Second virial coefficient reduced

Partial parameter, cubic equation of state 2d virial coefficient, density expansion Partial molar second virial coefficient Reduced second virial coefficient... 

B = reduced second virial coefficient from L-J potential B pp = second virial coefficient for "simple fluids reduced by and B pp = acentric correction term for second virial coefficient reduced by and T Bpy = second virial coefficient reduced by B-B = Beattie-Bridgeman B-W-R = Benedict-Webb-Rubin... 

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.

Figure A2.3.6 (a) Reduced second virial coefFicient fimction of and (b) In versus 1/Jj for...

The acentric factor, CO, was the third parameter used (20) in an equation based on the second virial coefficient. This equation was further modified and is suitable for reduced temperatures above 0.5. 

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). 

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. 

Fig. 12. Inverse of the reduced theta temperature for which the second virial coefficient vanishes from MC calculations on a cubic lattice for linear chains (squares) and f=6 stars (cir-clelike) broken lines (no symbols) stars with f=4 and 5. Reprinted with permission from [144]. Copyright (1991) American Chemical Society...

The concentration effects for the oligomers and also for the excluded (high) polymer species, are usually small or even negligible, k values depend also on the thermodynamic quality of eluent [108] and the correlation was found between product A2M and k, where A2 is the second virial coefficient of the particular polymer-solvent system (Section 16.2.2) and M is the polymer molar mass [109]. Concentration effects may slightly contribute to the reduction of the band broadening effects in SEC the retention volumes for species with the higher molar masses are more reduced than those for the lower molar masses. 

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 ... 

Note that the value of the intercept, the value of r/RTc at infinite dilution, obeys the van t Hoff equation, Equation (25). At infinite dilution even nonideal solutions reduce to this limit. The value of the slope is called the second virial coefficient by analogy with Equation (27). Note that the second virial coefficient is the composite of two factors, B and (1/2) Vx/M. The factor B describes the first deviation from ideality in a solution it equals unity in an ideal solution. The second cluster of constants in B arises from the conversion of practical concentration units to mole fractions. Although it is the nonideality correction in which we are primarily interested, we discuss it in terms of B rather than B since the former is the quantity that is measured directly. We return to an interpretation of the second virial coefficient in Section 3.4. 

We have already seen that the second virial coefficient may be determined experimentally from a plot of the reduced osmotic pressure versus concentration. Since all other quantities in Equation (99) are measurable, the charge of a macroion may be determined from the second virial coefficient of a solution with a known amount of salt. As an illustration of the use of Equation (99), we consider the data of Figure 3.6 in Example 3.5. 

Table 1.2 Reduced osmotic pressures (tt/c)c=o, number average molecular weights Mn and osmotic second virial coefficient A2 for poly(pentachlorophenyl methacrylate) fractions in toluene at 25°C and benzene at 40°C (tt in cm of benzene or toluene) (c in g dl-1). (From ref. [44])...

Second virial coefficients are functions of temperature only, and similarly B° and J31 are functions of reduced temperature only. They are well represented by the following simple equations t... 

Rudin combined Flory s theory of the dimension of polymer coils with Zimm s expression (Zimm, 1946) for the second virial coefficient in a dilute suspension of uniform spheres. He further assumed that the swelling factor (e) of the polymer coil (identical to a3 in Flory s formalism) reduces to 1 at a certain critical concentration cx, Ewhereas it tends to a value e0 (= [//] (/[//]w ) at infinite dilution. The functional relation between e and c, would therefore be... 

In the sections that follow the techniques used for reducing experimental quantities to weight fractions and weight fraction activity coefficients are described. For the solvents, pure component liquid densities and second virial coefficients were often required and were obtained from Daubert and Danner (1990). 

The negative value of the second virial coefficient and the low value of the reduced viscosity at pH = 2.1 confirm a globular conformation of the polymer particles at the lEP [212]. The pH-dependent swelling behavior of the PCIEAC gel is consistent with its linear analogue and a minimum at pHiEp 2.1-2.2. 

R = gas constant and T = absolute temperature). The so-called first virial coefficient, Bi, is simply equal to the reciprocal of the molecular weight, M. The evaluation of the second virial coefficient, Bi, has become one of the principal theoretical developments in recent years. In his theory of polymer solutions Flory (1953) has shown that Bi is proportional to a term (1 — 6/T), which vanishes T = 0. Accordingly 0 may be considered as the ideal temperature at which the above equation is reduced to the well-known van t Hoff s law, i.e.,... 

More recently Lapanje and Tanford (59) have reported osmotic pressure measurements for reduced protein polypeptide chains in 6M guanidine hydrochloride. Second virial coefficient data and intrinsic viscosity data are combined by these authors to yield unperturbed dimensions of randomly coiled proteins. The result is assentially identical with that obtained earlier from viscosity data alone. 

Derive an equation for the second virial coefficient in a solution of collapsed globules below their 0-temperature, in terms of the number of Kuhn monomers per chain N, the Kuhn monomer size b and the reduced temperature (0- T)jT. Can this second virial coefficient be related to the chain interaction parameter of Eq. (3.97) ... 

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