Kinetic theory dilute gases

Ocily n. - 1 of the n equations (4.1) are independent, since both sides vanish on suinming over r, so a further relation between the velocity vectors V is required. It is provided by the overall momentum balance for the mixture, and a well known result of dilute gas kinetic theory shows that this takes the form of the Navier-Stokes equation... 

This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

Theoretical predictions of surface viscosity have been made by Blank and Britten (6) who constructed a formalism based on fluctuations. Cooper and Mann (7) posed a dilute gas kinetic theory of monolayer transport that involved a modified Boltzmann equation which could be solved in detail. The substrate surface coupling could be easily accounted for quantitatively. The results of these calculations were similar in the sense that surface viscosities of the order of 10"10 g/sec were computed for typical monolayer systems. Even when the extensions involving the dense gas region were formulated and computed, the expected surface viscosity numbers were around 10"8 g/sec. 

This paper predicts spectra for a dilute gas of spherical particles, each containing a spin label moiety. The formalism developed by Cooper and Mann is not used although the generalization is certainly possible. Rather, we used the simplest model that retains the collisional character of the dilute gas kinetic theory. 

Equation (3.56) is the dilute gas kinetic theory definition of temperature where k is Boltzmann s constant. From these relationships, the constants in Eq. (3.53) can be uniquely determined, leading to (Prob. 3.10)... 

However, it should be recognized that although the theory of the transport properties of fluids is not completely developed, it can provide some guidance in the process of correlation. For example, all kinetic theories of transport reveal that it is the temperature and density that are the fundamental state variables and that pressure is of no direct significance. Since most measurements are carried out at specified pressures and not specified densities, this automatically means that a single, uniform equation of state must be used to convert any experimental data to p, T) space from the experimental p, T) space. Furthermore, the dilute-gas kinetic theory reveals a number of relationships between different properties of a gas that are exact or nearly exact so that these relationships provide consistency tests for experimental data as well as constraints that must be satisfied by the final correlation of the properties. 

Gas Kinetic Theory The Molecular Theory of Dilute Gases at Equilibrium... 

The gas kinetic theory of noninteracting molecules predicts the rate of wall collisions and the rate of effusion of a dilute gas. 

Molecular theories of transport processes in dilute gases are based on gas kinetic theory. 

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

We will almost always treat the case of a dilute gas, and almost always consider the approximation that the gas particles obey classical, Flarniltonian mechanics. The effects of quantirm properties and/or of higher densities will be briefly commented upon. A number of books have been devoted to the kinetic theory of gases. Flere we note that some... 

A kinetic theory for dilute polyatomic gases has been developed by Wang-Chang and Uhlenbeck (W3, U3). No calculations have been made of the diffusion coefficients on the basis of this theory, however. For most polyatomic gases the results of the Chapman-Enskog monatomic gas theory seem to be adequate. 

An interesting, but probably incorrect, application of the probabilistic master equation is the description of chemical kinetics in a dilute gas.5 Instead of using the classical deterministic theory, several investigators have introduced single time functions of the form P(n1,n2,t) where P(nu n2, t) is the probability that there are nl particles of type 1 and n2 particles of type 2 in the system at time t. They use the transition rate A(nt, n2 n2, n2, t) from the state with particles of type 1 and n2 particles of type 2 to the state with nt and n2 particles of types 1 and 2, respectively, at time t. The rates that are used are obtained by assuming that only uncorrelated binary collisions occur in the system. These rates, however, are only correct in the thermodynamic limit for a low density system. In this limit, the Boltzmann equation is valid from which the deterministic theory follows. Thus, there is no reason to attach any physical significance to the differences between the results of the stochastic theory and the deterministic theory.6... 

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