Boltzmann distribution function

The Maxwell-Boltzmann distribution function (Levine, 1983 Kauzmann, 1966) for atoms or molecules (particles) of a gaseous sample is... 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

The second assumption is that the concentration c, (particles per unit volume) of type,/ ions in the electrical Field is related to c°, the concentration at zero field, by the Maxwell-Boltzmann distribution function, ... 

In Chapter 10, we will derive the Maxwell-Boltzmann distribution function and describe its properties and applications. 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

Figure 2. Comparison of the simulated velocity distribution (histogram) with the Maxwell— Boltzmann distribution function (solid line) for kgT —. The system had volume V — 1003 cells of unit length and N = 107 particles with mass m = 1. Rotations (b were selected from the set Q — tt/2, — ti/2 about axes whose directions were chosen uniformly on the surface of a sphere.

Equation (30) is the Maxwell-Boltzmann distribution function in rectangular coordinates. Thus, in a system of N total molecules, the fraction of molecules, dN/ N, with velocity components in the ranges x component, vx to vx + dvx y component, vy to Vy + dvy, and z component, vz to vz + dvz is given by... 

The reactant molecule BC is specified to be in an initial vibrational v and rotational state 7, which determines p and allows R to be set to the maximum bond extension compatible with total vibrational energy. The initial relative velocity uR may be varied systematically or it may be chosen at random from Boltzmann distribution function. The orientation angle, which specify rotational phase and impact parameter b are selected at random. 

Electrons thermally excited from the valence band (VB) occupy successively the levels in the conduction band (CB) in accordance with the Fermi distribution function. Since the concentration of thermally excited electrons (10 to 10 cm" ) is much smaller than the state density of electrons (10 cm ) in the conduction band, the Fermi function may be approximated by the Boltzmann distribution function. The concentration of electrons in the conduction band is, then, given by the following integral [Blakemore, 1985 Sato, 1993] ... 

Fig. 6-46. Differential capacity observed and computed for an n-type semiconductor electrode of zinc oxide (conductivity 0. 59 S cm in an aqueous solution of 1 M KCl at pH 8.5 as a function of electrode potential solid curve s calculated capacity on Fermi distribution fimction dashed curve = calculated capacity on Boltzmann distribution function. [From Dewald, I960.]...

We consider a simple reaction composed of only a single elementary step of reacting particles that obey the Boltzmann distribution function. Then, the reaction rate, v, is given in Eqn. 7-13 [Rysselberghe, 1963] ... 

In contrast, the probability of activated flow is given by the Boltzmann distribution function in Eqn. 7-29 ... 

Consider a collection of particles with energies ei, 62, S3,Si, each corresponding to a single quantum state, starting with the lowest lying state and proceedith to the ith state. The number of particles in each state can be written symbolically as Ao, Ai, A72, As,... A, The Boltzmann distribution function relating the relative occupancy of two states is written as ... 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

At sufficiently high pressure, collisions with other molecules M in the system will stabilize the excited intermediate C ( ), producing a Boltzmann distribution of populations in the various n levels, characteristic of the temperature T. From the previous definition of the Boltzmann distribution function K(n, T),... 

In the related field of foam stability, De Vries (D3) has performed some very interesting studies. He suggests that the formation of a hole in the separating film and also the first expansion of this hole requires an increase of free energy. This activation energy, which must be supplied in order to make expansion possible, is proportional to the square of the film thickness and to the interfacial tension. The chance that this activation energy is indeed supplied is then described by a Boltzmann distribution function. 

The number of molecular species, nh occupying a given energy state, sit is estimated using the Boltzmann distribution function ... 

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. 

As an example let us consider a linear heteronuclear molecule of the point group such as CO. For this molecule, the degeneracy of states at the higher rotational levels (i 0) is Z/ + 1. The Boltzmann distribution function therefore leads to... 

The Maxwell-Boltzmann distribution function f(u) increases at small values of u and decreases at large values of u. Identify the parts of the function responsible for this behavior. 

At moderate temperatures, electrons in the conduction band have energies close to Ec, and holes in the valence band have energies close to E . Under the assumption that the Fermi energy is not close to the band edges, the Fermi-Dirac distribution, equation (12.1), can be approximated by Boltzmann distribution functions. The concentration of conduction-band electrons can be expressed as... 

Most MC studies of small clusters have used free-volume conditions. However, the choice of boundary conditions has a pronounced effect on the properties of small clusters. For a Leonard-Jones potential and a Boltzmann distribution function, the average potential energy given by Eq. (21) vanishes in the limit of an infinite number of configurations. Physically, this corresponds to the evaporation of particles from the surface and is not unexpected. If there is no constraining force on the cluster, there should be a net escape of... 

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