Virial third coefficient

The number h(z) which appears in (13.1.2) can serve to define the third virial coefficient. However, from a theoretical point of view h(z) is not as important as g(z). Moreover, the third virial coefficient is not easily measurable, and this is a pity because, in principle, the ratio h(z)/g2(z) can be measured independently of the swelling. 

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Equation (10a) is somewhat inconvenient first, because we prefer to use pressure rather than volume as our independent variable, and second, because little is known about third virial coefficients It is therefore more practical to substitute... 

Equations (7b) and (8) into Equation (6), neglecting all third virial coefficients. We then obtain... 

Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

The third virial coefficient C(7) depends upon tliree-body interactions, both additive and non-additive. The relationship is well understood [106. 107. 111]. If the pair potential is known precisely, then C(7) ought to serve as a good probe of the non-additive, tliree-body interaction energy. The importance of the non-additive contribution has been confimied by C(7) measurements. Unfortunately, large experimental uncertainties in C (7) have precluded unequivocal tests of details of the non-additive, tliree-body interaction. 

The nth virial coefficient can be written as sums of products of Mayer/-fiinctions integrated over the coordinates and orientations of n particles. The third virial coefficient for spherically syimnetric potentials is... 

This leads to the third virial coefficient for hard spheres. In general, the nth virial coefficient of pairwise additive potentials is related to the coefficient7) in the expansion of g(r), except for Coulombic systems for which the virial coefficients diverge and special teclmiques are necessary to resiim the series. 

Similarly, the third virial coefficients are defined by equation 26 ... 

Miscellaneous Generalized Correlations. Generalized charts and corresponding states equations have been pubhshed for many other properties in addition to those presented. Most produce accurate results over a wide range of conditions. Some of these properties include (/) transport properties (64,91) (2) second virial coefficients (80,92) (J) third virial coefficients (72) (4) Hquid mixture activity coefficients (93) (5) Henry s constant (94) and 6) diffusivity (95). 

The coefficient Bij characterizes a bimolecular interaction between molecules i and J, and therefore Bij = Bji. Two lands of second virial coefficient arise Bn and By, wherein the subscripts are the same (i =j) and Bij, wherein they are different (i j). The first is a virial coefficient for a pure species the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients Cm, Cjjj, and are for the pure species and Qy = Cyi = Cjn, and so on, are cross coefficients. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Figure A3.1 Examples of (a) the second virial coefficient and (b) the third virial coefficient [from equation (A3.3)] as a function of temperature for several gases.

If p is not high, terms beyond the second and third virial coefficients in equation (A3.3) and (A3.5) are usually small and can be neglected. This is fortunate, since experimental data are usually not accurate enough to give reliable values for the higher order terms. At low pressures, equation (A3.5) is often used and truncated after the second virial coefficient so that... 

This expression is called the virial equation. The coefficients B, C,. . . are called the second virial coefficient, third virial coefficient, and so on. The virial coefficients, which depend on the temperature, are found by fitting experimental data to the virial equation. 

The experimental second and third virial coefficients for steam are however widely available. But these experimental quantities should be used with more care than has been usual in the past. The prevailing notion asserts that a good two-body potential should yield the second virial in full agreement with the exper-... 

In Figure 1 we show the computed and the experimental second virial for the two potentials obtained in Ref. 26, for the most widely used semi-empirical ST2 potential, and for the Hartree-Fock potential. For the third virial coefficient we refer elsewhere. ... 

Factor relating the third virial coefficient Tz to Vl (Chaps. VII and XII). 

A3 AIBN c Cp DLS DLVO DSC EO GMA HS-DSC KPS LCST Osmotic third virial coefficient 2,2 -Azobis(isobutyronitrile) Polymer concentration Partial heat capacity Dynamic light scattering Derjaguin-Landau-Verwey-Overbeek Differential scanning calorimetry Ethylene oxide Glycidylmethacrylate High-sensitivity differential scanning calorimetry Potassium persulphate Lower critical solution temperature... 

Here, i, j, and k are subscripts representing the various species in solution and /dh is a function of ionic strength similar in form to the Debye-Hiickel equation. The terms Xy and Hijk are second and third virial coefficients, which are intended to account for short-range interactions among ions the second virial coefficients vary with ionic strength, whereas the third virial coefficients do not. 

Here, ydh is a Debye-Hiickel term, and Z) y and Ejjk are second and third virial coefficients, defined for each pair and triplet of ions in solution. As before, the values of D,j vary with ionic strength, whereas the terms Eijk are constant at a given temperature. 

STEP 5. The third virial coefficients for cation-anion pairs are... 

The coefficients B and C are the second and third virial coefficients, respectively, the first virial coefficient being 1. 

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. 

For most practical purposes the influence of the third virial coefficient A3 is slight in dilute solution so that the following form of Eq. (35) is adequate... 

The parameters -jj are second virial coefficients giving the effect of short-range forces between solutes i and j the parameters Pijk are corresponding third virial coefficients for the interaction of three solutes i, j, and k. The second virial coefficients are a function of ionic strength. 

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