Second virial coefficient definition

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. 

Tables VI and VII give results corresponding to two series of lignin fractions obtained with a flow-through reactor (3). (The units for dn/dc and A2 are respectively ml.g-1 and mole.ml.g-2). These results show that LALLS allows the determination of low Mw values. The dn/dc values differ from sample to sample but vary only slightly for a given set of fractions. The second virial coefficient exhibits no definite trend. Negative values indicate perhaps some association effects but light scattering alone is not sufficient to ascertain this point.

In Flory s theory (/< ), a polymer-solvent system is characterized by a temperature 0 at which (i) excluded-volume effects are just balanced by polymer-solvent interactions, so that os=l, (ii) the second virial coefficient is zero, irrespective of the MW of the polymer, and (iii) the polymer, of infinite molecular weight, is just completely miscible with the solvent The fundamental definition of the temperature is a macroscopic one, namely that for T near 0 the excess chemical potential of the solvent in a solution of polymer volume fraction v2 is of the form (18) ... 

For a gas satisfying the van der Waals equation of state, given in the definition, the second virial coefficient is related to the parameters a and b in the van der Waals equation by... 

This second definition of theta point is used to describe a solvent. In a solvent where Xi < 1/2, we have a better than theta solvent conditions with a second virial coefficient B2, which is positive. In a solvent where Xi > 1/2 we have less than theta solvent conditions with a second virial coefficient B2, which is negative. 

Figure 1 shows the representation of the experimental isotherm (B. G. Aristov, V. Bosacek, A. V. Kiselev, Trans. Faraday Soc. 1967 63, 2057) of xenon adsorption on partly decationized zeolite LiX-1 (the composition of this zeolite is given on p. 185) with the aid of the virial equation in the exponential form with a different number of coefficients in the series i = 1 (Henry constant), i = 2 (second virial coefficient of adsorbate in the adsorbent molecular field), i = 3, and i = 4 (coefficients determined at fixed values of the first and the second coefficients which are found by the method indicated for the adsorption of ethane, see Figure 4 on p. 41). In this case, the isotherm has an inflection point. The figure shows the role of each of these four constants in the description of this isotherm (as was also shown on Figure 3a, p. 41, for the adsorption of ethane on the same zeolite sample). The first two of these constants—Henry constant (the first virial constant) and second virial coefficient of adsorbate-adsorbate interaction in the field of the adsorbent —have definite physical meanings. 

A. V. Kiselev Zeolites are porous crystals. This means that we can find the molecular field distribution in their channels. The advantage of describing the adsorption on zeolites using the molecular theory consists in obtaining the constants which have a definite physical meaning (for example, the Henry constant and second virial coefficient). Further development of the theory needs a further improvement of the model based on the investigation of the adsorbate-zeolite systems by the use of modern physical methods. 

According to Flory s definition, a chain has its unperturbed dimensions in a 0 solvent where the osmotic second virial coefficient is zero (55). 

The second virial coefficient has the dimensions of a volume per mole. The unit often chosen is the Amagat unit which is by definition the molar volume of the gas at 0 °C and 1 atm. The exact value of this unit depends upon the gas considered but is approximately equal to 2-24 x 10 cm. /mole. Alternatively, it is now becoming customary to give B directly in terms of cm. /mole. 

The partial molar second virial coefficient is by definition... 

The decrease in Aj is attributed to two factors (1) poor solubility of solvent, since THF is not a good solvent for ionic groups (2) attraction between ion pairs in low dielectric constant medium. If the former is the only cause, we should obtain the same molecular weight irrespective of the ion content. However, as is shown in Figure 7, the molecular weight increases with ion content. Therefore, the decrease in definitely reflects the attractions between ion pairs in THF. The second virial coefficient of ionomer solutions is composed of two parts f32.331 i.e., — Ai +... 

The interpretation of this inequality is quite straightforward. First, because of the definition of the variable x, <5.x > 0 corresponds either to an increase of fluid substrate attractivity (i.e., an increase of ) or, alternatively, to a decrease in temperature T. Second, because x cannot become negative by definition, /(0 A, s o) given in Eq. (5.172) is an upper limit for the confinement-induced shift of the second virial coefficient of confined ideal quantum gases relative to its bulk value. The change in / (x A, Sj,o)... 

