Showing papers on "Boltzmann constant published in 1969"

Quantum Mechanical Second Virial Coefficient of a Lennard-Jones Gas. Helium

Abstract: The quantum‐mechanical second virial coefficients of Lennard‐Jones 3He and 4He gases with the De Boer parameters have been obtained over the complete temperature range from near absolute zero to the classical region. A formalism separating the virial into direct (Boltzmann) and exchange (spin and quantum statistics) contributions has been employed. The calculation is based on phase shifts except at the very highest temperatures where a Wigner–Kirkwood method has been used. Examination of the exchange term shows in detail the rapid suppression of the statistical effects with rising temperature, their contribution dropping to less than 0.001 cm3 by 7°K (4He). Comparison of the high‐temperature (Boltzmann) results with those obtained by a third‐order Wigner–Kirkwood expansion shows excellent agreement down to about 50°K for 4He and 60°K for 3He. The Wigner–Kirkwood expansion is shown to be unsuitable for determining the behavior of the exchange terms. Finally, results are compared with the available experimental data.

Grüneisen constant and thermal properties of crystalline and glassy polymers

Abstract: Polymer crystals are characterized by strong anisotropy of binding forces among units, i.e., the intrachain force constant f is much larger than the interchain force constant g. The anisotropic lattice is reduced to an isotropic one, in which each lattice point represents N* units (segment) along the chain axis of the anisotropic lattice [N = 2(f/g)1/2]. Vibrational modes of the reduced lattice correspond to interchain modes of the original lattice, i.e., modes whose frequencies are governed by interchain potential. Anharmonicity of crystalline force field is assumed to be related predominantly with interchain force alone. Thermodynamic and transport equations for a simple lattice are applied to the reduced, isotropic lattice, and numerical results are obtained for high-density polyethylene. The Gruneisen constant γ was obtained from the pressure dependence of sound velocity. The heat capacity of the reduced lattice, Cinter (interchain specific heat), was calculated from Gruneisen's equation, α = γβCinter (where α = thermal expansion coefficient, β = compressibility), and the mass of a segment m* was estimated from Dulong-Petit's equation, Cinter = 3ρk/m* (where ρ = density, k = Boltzmann constant). The value of m* is consistent with N* from force constants, m* = N*m (where m = mass of a unit in the original lattice). m*θ3 (where θ denotes the Debye temperature of the reduced lattice) is calculated from low temperature specific heat. The value of m* calculated from m*θ3 and θ from other sources agrees with that from the estimate by Dulong-Petit's equation. The high-temperature thermal conductivity K was calculated through Leibfried-Schloemann's equation by employing γ and m*θ3 as estimated as described above; satisfactory agreement was obtained with experiment. Poly(methyl methacrylate) and polystyrene were also studied by similar methods.

The Boltzmann equation for a polyatomic gas

Abstract: A formulation of the kinetic theory of dilute, classical polyatomic gases is given which parallels the Waldmann development for structureless molecules. In the first section the Boltzmann equation is written in terms of the specific rates of inelastic collision processes and then the properties of these rates and those of the corresponding collision cross sections are examined. The dependence of the distribution function on the dynamical variables is discussed and the equations of change for the gas are derived. Finally, a study is made of the properties of the linearized Boltzmann collision operation. In the second section the Boltzmann equation is deduced from a rigorous statistical-mechanical point of view and discussed in terms of the basic ideas of Bogoliubov. The computationally important special case of impulsive interactions is then considered.

Axially Symmetric Expansion of a Monatomic Gas from an Orifice into a Vacuum

Abstract: The problem of the steady axially symmetric expansion of a monatomic gas from an orifice into a vacuum is considered. The reservoir conditions are such that the local Knudsen number is initially small. It is noted that the near continuum solution, valid near the orifice, is not uniformly valid far downstream where the local mean free path may be comparable with some characteristic length. A valid solution of Boltzmann's equation, for Maxwell molecules, is deduced for this far field core region. Near the gas‐vacuum front, predicted by the equilibrium solution, this expansion procedure also breaks down. It is shown that a further scaling of the variables in Boltzmann's equation, consistent with this limit, can be found and the corresponding moment equations deduced. However, in contrast to the behavior in the core, these equations no longer form a closed set.

