Virial lines

Fig. 14. Comparison of the experimental constant term pVjRT) — b with the theoretical virial line for c = 1/1.10 for a variety of gases. water a methane O helium + xenon carbon dioxide.

If we represent the /bond by a line with two open circles to denote the coordinates of the particle 1 and 2, then the first two virial coefficients can be depicted graphically as... 

FIG. 8 PIMC results (symbols) of the imaginary-time correlations G r) versus imaginary time for densities p = 0.1,0.2,..., 0.7 from bottom to top the temperature is T = 1. The full line shows the results for Q r) according to the lowest-order virial expansion the dashed lines give the MF values of Q r) for the densities p = 0.7, 0.6, and 0.5 from top to bottom. (Reprinted with permission from Ref. 175, Fig. 1. 1996, American Physical Society.)... 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

The weight-average Molecular weightin the inverse of the intercept at c = 0 and q = 0 in the Zimm plot. The second virial coefficient can be calculated from the slope of the lines at constant angle by eq. 8.34, and to a first approximation is independent of angle. The radius of gyration is... 

Fig. 5.4.7. Sorption isotherms of C12-LAS on different sediments. The lines were calculated with the virial equation (from Westall et al. [20]).

Hm for steam + n-heptane calculated by the above method is shown by the dashed lines in figure 6. Considering the simplicity of the model and the fact that no adjustable parameters have been used, agreement with experiment is remarkable. For mixtures of steam + n-hexane, benzene and cyclohexane agreement with experiment is much the same. At low densities the model reproduces the curvature of the lines through the results better than the virial equation of state. The method fails to fully reproduce the downward turn of the experimental curves at pressures near saturation, but does marginally better in this region than the P-R equation with k. = -0.3. At supercritical temperatures the model seems to... 

Fig. 12. Inverse of the reduced theta temperature for which the second virial coefficient vanishes from MC calculations on a cubic lattice for linear chains (squares) and f=6 stars (cir-clelike) broken lines (no symbols) stars with f=4 and 5. Reprinted with permission from [144]. Copyright (1991) American Chemical Society...

Fig. 13. Chain length dependence of the second virial coefficient A2 for some star branched macromolecule, according to Casassa (full line). The data points correspond to measurements [89] (triangles 3-arm, circles 12-arm and rhombus 18-arm stars. Reprinted with permission from [89]. Copyright [1984] American Society...

A straight line is expected if all virial coefficients larger than A3 are neglected. The slope gives the third virial coefficient and the intercept the second one. The Japanese group found essentially the same values for with the exception that for very high molar masses a continuous increase of about 25% was found. The plot is essentially equivalent to a suggestion by Stockmayer and Casassa [155]. Roovers et al. [172] checked the Bawn procedure with the conventional one and found no difference between the various techniques. 

The MOLWT-II program calculates the molecular weight of species in retention volume v(M(v)), where v is one of 256 equivalent volumes defined by a convenient data acquisition time which spans elution of the sample. I oment of the molecular weight distribution (e.g., Mz. Mw. Mn ) are calculated from summation across the chromatogram. Along with injected mass and chromatographic data, such as the flow rate and LALLS instruments constants, one needs to supply a value for the optical constant K (Equation la), and second virial coefficient Ag (Equation 1). The value of K was calculated for each of the samples after determination of the specific refractive index increment (dn/dc) for the sample in the appropriate solvent. Values of Ag were derived from off-line (static) determinations of Mw. 

The slope of the lines in Figure 3.10, i.e., the virial constant B, is related to the CED. The value for B would be zero at the theta temperature. Since this slope increases with solvency, it is advantageous to use a dilute solution consisting of a polymer and a poor solvent to minimize extrapolation errors. 

Since the surface is not crossed by any gradient lines, it is referred to as the surface of zero flux. As further discussed below, the virial theorem is satisfied for each of the regions of space satisfying the zero-flux boundary condition. 

When we make a plot of Z versus 1/V, we expect that the value of Z = 1 when 1/y = 0 or at infinite volume. The first virial coefficient B T) is the slope of the line at 1/y = 0. B T) is usually negative at very low temperatures due to the attractive forces. But when the temperature increases, B T) will turn positive, and the temperature at which B becomes zero is called the Boyle temperature, since that is the temperature where Boyle s law applies exactly (at infinite volume). 

Time-dependent correlation functions. Similar pair and triplet distributions, which describe the time evolution of a system, are also known [318]. These have found interesting uses for the theory of virial expansions of spectral line shapes, pp. 225 ff. below [297, 298],... 

Collision-induced dipoles manifest themselves mainly in collision-induced spectra, in the spectra and the properties of van der Waals molecules, and in certain virial dielectric properties. Dipole moments of a number of van der Waals complexes have been measured directly by molecular beam deflection and other techniques. Empirical models of induced dipole moments have been obtained from such measurements that are consistent with spectral moments, spectral line shapes, virial coefficients, etc. We will briefly review the methods and results obtained. 

We start with the basic relationships ( Ansatz ) of collision-induced spectra (Section 5.1). Next we consider spectral moments and their virial expansions (Section 5.2) two- and three-body moments of low order will be discussed in some detail. An analogous virial expansion of the line shape follows (Section 5.3). Quantum and classical computations of binary line shapes are presented in Sections 5.4 and 5.5, which are followed by a discussion of the symmetry of the spectral profiles (Section 5.6). Many-body effects on line shape are discussed in Sections 5.7 and 5.8, particularly the intercollisional dip. We conclude this Chapter with a brief discussion of model line shapes (Section 5.10). 

Summarizing, it may be said that virial expansions of spectral line shapes of induced spectra exist for frequencies much greater than the reciprocal mean free time between collisions. The coefficients of the density squared and density cubed terms represent the effects of purely binary and ternary collisions, respectively. At the present time, computations of the spectral component do not exist except in the form of the spectral moments see the previous Section for details. 

M. Moraldi, Virial expansion of correlation functions for collision-induced spectroscopies, in Spectral Line Shapes 6, L. Frommhold and J. W. Keto, eds., American Institute of Physics 1990. 

The theory of line shapes of systems involving one or more molecules starts from the same relationships mentioned in Chapter 5. We will not repeat here the basic developments, e.g., the virial expansion, and proceed directly to the discussion of binary molecular systems. It has been amply demonstrated that at not too high gas densities the intensities of most parts of the induced absorption spectra vary as density squared, which suggests a binary origin. However, in certain narrow frequency bands, especially in the Q branches, this intensity variation with density q differs from the q2 behavior (intercollisional effect) the binary line shape theory does not describe the observed spectra where many-body processes are significant. In the absence of a workable theory that covers all frequencies at once, even in the low-density limit one has to treat the intercollisional parts of the spectra separately and remember that the binary theory fails at certain narrow frequency bands [318],... 

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