Gases Maxwell-Boltzmann distribution

FIGURE 4.21 Distribution of speeds in a gas as predicted by the kinetic theory of gases (Maxwell-Boltzmann distribution). As T increases or M decreases, the distribution flattens and spreads to higher overall speeds. 

Distribution of Speeds in Gas— Maxwell-Boltzmann Distribution (Rae D6jur)... 

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. 

The distribution function (24) for an ideal gas, shown in figure 6 is known as the Maxwell-Boltzmann distribution and is specified more commonly [118] in terms of molecular speed, as... 

D) Whether you can answer this question depends on whether you are acquainted with what is known as the Maxwell-Boltzmann distribution. This distribution describes the way that molecular speeds or energies are shared among the molecules of a gas. If you missed this question, examine the following figure and refer to your textbook for a complete description of the Maxwell-Boltzmann distribution. 

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 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 Maxwell-Boltzmann distributions in one and three dimensions will be used next to find the frequency with which molecules undergo collisions, both with other gas-phase molecules as well as with a wall. 

In a hot gas the velocity distribution of each component will be given by a Maxwell-Boltzmann distribution ... 

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. 

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. 

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

Ill) If we consider a motion of the gas model which is of unlimited duration, the Maxwell-Boltzmann distribution will predominate overwhelmingly in time over all other appreciably different state distributions. 

Let f(q, p) be a phase function that does not change its value upon the interchange of molecules. It will therefore assume a definite value if the Maxwell-Boltzmann distribution (Eq. 46) with E0,ri, , r holds for the molecules of the gas. This value will be denoted by... 

If we now supplement Gibbs s discussion with the investigations of Boltzmann as presented in Section 13(1), we come to the following conclusion In a canonically distributed ensemble of gas models the overwhelming majority of the individual members are in a state described by the Maxwell-Boltzmann distribution given in Eq. (46) with the parameters n, , rm, and with the energy E—E. 

Boltzmann One considers in T-space the shell of T-points which correspond to the given total energy Eo. The overwhelming majority of these phase points correspond closely to a Maxwell-Boltzmann distribution (Eq. 53 ) of the molecules of the gas model (cf. Section 13, I). Then from Eqs. (57 ), (60), etc., one calculates for this distribution of state the pressure and the other reactive forces, the kinetic energy per molecular degree of freedom, etc. 

Boltzmann If the external interaction with the gas model is infinitely slow, we may calculate as if the molecules had at each time a Maxwell-Boltzmann distribution corresponding to the instantaneous values of Ei, r , , rm. With this assumption we get for the sum of the energy increase and the work performed for an infinitesimal transition, i.e., for the supplied heat, ... 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

The barometer formula can be derived by elementary methods, thus checking this part of the Maxwell-Boltzmann distribution law. Consider a column of atmosphere 1 sq. cm. in cross section, and take a section of this column bounded by horizontal planes at heights ft and ft + dh. Let the pressure in this section be P we are interested in the variation of P with ft. Now it is just the fact that the pressure is greater on the lower face of the section than on the upper one which holds the gas up against gravity. That is, if P is the upward pressure on the lower face, P + dP the downward pressure on the upper face, the net downward force is dP,... 

In Section 5.2, we will derive the three-dimensional Maxwell-Boltzmann distribution n(v)dv of molecular speeds between v and v + dv in the gas phase ... 

Most of the chemical reactions occur in the condensed phase or in the gas phase under conditions such that the number of intermolecular collisions during the reaction time is enormous. Internal energy is quickly distributed by these collisions over all the molecules according to the Maxwell-Boltzmann distribution curve. 

In actuality, molecular velocities are not all the same. At any time some molecules are moving much faster than the average while others are moving more slowly than the average. For a perfect gas the velocity distribution (in one dimension) is given by the Maxwell-Boltzmann distribution function,... 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

The distribution of velocities of the particles in an ideal gas is described by the Maxwell-Boltzmann distribution law ... 

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