The second virial coefficient f3i is actually the excluded volume of the at molecule to another rod with orientation a3. According to the definition, the exclusion volume is the volume occupied by one molecule in which the mass centers of other molecules are not allowed to touch. For cylindrical rods with length L and diameter I). the exclusion volume is schematically shown in Figure 2.2. In this figure the two particular cases are depicted, i.e., two cylinders are parallel or perpendicular to each other, fti is dependent on the shape of the rigid rods. The expression for cylinders can be approximately expressed by... 

On the contrary, with Daoud and Jannink,49 we can consider the length of the polymers as fixed and assume that the temperature and consequently the solubility of the solution may vary. In this case, a polymer solution is represented by a point in the (C, T) plane. The good-solubility domain corresponds to temperatures T > TF where Tr in the Flory temperature. By definition TrF(N) is the temperature at which the second virial coefficient vanishes and TF = lim Trf (N - qo ). Anyway, if N is large TrF(N) is close to TF. 

It is convenient to consider c as a constant and b as a linear function of temperature. Thus, a question arises which values of z and y correspond to the Flory temperature TFT. The simplest and most physical definition of TF is as follows it is the temperature at which the second virial coefficient vanishes. This condition can be written in the form... 

This method is related to the measurement of the second virial coefficient. From the definitions given in Section 3.1.2... 

Second virial coefficients, B, at several temperatures are the most commonly available experimental data for polar solutes. Substitution of Equation 5 into the definition of second virial coefficients (3) gives ... 

Second virial cross efficients between the polar chemical and the light gases appear in the literature less often than second virial coefficients for pure polar substances. However, these coefficients (B ) can be used to determine AP(TC) of the polar solute. By substituting Equations 5, 12, and 13 in Equation 16 and by using the definition of the second virial cross coefficient (3) ... 

The simplification introduced by restricting the rods to three discrete orientations will now be apparent. The rods must intersect only in those three directions, so that the cluster integrals in three dimensions can be factorized into products of cluster integrals in one dimension. Consider for example the second virial coefficient for parallel rods. By definition... 

Roovers and Toporowski [59] found that A it) for polystyrene in cyclohexane has a definite positive value at B 1). This was a veiy important finding, because it demonstrated thcit repulsion arises between a pair of unperturbed macrorings. Casassa [68] calculated A2(t) to first order in 2. In terms of the reduced second virial coefficient h defined in Section 2.2 of Chapter 2 his result is expressed as... 

By definition, the second virial coefficient comes from the connected two chain partition function with all the ends free. An expansion in terms of the coupling constant would involve polymer configurations as shown in Fig. 5. This is incidentally identical to Eq. (27). Each line represents the probability of free polymer going from r, z to r, z,... 

By definition, the virial coefficients B, C,... or B, C,... of the pure substances depend solely on temperature. The values of the coefficients are very different for each of the forms, but they are convertible into each other, for example, the second virial coefficients may be converted as... 

Introduction of ternary interactions is of principal significance for a shift effect of the 0 point. By definition, this point implies that the second virial coefficient is equal to zero. If Equation 301 is represented as... 

Another practical definition of 0 can be proposed. We may call 0 the temperature at which the second virial coefficient between two very large coils vanishes. Fortunately, these two definitions coincide. When we are on the dividing line, the parameter u (at the m-th iteration) gives (in dimensionless units) the virial coefficient between two subunits. Since the dividing line ends at 0, where u = 0, this coefficient vanishes when the subunits are large enough. The distinction between 0 ai 0 is essentially absent from the polymer literature (which has been written mainly on the mean field level). 

By definition, the compressibility of an ideal gas is 1. By approximately what percentage does this change for hydrogen upon inclusion of the second virial coefficient term How about for water vapor Give the conditions under which you make this estimate. 

Very recently. Beer [283] reinvestigated identical samples and similarly prepar ones even more carefully and could impressively demonstrate the increase of Rg,app at low Cp to follow from the definition of the apparent radius of gyration by Eq. (4.22) if the second virial coefficient becomes anomalously large, i.e., Ke/Re at c = Cp is much larger than Kc/Re = 0 at Cp = 0. Defining an apparent radius of gyration as... 

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