Second virial coefficients temperature dependence

To compare experimental values of the second virial coefficients of A, Ng, and C2Hg with the reduced equation 

The following table 1 gives experimental values of the second virial coefficient B (Holbom and Otto, Z. Phys. 1926, 33 1 Reamer, Olds, Sage, and Lacey, Ind. Eng. Chem. 1944, 36, 956) and also experimental values of the critical temperature Tg and critical volume Vg (Onnes and Crommelin, Comm. Phjre. Lab. Leiden, 1912, 131a Mathias, Oimes, and Crommelin, Comm. Phys. Lab. Leiden, 1914, 145c White, Friedman and Johnston, J. Amer. Chem. Soc. 1951, 73, 5713 Sage, Webster, and Lacey, Ind. Eng. Chem. 1937, 29, 658). 

For each substance at each experimental temperature we calculate a value of the reduced virial coefficient BiV by means of (1) and compare with the experimental value. 

We construct table 2. We see that the agreement between calculated and experimental values is satisfactory. 

The temperature dependence of the second virial coefficient can be fitted by two-constant equations, e.g. that of Lennard-Jones, but these have not a simple algebraic form. It can be fitted by several alternative three-constant equations. The form here used is the simplest derivable from a well defined model, namely a square-well potential (see Guggenheim, Australian Rev. Pure and App. Chem. 1953, 3,1). 

---

The values of the virial coefficients for a gas at a given temperature can be determined from the dependence of p on Fm at this temperature. The value of the second virial coefficient B depends on pairwise interactions between the atoms or molecules of the gas, and in some cases can be calculated to good accuracy from statistical mechanics theory and a realistic intermolecular potential function. 

Individual contributions to the second virial coefficient are calculated from temperature-dependent correlations ... 

CALCULATE THE TEMPERATURE DEPENDENT SECOND VIRIAL COEFFICIENTS. 

This expression is called the virial equation. The coefficients B, C,. . . are called the second virial coefficient, third virial coefficient, and so on. The virial coefficients, which depend on the temperature, are found by fitting experimental data to the virial equation. 

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. 

To overcome the problem of non-ideality the work be carried out at the Q temperature because in nonideal solutions the apparent Molecular weight is a linear function of concentration at temperatures near Q and the slope depending primarily on the second virial coefficient. 

Other dilute solution properties depend also on LCB. For example, the second virial coefficient (A2) is reduced due to LCB. However, near the Flory 0 temperature, where A2 = 0 for linear polymers, branched polymers are observed to have apparent positive values of A2 [35]. This is now understood to be due to a more important contribution of the third virial coefficient near the 0 point in branched than in linear polymers. As a consequence, the experimental 0 temperature, defined as the temperature where A2 = 0 is lower in branched than in linear polymers [36, 37]. Branched polymers have also been found to have a wider miscibility range than linear polymers [38], As a consequence, high MW highly branched polymers will tend to coprecipitate with lower MW more lightly branched or linear polymers in solvent/non-solvent fractionation experiments. This makes fractionation according to the extent of branching less effective. 

At moderate pressures, the virial equation of state, truncated after the second virial coefficient, can be used to describe the vapor phase. As suggested by Hirschfelder, et. al. (1 3) the temperature dependence of the virial coefficients is expressed... 

The properties of solutions of macromolecular substances depend on the solvent, the temperature, and the molecular weight of the chain molecules. Hence, the (average) molecular weight of polymers can be determined by measuring the solution properties such as the viscosity of dilute solutions. However, prior to this, some details have to be known about the solubility of the polymer to be analyzed. When the solubility of a polymer has to be determined, it is important to realize that macromolecules often show behavioral extremes they may be either infinitely soluble in a solvent, completely insoluble, or only swellable to a well-defined extent. Saturated solutions in contact with a nonswollen solid phase, as is normally observed with low-molecular-weight compounds, do not occur in the case of polymeric materials. The suitability of a solvent for a specific polymer, therefore, cannot be quantified in terms of a classic saturated solution. It is much better expressed in terms of the amount of a precipitant that must be added to the polymer solution to initiate precipitation (cloud point). A more exact measure for the quality of a solvent is the second virial coefficient of the osmotic pressure determined for the corresponding solution, or the viscosity numbers in different solvents. 

A2 from equation (5.16) or the cross second virial coefficient from equation (5.17). In turn, this knowledge of the second virial coefficients and their temperature dependence allows calculation of the values of the chemical potentials of all components of the biopolymer solution or colloidal system, as well as enthalpic and entropic contributions to those chemical potentials. On the basis of this information, a full description and prediction of the thermodynamic behaviour can be realised (see chapter 3 and the first paragraph of this chapter for the details). 