The Boltzmann collision integrals for a combination of Maxwellians

Abstract: The gain and loss integrals in the Boltzmann equation for a rigid sphere gas are evaluated in closed form for a distribution which can be expressed as a linear combination of Maxwellians. Application to the Mott-Smith bimodal distribution shows that the gain is also bimodal, but the two modes in the gain are less pronounced than in the distribution. Implications of these results for simple collision models in non-equilibrium flow are discussed.

Theory of the propagation of the plane-wave disturbances in a distribution of thermal neutrons.

Abstract: The propagation of plane‐wave disturbances in a neutron gas in thermal equilibrium with a moderator is studied using the linearized Boltzmann equation. The eigenvalue spectrum of the appropriate Boltzmann transport operator is investigated for both polycrystalline and noncrystalline media, and several necessary conditions for the existence of a point spectrum (and hence for the existence of plane‐wave propagation) are discussed. Techniques are presented which allow the solution of various boundary‐value problems using this spectral representation.

Electron Distribution Function in a Slightly Ionized Moving Gas

Abstract: The Lorentz gas approximation to the Boltzmann collision operator is generalized to the case of the heavy‐molecule gas distribution function being non‐Maxwellian; results are obtained to first order in the mass ratio, but can easily be generalized to higher orders. The equilibrium distribution for the light‐molecule gas is then found. The results are applicable to any light‐molecule trace gas in a heavy‐molecule non‐Maxwellian bath gas, and, in particular, to a slightly ionized gas.

Response of a Lorentzian gas to an a. c. pulse

Abstract: The response of a “Lorentz gas” to a pulsed a.c. electric field and its relaxation after the cessation of this field has been studied by solving the Boltzmann's transfer equation. Explicit expressions for the electron distribution function and the current density are obtained under the assumption that the collision frequency is independent of the electron velocity.

Ideal equation of state for charged monolayer at the liquid interface

Abstract: An attempt has been made to estimate the value of the kT -coefficient of the ideal equation of state for the charged monolayer in the presence and absence of the neutral salt. Guggenheim's concept of the surface phase is taken as the basis for such calculation. The value of the kT -coefficient has been found to be “two” both in the presence and absence of the neutral salt when integrated forms of the Boltzmann equations are used for the calculation of the surface concentrations of the inorganic ions. The result is in agreement with that previously suggested by Phillips and Rideal. However, if the nonintegrated forms of the Boltzmann equations are used for this calculation, p is found to be much less than two in agreement with the expectation of Davies.

Kinetic theory of inhomogeneous systems

Abstract: A theory is developed which treats the coupled equations of the various hierarchies as simultaneous equations in time. This scheme proceeds by successive approximations rather than a power series expansion in the small parameter ([nλd3]−1 in a plasma, nr03 in a Boltzmann gas). The theory is suitable for the derivation of equations for nonuniform and force driven systems. Examples are given for a plasma and a Boltzmann gas.

Spectral temperatures of nonuniform plasmas

Abstract: Boltzmann plots derived from spatially integrated spectral line intensities emitted from nonuniform gaseous plasmas are investigated. Emphasis is placed on the meaning of temperatures inferred from these Boltzmann plots and also on the effect of relaxing the LTE requirement. It is shown that for plasmas with sufficiently small electron density and electron temperature gradients, the line-of-sight Boltzmann temperature is nearly equal to the spatially averaged electron temperature if sufficient LTE restrictions are applied.

Revised formulation of the kinetic plasma theory

Abstract: Using the Maxwellian macroscopic approach and analysing the formulation of the dielectric constant, it is shown that the concept of energy has not been properly incorporated into the current kinetic plasma theory. The difficulties are due to the Boltzmann collisional term (∂F/∂t)coll which accounts for a change in the velocity distribution due to collisions alone. If one attempts to rephrase the Boltzmann-Vlasov theory in terms of the Maxwcllian macroscopic formulation, one obtains an expression for energy which is not consistent with the meaning of this concept in generalized dynamics. In a revised version developed in this analysis the Boltzmann collisional term has been eliminated and an appropriate collisional operator is introduced which is believed to describe more adequately collisional processes in a plasma. It is assumed that the collisional operator can be applied directly to the electrical intensity of the field interacting with the plasma and is effective in transforming the intensity in a col...