Reduced virial coefficient

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. 

From equations (13a) and (13b) w e see that it is possible to define reduced virial coefficients B = 5/6q and C = Cjbl, which, for a given Ay are functions... 

Figure 1 shows reduced virial coefficients calculated for a more realistic two-parameter potential, the Lennard-Jones 6—12 potential. The only significant difference between the Lennard-Jones and square-well second virial coefficients occurs at high reduced temperatures, where experimentally B is observed to have a maximum value. This maximum is related to the softness of the repulsive portion of the potential it therefore cannot be reproduced by the infinitely steep square well. The third virial coefficients calculated from the square well are qualitatively correct but show significant departures from the Lennard-Jones values. 

More reliable procedures are based on the principle of corresponding states. As noted in the earlier section, it is possible to define reduced virial coefficients that are universal functions of the reduced temperature. Pitzer has shown that this corresponding states principle is strictly valid for substances that interact with a potential of the form... 

As the above expressions for Uk meike clear, the reduced virial coefficients are dimensionless, and therefore independent of the sphere diameter a. To simplify the following discussion, we will therefore set <7=1. With this choice, the product of /-functions in the denominator is nonzero in the region where r/ < 1 for all / n. The maximum volume assumed by a parallelotope within this region is just 1. (If the edges incident with vertex n are used to define the parallelotope, it is just a unit cube for other choices of reference vertex the parallelotope is skewed, but still has unit voltune.) For the numerator, the product of /- functions is nonzero when rim < 1 for all /m n. The maximum parallelotope volume within this region can be determined simply by maximizing the expression for the parallelotope volume, subject to the... 

The temperature dependence of several first virial coefficients is calculated for the Lennard-Johns (12,6) model potential (Equation 1.7 0). Figure 1.41 contains reduced virial coefficients (where 6 is the excluded volume of hard spheres, see... 

CALCULATE THE MODIFIED REDUCED DIPOLE TO BE USED IN CALCULATING THE FREE-POLAR CONTRIBUTION TO THE VIRIAL COEFFICIENT. 

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...

Statistical mechanics provides physical significance to the virial coefficients (18). For the expansion in 1/ the term BjV arises because of interactions between pairs of molecules (eq. 11), the term C/ k, because of three-molecule interactions, etc. Because two-body interactions are much more common than higher order interactions, tmncated forms of the virial expansion are typically used. If no interactions existed, the virial coefficients would be 2ero and the virial expansion would reduce to the ideal gas law Z = 1). 

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). 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

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. 

Both the value of the intrinsic viscosity and virial coefficients of the concentration dependence terms decrease with time. The reduced specific viscosity... 

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])...

Since the terms B/V, CfV2, etc., of the virial expansion [Eq. (3.11)] arise on account of molecular interactions, the virial coefficients B, C, etc., would be zero if no such interactions existed. The virial expansion would then reduce to... 

For a real gas, molecular interactions do exist, and exert an influence on the observed behavior of the gas. As the pressure of a real gas is reduced at constant temperature, V increases and the contributions of the terms B/V, C/V2, etc., decrease. For a pressure approaching zero, Z approaches unity, not because of any change in the virial coefficients, but because V becomes infinite. Thus in the limit as the pressure approaches zero, the equation of state assumes the same simple form as for the hypothetical case of B = C = = 0 that is... 

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... 

---