the translational kinetic energy U of molecules or other particles. N is number of molecules, kit is Boltzmann s constant, and T is temperature (K). (a) Normalized distribution, (b) Distribution for two temperatures. 

The classical rate theory due to Arrhenius proceeds on the Maxwell-Boltzmann distribution of the velocity, and thereby the kinetic energy, of molecules or particles their average kinetic energy equals (2/2)k-QT. If two molecules collide with a kinetic energy larger than an activation energy Ea for a reaction between them to proceed, they are assumed to react. The... 

If into the system an equilibrium Maxwell-Boltzmann distribution of the energy between the particles and their freedom degrees is kept, then in accordance with the law of mass action the following ratios should be performed ... 

Consideration of the Maxwell-Boltzmann distribution of the kinetic energies of the particles in a sample also helps us interpret the action of a catalyst in speeding up a chemical reaction (Figure 6.19). A catalyst acts by allowing a reaction to take place by an alternative reaction pathway that has a lower activation energy. This means that a greater proportion of the sample particles have sufficient energy to react on collision. 

Figure D.l. Average occupation numbers in the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions, for = 10 (left) and f = 100 (right), on an energy scale in which fXfo = p-be = 1 for the FD and BE distributions. The factor nl(2nmfl in the Maxwell-Boltzmann distribution of the ideal gas is equal to unity.

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. 

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. 

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 ... 

A study of the Maxwell-Boltzmann distribution of molecular energies at temperatures T Kelvins and at (T + 10) K illustrates this. 

The Maxwell-Boltzmann distribution of molecular energies can be used to explain how a catalyst works at constant temperature. 

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 ... 

The thermodynamic temperature is the sole variable required to define the Maxwell-Boltzmann distribution raising the temperature increases the spread of energies. 

At this point, it is worthwhile to return on the theoretical basis of the kinetic method, and make some considerations on the assumptions made, in order to better investigate the validity of the information provided by the method. In particular some words have to been spent on the effective temperamre The use of effective parameters is common in chemistry. This usually implies that one wishes to use the form of an established equation under conditions when it is not strictly valid. The effective parameter is always an empirical value, closely related to and defined by the equation one wishes to approximate. Clearly, is not a thermodynamic quantity reflecting a Maxwell-Boltzmann distribution of energies. Rather, represents only a small fraction of the complexes generated that happen to dissociate during the instrumental time window (which can vary from apparatus to apparatus). 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann distribution of reactants is postulated to persist during a reaction.68 The equilibrium theory first passage time is the TV -> oo limit in Eq. (6), Corrections to it then are to be expected when the second term in this equation is no longer negligible, i.e., when N is not much greater than e — e- )-1. The mean first passage time and rate of activation deviate from their equilibrium value by more than 10% when... 

The velocity probability distribution function of Eq. 10.20 is the well-known Maxwell-Boltzmann distribution of velocities. Integrating over vx = —cc — oo shows that P(vx) is normalized. It is also easy to calculate the expectation value for the one-dimensional translational energy of a mole of gas as... 

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

The formulas that we have derived in this chapter and in Chapter 8 to describe energy and velocity distributions also apply to the center of mass and relative velocities. In particular, the distribution of relative velocities obeys the Maxwell-Boltzmann distribution of Eq. 10.27, with the mass replaced by the reduced mass /W 2 ... 

There is a Maxwell-Boltzmann distribution of collision velocities, given by Eq. 10.27, so to obtain the average collision frequency integrate Eq. 10.46 over all possible collision... 

Begin by considering the translational motion of molecules in a container that has a total concentration of molecules per unit volume [c]. The distribution of velocities in the x, y, and z directions is given by the one-dimensional Maxwell-Boltzmann distribution of... 

The one-dimensional Maxwell-Boltzmann distribution gives the fraction of molecules in the velocity range vx - vx + dvx, so the product [c] P(vx) gives the number of molecules per unit volume in that velocity range. The total number of molecules passing through the plane in a time At that have velocities in the range vx - vx + dvx is... 

The Maxwell-Boltzmann distribution defines the most probable route. Here the Maxwell-Boltzmann distribution is being used in terms of a probability, rather than a fraction of molecules with energy at least a certain critical energy. The probability that a molecule has a given potential energy at any point on the surface is proportional to exp(—s/kT), where e is the PE at the point. 

That is, the Maxwell-Boltzmann distribution for the two molecules can be written as a product of two terms, where the terms are related to the relative motion and the center-of-mass motion, respectively. After substitution into Eq. (2.18) we can perform the integration over the center-of-mass velocity Vx. This gives the factor y/2iVksTjM (IZo eXP( —ax2)dx = sjnja) and, from the equation above, we obtain the probability distribution for the relative velocity, irrespective of the center-of-mass motion. 

The average velocity for the motion from the left to the right over the barrier is then evaluated. From the one-dimensional Maxwell-Boltzmann distribution of velocities, Eq. (2.26),... 

The well-known Maxwell-Boltzmann distribution for the velocity or momentum associated with the translational motion of a molecule is valid not only for free molecules but also for interacting molecules in a liquid phase (see Appendix A.2.1). The average kinetic energy of a molecule at temperature T is, accordingly, (3/2)ksT. For the molecules to react in a bimolecular reaction they should be brought into contact with each other. This happens by diffusion when the reactants are dispersed in a solution, which is a quite different process from the one in the gas phase. For fast reactions, the diffusion rate of reactant molecules may even be the limiting factor in the rate of reaction. 

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. 

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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