Interaction parameter from virial coefficient

The same intermolecular potential-energy functions may be applied to correlate interaction virial coefficients. In the case where there are no, or insufficient, experimental data, but adequate data exist for the pure components, it may be possible to estimate with useful accuracy the unlike interaction parameters from combining rules such as ... 

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). 

Binary interaction parameters are determined for each pq pair p q) from experimental data. Note that = k and k = k = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. 

This stipulation of the interaction parameter to be equal to 0.5 at the theta temperature is found to hold with values of Xh and Xs equal to 0.5 - x < 2.7 x lO-s, and this value tends to decrease with increasing temperature. The values of = 308.6 K were found from the temperature dependence of the interaction parameter for gelatin B. Naturally, determination of the correct theta temperature of a chosen polymer/solvent system has a great physic-chemical importance for polymer solutions thermodynamically. It is quite well known that the second viiial coefficient can also be evaluated from osmometry and light scattering measurements which consequently exhibits temperature dependence, finally yielding the theta temperature for the system under study. However, the evaluation of second virial... 

The calculations were carried out for various values of the parameters, the aspect ratio of segment p, and the number ratio of ionizable groups in the chain f. The other parameters were estimated for NIPA gel. All the values of parameters used are summarized in Table 3. The value of v0 was determined by taking the intermediate value between water and NIPA [20]. The parameters Ca, Cb and Cc for the hydrophobic interaction were determined from the values of isobutyl substituents of amino acids, determined by Nemethy and Scheraga [19]. Since there are no data for the 6 temperature and the virial coefficients of this system, we assumed Te to be 273.15 K, and estimated the virial coefficients... 

Once the second virial coefficient has been obtained for a given polymer - solvent system one can calculate the corresponding Flory - Huggins interaction parameter, X, from the equation ... 

Rudin s aim was to predict the size of dissolved polymer molecules and the colligative properties of polymer solutions (hydrodynamic volume, second virial coefficient, interaction parameter, osmotic pressure, etc) from viscometric data (average molar mass, intrinsic viscosity, etc.). 

Fig. 36a and b. Second virial coefficient A2 of a) cellulose nitrate (CN) (Nc = 13.9%) and b) CN (Nc = 12.9%), in acetone78 79> O experimental data 2). Lines are calculated by using the penetration function j/ from short and long range interaction parameters A and B, which are estimated by methods 2C (full line), 2D (broken line), 2E (dotted line), and 2G (chain line), together with experimental [Pg.41]

This is a virial equation, the word virial being taken from the Latin word for force and thus indicating that forces between the molecules are having an effect. It turns out that statistical mechanical models also give equations that can be written in this form with the virial coefficients, B C > etc., being related to various interaction parameters. 

Figure 3. Dependence of the second virial coefficient on the concentration of cosolvent in the water (1)—lysozyme (2)—arginine (3) mixture. , experimental data solid line, values predicted using eq 10 with722 (see Table 2) as an adjustable parameter , protein—solvent interaction contribution B " calculated using eq IIB O, ideal mixture contribution B (eq 11 A) A, protein—protein interaction contribution B " P predicted by eq 11C with 722 as an adjustable parameter from Table 2. See details in Figure 1.

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. 

It is to be expected from Eq. (4.52). that measurements of the osmotic pressures of the solutions of the same polymer in different solvents should yield plots with a common intercept (at c = 0) but different slopes (see Fig. 4.6), since the second virial coefficient, which reflects polymer-solvent interactions, will be different in solvents of differing solvent power. For example, the second virial coefficient can be related to the Flory-Huggins interaction parameter x (see p. 162) by... 

It follows that atoms or molecules interacting with the same pair potential s( )(rya), but with different s and cj, have the same thermodynamic properties, derived from A INkT, at the same scaled temperature T and scaled density p. They obey the same scaled equation of state, with identical coexistence curves in scaled variables below the critical point, and have the same scaled vapour pressures and second virial coefficients as a function of the scaled temperature. The critical compressibility factor P JRT is the same for all substances obeying this law and provides a test of the hypothesis. Table A2.3.3 lists the critical parameters and the compressibility factors of rare gases and other simple substances. 

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