Very dilute gas

A direct way of obtaining (2.55) from the definition (2.48) (and not through the density expansion) is to consider the case of a system containing only two particles. 

It should be noted that for any gas with any intermolecular interactions, when p — 0, we obtain the ideal-gas behavior. For instance, the equation of state has the typical and well-known form. One should distinguish between the ideal-gas behavior of a real gas as p — 0, and a theoretical ideal gas which is a model system, where no interactions exist. Such a system does not exist however, the equation of state of such a model system is the same as the equation of state of a real system as p — 0. 

In this section, we have seen that in the limit p — 0, the pair correlation is (2.55). This is different from the theoretical ideal gas case obtained in section (2.5.1). There, the form of g(R) is valid for any density provided that all 

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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... 

Direct knowledge of the amount of excitation energy and the modes of decomposition is found by mass spectrometric studies in the very dilute gas phase. 

The effect of the collisional force due to the impact of particles should be included when accounting for the motion of a particle except in a very dilute gas-solid flow situation. Basic mechanisms of collision between two particles or between a particle and a solid wall are discussed in Chapter 2. The collisional force between a particle and a group of neighboring particles in a shear suspension is discussed in 5.3.4.3. In a very dense system where particle collisions dominate the flow behavior, collisional forces can be described by using kinetic theory, as detailed in 5.5. The key equations derived in other chapters pertaining to the collisional forces can be summarized in the following. 

If we suppose a very dilute gas of random flight chains. As a result of thermal rotation of chain segments each chain will take up a great number of conformations, with a very short interval of time being spent for the passage from one conformation to another. In so doing, it automatically avoids taking those conformations in which... 

Here we show that an application of bichromatic control (Section 3.1.1) allows us to control both the real and imaginary parts of the refractive index. In doing so we consider isolated molecules [213, 214], or molecules in a very dilute gas, where collisional effects can be ignored and time scales over which radiative decay occurs can be ignored. 

Consider a very dilute gas solute in a liquid phase. Henry s law relates the mole fraction of the solute i in the gas phase to the mole fraction of the solute / in the liquid phase... 

First, consider a very dilute gas of molecules [16]. The conventional theory of the static dielectric susceptibility % of such a gas invokes the notion of polarizable molecules with permanent dipole moments that are partially aligned by the external electric field . Standard techniques of statistical thermodynamics produce the Langevin-Debye formula for x Per molecule that reads... 

The derivation of (2.65) illustrates the origin of the coefficient B(R R"), which in principle results from the simultaneous interaction of three particles [compare this result with (2.58)]. This is actually the meaning of the term slightly dense gas. Whereas in a very dilute gas we take account of interactions between pairs only, here we also consider the effect of interactions among three particles, but not more. For hard spheres (HS), we can calculate B(R, R") exactly in this case we have... 

At pressures below the first threshold (see the figure), the radical intermediates are more Hkely to react with the container surface than with the very dilute gas-phase reactants, and there is no explosion. At pressures between the second and third explosion thresholds, the concentration of gas molecules is enough to lower the concentration of radicals sufficiently to slow the rate of reaction, hence no explosion. For these studies, Semenov and Hinshelwood shared the 1956 Nobel Prize in chemistry. 

The simple Skarstrom cycle for PSA shown in Figure 18-llA has constant pressure (isobaric) periods and periods when pressure is changing. We will assume that a very dilute gas stream containing trace amounts of adsorbate A in an weakly adsorbed carrier gas is being processed and that over the concentration range of interest the linear isotherm, Eq. fl8-5bl. is accurate. If mass transfer is very rapid, then the solute movement theory can be applied. Since the system is very dilute, the gas velocity is constant and the system is assumed to be isothermal. In more concentrated PSA systems neither of these assunptions are true, and a more conplicated theory must be used fRuthvenetal.. 19941. 

It is accordingly not surprising that sulfur monoxide is not known as a stable substance, but only as a highly reactive molecule in a very dilute gas or frozen in a matrix. It has the structure S-HtO , with two electrons with parallel spins, thus resembling the molecules O2 and S2. 

Now consider a gas in extremely low concentration, a very dilute gas. In this limit, the gas molecules are mostly noninteracting or independent. With that, we can take each molecule as one of the M systems, or in other words, take N = 1. In this limit, the ensemble partition... 

The transport experiments highlight some important differences between the solids, shown in Table 3. This table presents the results of measurements of very dilute, gas phase argon in nitrogen diffiisivities for a set of porous solids, along with morphological properties obtained from mercury porosimetry. 

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