Virial coefficients, values

Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, ATChP J., 25, pp. 931-938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B given by Eq. (4-188), is usually preferred ... 

Values of the second virial coefficient of ethylene for temperatures between 0° and 175°C have been determined to an estimated accuracy of 0.2 cm3/mol or less from low-pressure Burnett PVT measurements. Our values, from —167 to —52 cm3/mol, agree within an average of 0.2 cm3/mol with those recently obtained by Douslin and Harrison from a distinctly different experiment. This close agreement reflects the current state of the art for the determination of second virial coefficient values. The data and error analysis of the Burnett method are discussed. 

These functions have no theoretical basis. They furnish means of calculating tables of smoothed values of the second virial coefficients and their derivatives and of visualising the scatter of experimental values. These functions did not fit some data sets adequately. In this case it was necessary to split the data for a substance into two regions of temperatme, and fit the two sets independently. The virial coefficient values merged at the boundary. 

It is necessary to consider different regimes dependent on the dimensions of polymer chains. A long chain collapses into a globule consisting of close-packed blobs of size The blobs are either Gaussian or swollen depending on the second virial coefficient value. In a 0 solvent, the chain radius in the collapsed state is given by ... 

Estimation of fugacities with the virial EoS truncated after B uses Eq. 9.11.10, with second virial coefficient values obtained from the empiric correlations of Tsonopoulosor of Hayden and O Connell (Chapter 8). Use of the SRK EoS into Eq.9.11.6 leads to ... 

At low pressures, the virial equation truncated after B, with the second virial coefficient values pr icted by the Tsonopoulos or the Hayden-0 Connell correlations, gives the best results. Cubic EoS are also successful, but for nonpolar/weakly polar compounds only. 

PARAMETER USED TO CALCULATE PART OF CHEMICAL CONTRIBUTION TO THE SECOND VIRIAL COEFFICIENT. CALCULATED ONE OF TWO WAYS DEPENDING ON THE VALUE OF ETA(IJ). 

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

Krigbaumf measured the second virial coefficient of polystyrene in cyclohexane at several different temperatures. The observed values of B as well as some pertinent volumes at those temperatures are listed below ... 

The value of B given by Eq. (10.55) has exactly the same significance that we discussed for the second virial coefficient in Chap. 8. 

When i = J, all equations reduce to the appropriate values for a pure species. When i j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of By and dBjj/dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4-183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of and for the mixture are then given by Eqs. (4-193) and (4-194), and values of In i for the component fugacity coefficients are given by Eq. (4-196). 

The two values kp and k are usually not very different, and kp is not strongly composition dependent. Nevertheless, the quadratic dependence of Z — a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the composition-dependent kp leads to better results than does use of a constant value. 

If p is not high, terms beyond the second and third virial coefficients in equation (A3.3) and (A3.5) are usually small and can be neglected. This is fortunate, since experimental data are usually not accurate enough to give reliable values for the higher order terms. At low pressures, equation (A3.5) is often used and truncated after the second virial coefficient so that... 

The same measurements also provide values of the second virial coefficient, which corresponds to the repulsive energy between micelles. The coefficient of the Na methyl a-sulfomyristate decreases from 7.30 x 10 3 to 3.05 x 10"4 ml/ g with increasing concentration of the electrolyte. The second virial coefficient of the calcium salt is small and changes to a negative value in 0.01 N Ca(N03)2. 

The second virial coefficient B in Eq. 17 refers to the static case. In the ultracentrifuge the measured value can show a speed dependence [39], an effect which can be minimized by using low speeds and short solution columns. If present it will not affect the value of after extrapolation to zero concentration. 

In some extreme cases, third or even higher virial coefficient(s) may be necessary to adequately represent the data for example /c-carrageenan [90] and alginate [91]. In a further study on alginates, Straatman and Bor-chard [92] demonstrated excellent agreement between and B values ob-... 

For dilute, teal gases, where ternary and higher collisions can be neglected, the angle of deflection can be employed to evaluate a number of physical properties. Of course appropriate distributions of the values of g and b must be introduced. The resulting expressions for the virial coefficients and the transport properties (viscosity, diffusion and thermal conductivity) are quite complicated. The interested reader is referred to advanced books on this subject... 

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

The stability of the dispersions upon dilution at 50 °C and zero value of the osmotic second virial coefficient suggests that the surface of the particles at temperatures above the LCST may possess some hydrophilic character the macromolecules self-organise and build up particles with hydrated po-... 

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