The third virial coefficient

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 ... 

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. 

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.

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). 

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. 

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

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... 

Dependence of the third virial coefficients on ionic strength is neglected. The and u matrices are taken to be symmetric. 

The third virial coefficients for molecule-molecule interactions can be taken as zero for aqueous systems containing molecular solutes at low concentration. The remaining term for the molecule-molecule interaction contribution is equivalent to the unsymmetric two-suffix Margules model. 

The factors were determined in these cases by a virial expansion that was truncated with the third virial coefficient. In this case one has... 

The second remark concerns the results for the third virial coefficients obtained by Japanese colleagues around Teramoto [168-170]. The authors applied... 

A straight line is expected if all virial coefficients larger than A3 are neglected. The slope gives the third virial coefficient and the intercept the second one. The Japanese group found essentially the same values for with the exception that for very high molar masses a continuous increase of about 25% was found. The plot is essentially equivalent to a suggestion by Stockmayer and Casassa [155]. Roovers et al. [172] checked the Bawn procedure with the conventional one and found no difference between the various techniques. 

An Approximate Method. When the third virial coefficient is sufficiently small, it frequently happens that a is roughly constant, particularly at relatively low pressures. A good example is hydrogen gas (Fig. 10.7). When this is the case, we can integrate Equation (10.51) analytically and obtain... 

For a theta solvent (V2 = 0) the relevant interaction is described by the third virial coefficient using a simple Alexander approach similar to the one leading to Eq. 13, the brush height is predicted to vary with the grafting density as h pa in agreement with computer simulations [65]. 

An important assumption made in truncating Equation (34) is that the third virial coefficient is small. It is known that the third virial coefficient depends strongly on the second so that it approaches zero in poor solvents even faster than B does. In fact, if T2 is defined as the product BM2, it is known that T3 is approximately 0.25T2 in a good solvent that is, Equation (34) may be written with one additional term as... 

U. Bafile, L. Ulivi, M. Zoppi, M. Moraldi, and L. Frommhold. The third virial coefficients of collision-induced, depolarized light scattering of hydrogen. Phys. Rev. A, 44 4450, 1991. 

Here B T) is the second virial coefficient, C(T) the third virial coefficient, and so forth. Formally, the virial coefficients can be defined as successive partial derivatives of Z with respect to inverse molar volume (density) under isothermal conditions for example, B(T) is given by... 

The virial coefficients B(T), C(T), D(T),... are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B T) to be explicitly evaluated by statistical mechanical methods as... 

It is suggested by several equations describing the solution behavior of flexible polymers in good solvents that the second and third virial coefficients of the concentration dependence have a common parameter dependence 12-14). This behavior may be attributed to interchain entanglements for some polymers. In contrast, rigid rods can show completely linear concentration dependence in moderately dilute solution. Here the third virial coefficient is either negligibly small, or it is non-existent—in which case no relation exists between the second and third coefficient. Viscometry of some polyisocyanide solutions shows time-dependent transformations, reflected in pronounced changeovers from parabolic to linear concentration dependencies, and it would be of interest to define the transformations, and to detail their physical descriptions. 

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