Gases dilute monatomic

In Sec. II,A the equations of change are derived by assuming that the fluid is a continuum. A physically more satisfying derivation may be performed in which one starts directly from considerations of the fundamental molecular-collision processes occurring in the fluid. For dilute monatomic gases and gas mixtures one can start... 

The Chapman-Enskog theory was developed for dilute, monatomic gases for pure substances and for binary mixtures. The extension to multicomponent gas mixtures was performed by Curtiss and Hirschfelder (C12, Hll), who in addition have shown that the Chapman-Enskog results may also be obtained by means of an alternate variational method. Recently Kihara (K3) has shown how expressions for the higher approximations to the transport coefficients may be obtained, which are considerably simpler than those previously proposed by Chapman and Cowling these simpler formulas are particularly advantageous for calculating the coefficients of diffusion and thermal diffusion (M3, M4). 

The thermal conductivities of dilute monatomic gases are well understood. The thermal conductivity of a dilute gas composed of rigid spheres of diameter d is expressed as... 

Exact results for the thermal force for a highly viscous, stable particle in dilute, monatomic host gas are available in the limits Kn. oo, Kn 0. The particle is in... 

In the derivation of the most probable distribution there was no restriction to a particular dilute gas. We showed that p = l/(k T) for a dilute monatomic gas with negligible electronic excitation, but this relation must be valid for all dilute gases. With this identification the probability of a molecular energy level of any dilute gas is... 

Wehave already determined that the molecular partition function for a dilute monatomic gas is the product of a translational partition function and an electronic partition function. We obtained a formula for the translational partition function in Eq. (25.3-21) ... 

The result of this example justifies our earlier assumption that there was no significant electronic excitation in our dilute monatomic gas. 

As previously stated, the classical molecular partition function has units of kg s raised to some power, so a divisor with units must be included to make the argument of the logarithm dimensionless. If a divisor of lkgm s is used, values are obtained for the entropy and the Helmholtz energy that differ from the experimental values. However, when the classical canonical translational partition function is divided by h A and Stirling s approximation is used for ln(iV ), the same formulas are obtained as Chapter 26. For a dilute monatomic gas the corrected classical formula is... 

Thermal motions A molecule has three translational degrees of freedom. Let us consider a system of M ideal monatomic gas molecules in a cubic box kept at a constant temperature. For a very dilute gas, where the molecules do not interact with one another, the quantum mechanical solution is a number of wave functions with three quantum numbers, nx, riy, and n, for the translational energies in three dimensions. The energy of a molecule in a cubic box with side length a is given by... 

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. 

Since the forms of the Chapman-Enskog expressions for dilute-gas viscosity and conductivity are so similar, it might be expected that there is a simple relationship between thermal conductivity and viscosity. Indeed, for monatomic gases, combining Eqs. 3.41 and 3.136 yields... 

Sato, Tsuchiya, and Kuratani [216] have solved the relaxation equation for the vibrational energies of two diatomic gases A and B diluted in an inert monatomic gas M, and have applied the solution to shock-wave relaxation profiles in order to obtain V-V transfer rates. Their solution shows that the relaxation of each of the component molecules proceeds as if it possessed two relaxation times. At the onset of the relaxation process, both components begin to relax with their respective V-T rates, whereupon the relaxation rate of that component having the smaller V-T relaxation time begins to decrease, while the relaxation rate of the other component increases. Finally, both components relax with the same rate toward their equilibrium states. By observed infrared emission from the CO fundamental behind shock waves in mixtures of CO-N2, C0-02, CO-D2, and CO-H2, they were able to determine Pil as a function of temperature. Argon was used as inert buffer gas. 

The use of the master equation to describe the relaxation of internal energy in molecules is, in fact, nothing more than the writing of a set of kinetic rate equations, one equation for each individual rotation-vibration state of the molecule. The simplest case we can consider is that of an assembly of diatomic molecules highly diluted in a monatomic gas under these conditions, we only need to consider the set of processes... 

Here, as earlier, we are restricting our attention to systems of pure monatomic gases. The theory as well as the comparison with experiment can be generalized to include dilute gas mixtures and dilute gases of polyatomic molecules (cf. Sections 2.4.4.1 and 2.4.4.2). 

The dilute-gas theory is presented here for the first time in terms of effective collision cross sections in a comprehensive readily usable form which applies to both polyatomic fluids and monatomic fluids. This description should now be used exclusively but, because it is relatively new, expressions are given for the macroscopic quantities in terms of these effective cross sections, and certain simple relationships between these effective cross sections and the previously used collision integrals are also described. 

Here, X is the transport property associated with the particular process under consideration. It follows that the transport coefficient, which itself may be a function of the temperature and density of the fluid, will reflect the interactions between the molecules of the dilute gas. For that reason there has been, for tq)proximately 150 years, a purely scientific interest in the transport properties of fluids as a means of probing the forces between pairs of molecules. Within the last twenty years, at least for the interactions of the monatomic, spherically symmetric inert gases, the transport prq>erties have played a significant role in the elucidation of these forces. 

Subsequently, and of greater significance in the context of this volume, it was shown that it was possible to determine the pair potential of monatomic species directly from measurements of the viscosity of the dilute gas, by a process of iterative inversion (Maitland et al. 1987). As an illustration of the success that can be achieved. Figure 2.1 compares the pair potential that is obtained by application of the inversion process to the viscosity data for argon with that currently thought to be the best available pair potential for argon which is consistent with a wide variety of experimental and theoretical information. 

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