Virial coefficients definitions

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) ... 

A somewhat more subtle ion pair definition was introduced by Ebeling (Eb) [210-212]. Ebeling s definition of the association constant ensures consistency up to the level of the second ionic virial coefficient between the... 

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)... 

Generally speaking, positive values of A2 mean net repulsion between the particles, while negative values of A2 correspond to attraction. Eor more detailed analysis of the values of the second osmotic virial coefficient, the use of other definitions of the particle concentration is more convenient. The common virial expansion"... 

Interpretation of the second and third virial coefficients, A2 and A3, in terms of Floiy-Huggins theory is apparent from Eq. (3.82). The second virial coefl[icient A2 evidently is a measure of the interaction between a solvent and a polymer. When A2 happens to be zero, Eq. (3.82) simplifies greatly and many thermodynamic measurements become much easier to interpret. Such solutions with vanishing A2 may, however, be called pseudoideal solutions, to distinguish them from ideal solutions for which activities are equal to the molar fractions. Inspection of Eq. (3.83) reveals that A2 vanishes when the interaction parameter X equal to. We should also recall that %, according to its definition given by Eq. (3.40), is inversely proportional to temperature T. Since x is positive for most polymer-solvent systems, it should acquire the value at some specific temperature. 

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... 

This equation clearly is consistent with Eq, (1.11), which contains the definition of the absolute temperature. The limiting process indicated in Eq. (1.11) rigorously eliminates the virial coefficients to provide a precise definition Eq. (1-48) covers a useful range of densities for practical calculations and is called the ideal-gas law. 

The simple Quadratic Representation 4 of f(p T) is definitely applicable in a large neighborhood of the coexistence boundary, and evolves to a form equivalent to a virial expansion with second and third virial coefficients at supercritical temperatures. 

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) ... 

For this experiment, the treatment of N as a constant results in the best approximation of the series coefficients as virial coefficients. The most accurate as a true virial coefficient is, of course, the second (linear) coefficient since the experiment was optimized for the determination of the linear behavior and not for the nonlinear behavior of the sample gas. However, the role of the nonlinear terms is more than that of virial remainders terms to accommodate only the analytical fit to the data. The nonlinear coefficients are characteristic of the true virial coefficients to the extent suggested to us by the variations of their values and by the definitiveness of their temperature behavior. Our listed values of the third virial coefficient agree within 2% with those determined by Douslin and Harrison (2). In Table II, we present what we consider to be our best values for the virial coefficients for N constant. Where several groups of... 

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