Maxwell-Boltzmann velocity distribution derivations

Derive Eq. (5.38) on the basis of the Maxwell-Boltzmann velocity distribution. 

It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at T through the use of the Maxwell-Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature T. The quantity K(fhl, ij) may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium. 

To understand how collision theory has been derived, we need to know the velocity distribution of molecules at a given temperature, as it is given by the Maxwell-Boltzmann distribution. To use transition state theory we need the partition functions that follow from the Boltzmann distribution. Hence, we must devote a section of this chapter to statistical thermodynamics. 

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 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 ... 

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... 

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. ... 

The calculations in this case are clearly analogous to those required to prove the Bernoulli theorem. In order to show the first part of the statement, all we have to do is to determine the maximum of Eq. (36), i.e., the minimum of Eq. (43), given the auxiliary condition of Eq. (45). Boltzmann makes use of the second half of the statement in all those cases when he calls the Maxwell velocity distribution overwhelmingly the most probable one." A more quantitative formulation and derivation of this part of the statement is sketched by Jeans in [2, 22-26] and in Dynamical Theory, 44-46 and 56. 

In deriving Eq. (8), it is assumed that molecules A and B collide with a single relative velocity g. In a real gaseous sample containing both A and B molecules at thermal equilibrium, the distribution of relative velocities is described by the Maxwell-Boltzmann Distribution Law ... 

These equations express the quantal analogue of the Maxwell-Boltzmann distribution law, and they are more general in their scope than the Maxwell expression for the distribution of velocities (the latter will be derived from (12 72) in 12 11). 

The Gaussian distribution is derived from the binomial distribution for large N [5]. It is important for statistics, error analysis, diffusion, conformations of polymer chains, and the Maxwell Boltzmann distribution law of gas velocities. 

From the kinetic theory of gases, an expression for the net-mass flux at the interphase can be derived based on the works of Hertz [223] and Knudsen [224]. From a statistical consideration under the assumption of a Maxwell-Boltzmann distribution for the velocity of the gas molecules, the maximum condensation mass flux can be calculated. The evaporation mass flux has to equal the condensation mass flux at equilibrium. The resulting Hertz-Knudsen equation for calculating the area specific net-mass flux is given below ... 

Maxwell-Boltzmann distribution A mathematical function used in the kinetic theory of gases. It is derived on the basis of statistical mechanics and gives the distribution of particle velocities in a gas at a particular temperamre. 

In the thermal region of the neutron-energy spectrum, the kinetic energies of the neutrons will be distributed statistically according to the Maxwell-Boltzmann distribution law. This distribution, derived from the kinetic theory of gases, is applicable to the neutron gas only if the medium in which the neutrons are present is weakly absorbing. The velocity distribution is... 

A macroscopic sample of a gas contains an enormous number of molecules continually colliding with one another and with the walls of the container. The effect of these collisions is to change the velocity of the colliding particles. The fundamental problem in kinetic theory is to determine the distribution of velocities of the molecules in a gas. The answer was first given by Maxwell for a gas in thermal equilibrium. Indicative of the importance of this problem is the fact that, over a period of years. Maxwell and Boltzmann presented four different derivations of the basic equation, each somewhat more sophisticated than its predecessor. We shall consider the two simplest ones. 

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