Figure 6.5 Temperature dependence of the characteristics of sodium k-carrageenan particles dissolved in an aqueous salt solution (0.1 M NaCl). The cooling rate is 1.5 °C min-1, (a) ( ) Weight-average molar weight, Mw, and (A) second virial coefficient, A2. (b) ( ) Specific optical rotation at 436 nm, and ( ) penetration parameter, y, defined as tlie ratio of the radius of the equivalent hard sphere to the radius of gyration of the dissolved particles (see equation (5.33) in chapter 5). See the text for explanations of different regions I, II, III and IV.

Equations of this type are known as virial equations, and the constants they contain are called the virial coefficients. It is the second virial coefficient B that describes the earliest deviations from ideality. It should be noted that B would have different but related values in Equations (26) and (27), even though the same symbol is used in both cases. One must be especially attentive to the form of the equation involved, particularly with respect to units, when using literature values of quantities such as B. The virial coefficients are temperature dependent and vary from gas to gas. Clearly, Equations (26) and (27) reduce to the ideal gas law as p - 0 or as n/V - 0. Finally, it might be recalled that the second virial coefficient in Equation (27) is related to the van der Waals constants a and b as follows ... 

The leading correction to ideality arises from the second virial coefficient B(T), whose qualitative T dependence is shown in Fig. 2.6 (approximating the experimental data for C02). As shown in the figure, B(T) rises from strongly negative values near the low-T condensation limit to weakly positive values at very high T. At an intermediate T known as the Boyle temperature rBoyle, the second virial coefficient vanishes ... 

Figure 2.6 Representative T dependence of the second virial coefficient B(T), showing the strong negative deviations from ideality at small T, the weak positive deviations at high Tand the Boyle temperature (TBoyie — 750K for C02) where B(TBoylc) vanishes.

This measure, however, pertains to the normal boiling point rather than to ambient conditions. The deficit of the entropy of the liquid solvent relative to the solvent vapour and to a similar non-structured solvent at any temperature, such as 25 °C, has also been derived (Marcus 1996). An alkane with the same skeleton as the solvent, i.e., with atoms such as halogen, O, N, etc. being exchanged for CH3, CH2, and CH, etc., respectively, can be taken as the non-structured solvent. Since the vapour may also be associated, the temperature dependence of the second virial coefficient, B, of the vapour of both the solvent and the corresponding alkane, must also be taken into account. The entropy of vaporization at the temperature T, wherep P°, is given by ... 

The lateral forces depend on temperature at high temperatures the repulsion interactions between particles prevail on the contrary, at low temperatures the attraction interactions prevail, so that there is a temperature at which the repulsion and attraction effects exactly compensate each other. This is the 0-temperature at which the second virial coefficient is equal to zero. It is convenient to consider the macromolecular coil at 0-temperature to be described by expressions for an ideal chain, those demonstrated in Sections 1.1-1.4. However, the old and more recent investigations (Grassberger and Hegger 1996 Yong et al. 1996) demonstrate that the last statement can only be a very convenient approximation. In fact, the concept of 0-temperature appears to be immensely more complex than the above picture (Flory 1953 Grossberg and Khokhlov 1994). 

The second virial coefficient of the macromolecular coil B(T) depends not only on temperature but on the nature of the solvent. If one can find a solvent such that B(T) = 0 at a given temperature, then the solvent is called the 0-solvent. In such solvents, roughly speaking, the dimensions of the macromolecular coil are equal to those of an ideal macromolecular coil, that is the coil without particle interactions, so that relations of Sections 1.1—1.4 can be applied to this case, as a simplified description of the phenomenon. 

Values of C, like those of B, depend on the identity of the gas and on the temperature. However, much less is known about third virial coefficients than about second virial coefficients, though data for y-number of gases can be found in the literature. Since virial coefficients beyond the third are rarely known and since the virial expansion with more than three terms becomes unwieldy, virial equations of more than three terms are rarely used. Alternative equations are described in Secs. 3.S and 3.6, which follow. 

A further issue for review is the treatment of attractive interactions. The treatment here was limited to consideration of the second virial coefficient as in Eq. (4.46), and this implies the composition and temperature dependences exhibited in Eq. (4.49). Those composition and temperature dependences are certainly the leading factors, but a more general evaluahon of first-order perturbahon theory could result in subtle corrections to those dependences. Additionally, some implicit temperature and pressure dependence is implied by the variations of the pure liquid properties in those factors. Finally, the limitation of the hrst-order perturbation theory must also be borne in mind there are experimental cases where hrst-order perturbation theory appears to be unsatisfactory (Lefebvre et al, 1999). 

---