Maxwell-Boltzmann equilibrium

The Boltzmann equation is a nonlinear, integrodifferential equation. As such it is extremely difficult to solve and, in fact, almost no exact solutions are known, apart from the Maxwell-Boltzmann equilibrium solution. Furthermore, only a few existence theorems are known notable are the theorems of Carleman, later extended by Wild and by Morgenstern, proving the existence of a solution of the nonlinear Boltzmann equation for special intermolecular potentials in the case that the system is spatially uniform, i.e., that the distribution function does not depend on r. However, there are a number of circumstances where the system is close enough to equilibrium that the distribution function may be written... 

An expansion of the Maxwell-Boltzmann equilibrium distribution (85) at low velocities suggests the following ansatz [97,106] for the discrete velocity equilibrium,... 

As stated earlier, within C(t) there is also an equilibrium average over translational motion of the molecules. For a gas-phase sample undergoing random collisions and at thermal equilibrium, this average is characterized by the well known Maxwell-Boltzmann velocity distribution ... 

The relative velocity between the molecules not only determines whether A and B collide, but also if the kinetic energy involved in the collision is sufficient to surmount the reaction barrier. Velocities in a mixture of particles in equilibrium are distributed according to the Maxwell-Boltzmann distribution in spherical coordinates ... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

As stated earlier, in the state of thermal equilibrium at room temperature, dihydrogen (H2) contains 25.1% parahydrogen (nuclear singlet state) and 74.9% orthohydrogen (nuclear triplet state) [19]. This behavior reflects the three-fold degeneracy of the triplet state and the almost equal population of the energy levels, as demanded by the Maxwell-Boltzmann distribution. At lower temperatures, different ratios prevail (Fig. 12.5) due to the different symmetry of the singlet and the triplet state [19]. 

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... 

The rate constant is measured in units of moles dnr3 sec /(moles dnr3)", where n = a + b. Time may also be in minutes or hours. It should be noted that in case where the reaction is slow enough, the thermal equilibrium will be maintained due to constant collisions between the molecules and k remains constant at a given temperature. However, if the reaction is very fast the tail part of the Maxwell-Boltzmann distribution will be depleted so rapidly that thermal equilibrium will not be re-established. In such cases rate constant will not truly be constant and it should be called a rate coefficient. 

A fluid composed of a single species is described by five fields the three components of the velocity, the mass density, and the temperature. This is a drastic reduction of the full description in terms of all the degrees of freedom of the particles. This reduction is possible by assuming the local thermodynamic equilibrium according to which the particles of each fluid element have a Maxwell-Boltzmann velocity distribution with local temperature, velocity, and density. This local equilibrium is reached on time scales longer than the intercollisional time. On shorter time scales, the degrees of freedom other than the five fields manifest themselves and the reduction is no longer possible. 

If we are going to relate the properties of our system to a physical situation, we need to be able to characterize the system s temperature, T. In a macroscopic collection of atoms that is in equilibrium at temperature T, the velocities of the atoms are distributed according to the Maxwell -Boltzmann distribution. One of the key properties of this distribution is that the average kinetic energy of each degree of freedom is... 

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... 

Today, non-equilibrium reaction theory has been developed. Unlike the absolute rate theory, it does not require the fulfilment of the Maxwell-Boltzmann distribution. Calculations are carried out on large computers, enabling one to obtain abundant information on the dynamics of elementary chemical acts. The present situation is extensively clarified in the proceed-dings of two symposia in the U.S.A. [23, 24]. 

In the kinetic theory of gases, the molecules are assumed to be smooth, rigid, and elastic spheres. The only kinetic energy considered is that from the translational motion of the molecules. In addition, the gas is assumed to be in an equilibrium state in a container where the gas molecules are uniformly distributed and all directions of the molecular motion are equally probable. Furthermore, velocities of the molecules are assumed to obey the Maxwell-Boltzmann distribution, which is described in the following section. 

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 ... 

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. 

In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduction, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of "equilibrium" and "reversibility," "equilibrium processes" and "reversible processes" started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the "time arrow."... 

Although Boltzmann did not fully succeed in proving the tendency of the world to go to a final equilibrium state, there remain after all criticisms the following valuable results first, the derivation of the Maxwell-Boltzmann distribution for equilibrium states, then the kinetic interpretation of the entropy by the //-function, and finally the explanation of the existence of an integrating factor for dU+dA. In thermodynamics the existence of such a factor is always based on an unexplained hypothesis. 

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