Maxwell-Boltzmann expression

Before deriving the expressions for gas-phase collision frequency, we need to discuss the relative velocity that is important in collisions. The reduced mass mn is also obtained from this analysis, and will be the appropriate mass to use in the Maxwell-Boltzmann expression for collision velocities. 

This is called an energy in two squared terms. The Maxwell-Boltzmann expression for such a situation is... 

At equilibrium the distribution of molecules among the various energy states E is given by the Maxwell-Boltzmann expression. Thus for a molecule with n classical internal, harmonic oscillators, the fraction of molecules with energy J i, E2,. , En present in these oscillators is... 

The problem is to get some device which would substitute the average distribution of the discrete ions in the ionic atmosphere around the centralj-ion, given by n, in the Maxwell-Boltzmann expression, by a continuous charge density which could be taken to be equivalent to pj in the Poisson equation. This would enable Poisson s equation to be combined with a Maxwell-Boltzmann distribution. 

The relative populations of ground-state (Nq) and excited-state (iVJ populations at a given flame temperature can be estimated from the Maxwell-Boltzmann expression ... 

It is of interest to know the number of thermally excited atoms relative to the number of ground state atoms at a given flame temperature. In a quantity of atoms, under the same external conditions, the electrons are not all in the same energy level but are statistically distributed among the levels. At a flame temperature T (in K), the ratio of the number of atoms in an excited (upper) state u to the number of atoms Ao in the ground state is given by the Maxwell-Boltzmann expression... 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

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 relative populations of energy levels, that is the proportions of the analyte species occupying them, have a direct bearing on line intensities and are determined by the spacings of the levels and the thermodynamic temperature. The relation is expressed in th q Maxwell-Boltzmann equation,... 

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...

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

Insertion of equation 3 into equation 1, approximation of the Fermi distribution by a classical Maxwell-Boltzmann distribution, and integration of equation 1 yield the expression for the total number of electrons in the conduction band ... 

Prove that the Maxwell-Boltzmann distribution can be expressed as Eq. (5.24). Several steps are suggested ... 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

We now proceed to develop a specific expression for the rate constant for reactants where the velocity distributions /a( )(va) and /B(J)(vB) for the translational motion are independent of the internal quantum state (i and j) and correspond to thermal equilibrium.4 Then, according to the kinetic theory of gases or statistical mechanics, see Appendix A.2.1, Eq. (A.65), the velocity distributions associated with the center-of-mass motion of molecules are the Maxwell-Boltzmann distribution, a special case of the general Boltzmann distribution law ... 

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... 

The well-known Maxwell-Boltzmann distribution for the velocity or momentum distribution associated with the translational motion of a molecule is valid not only for free molecules but also for interacting molecules, say in a liquid phase. We start with the general expression, Eq. (A.40), i.e., the Boltzmann distribution for N identical molecules each with s degrees of freedom ... 

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. 

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. ... 

Equation (1.4) expresses what is called the Maxwell-Boltzmann distribution law. If Eq. (1.4) gives the probability of finding any particular molecule in the fctii state, it is clear lhat it also gives the fraction of all molecules to be found in that state, averaged through the assembly. 

The Boltzmann rate coefficient, k T), at temperature T is then obtained by averaging this expression over the Maxwell-Boltzmann distribution of relative velocities,1 or relative energies. In terms of E,... 

Equation (4) is known as the law of Stefan and Boltzmann. Stefan arrived at this law in an empirical manner from the rough experimental material then at his disposal. Boltzmann gave the above thermodynamical proof involving Maxwell s expression for the radiation pressure. 

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