Maxwell-Boltzmann Distribution of Velocities

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  

This distribution is readily generalized to three dimensions where it takes the form 

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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. 

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 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... 

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),... 

Molecules travel at different velocities. The Maxwell-Boltzmann distribution of velocity is used to define the velocity profile of molecules and is written as... 

We have illustrated the calculation of the averages from the Langevin equation for sharp initial conditions. The solution of the Langevin equation subject to a Maxwell-Boltzmann distribution of velocities is called the stationary solution. Clearly for the stationary solution... 

It must be noted that this is a schematic diagram where the abscissa is not a linear distance scale instead it represents the trajectory pathway of an incoming molecule to a surface. Dissociative adsorption can occur from a weakly held molecular state if the net barrier to adsorption is low (precursor mediated) but is of low probability if it is high. Then it is only the hot molecules of the Maxwell Boltzmann distribution of velocities (fig. 9) which can dissociate and they do this by direct passage over the energy barrier (direct activated). The rate of dissociation from a precursor state can be written as follows for the simple case in fig. 9,... 

Fig. 8 Collision-induced rotational transfer in collisions between Ar and N2(0 10) squares), O2(0 12) circles) and OF1(0 3) triangles). Collision conditions are Maxwell-Boltzmann distribution of velocities at 300 K. Note that there is a Aj = 2n selection rule in collision-induced processes for homonuclear diatomic molecules...

The integration variable E in equation (26) is effectively E, ,. The condition for the validity of these equations for a thermally averaged rate constant kba(T) is the existence of a well defined Maxwell-Boltzmann distribution of velocities of collision partners or relative collision energies (E — a) at temperature T, which remains unperturbed by the reaction process. If, furthermore the internal state distributions of the reactants also remain at an unperturbed Boltzmann distribution at temperature T, one finds a thermal rate constant for complex formation (or capture ) given by equation (28) ... 

A well-known example of a Gaussian distribution is the celebrated Maxwell-Boltzmann distribution of velocities of gas particles. In one dimension, we write... 

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