Virial expansions

32 Virial Expansion To compare the theory with experiments, II needs to be expressed in terms of mass concentration c, typically in g/L or mg/L. Using the identity 

In this virial expansion, Aj is the (osmotic) second virial coefficient, and A3 is the third virial coefficient. A positive A2 deviates II upward compared with that of the ideal solution = c/M). When Aj = 0, the solution is close to 

This eqnality applies to a sufficiently good solvent only in which AjM dominates over the third term. Because c = M/Np /R (Eq. 1.108) and = b(M/M y with My, being the molecular weight of the segment, c s With Eq. 

The exponent on M is 1/5 or -0.23. We thus find that A2 decreases with M, but its dependence is weak. This dependence was verified in experiments. 

Likewise the virial expansion of Il/Ilideai in terms of f allows us to find the overlap volume fraction 4 as 4 = [N( /2-x) - This result is, however, wrong. We know that 4 should rather be or for real chains in a good sol- 

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The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

In the thennodynamic limit (N x, F -> oo withA7F= p), this is just the virial expansion for the pressure, with 7,(7) identified as the second virial coefficient... 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

The PirialExpansion. Many equations of state have been proposed for gases, but the virial equation is the only one having a firm basis in theory (1,3). The pressure-expHcit form of the virial expansion is... 

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 equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the Hquid region. The compressibihty factor can be used to accurately determine gas volumes when used in conjunction with a virial expansion or an equation such as equation 53 (77). However, the prediction of saturated Hquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibihty factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

For supercritical temperatures, it is satisfactory to ever-higher pressures as the temperature increases. For pressures above the range where Eq. (4-190) is useful, but below the critical pressure, the virial expansion in density truncated to three terms is usually suitable ... 

The first term of a virial expansion [296] of the correlation function is... 

FIG. 8 PIMC results (symbols) of the imaginary-time correlations G r) versus imaginary time for densities p = 0.1,0.2,..., 0.7 from bottom to top the temperature is T = 1. The full line shows the results for Q r) according to the lowest-order virial expansion the dashed lines give the MF values of Q r) for the densities p = 0.7, 0.6, and 0.5 from top to bottom. (Reprinted with permission from Ref. 175, Fig. 1. 1996, American Physical Society.)... 

In the inset of Fig. 9 we show the mean field frequency 0 = 0// as a function of density for T = 1. At this temperature the system undergoes a phase transition from a paramagnetic to a ferromagnetic fluid at a density whose mean field value is p mf = 0-4- For densities below this value we obtain 0 = cjq, which agrees with the frequency value of the low-order virial expansion (see Eq. (34)). For p > Pc,mF) increases with the density due to increase of the magnetization. 

By analogy, the virial expansion of the bulk molecular property X (such as the dielectric polarization) is written... 

The equilibrium between a pure solid and a gaseous mixture is one of very few classes of solution for which an exact treatment can be made by the methods of statistical mechanics. The earliest work on the theory of such solutions was based on empirical equations, such as those of van der Waals,45 of Keyes,44 and of Beattie and Bridgemann.3 However, the only equation of state of a gas mixture that can be derived rigorously is the virial expansion,46 66... 

Vinylidene dicyamide, 155 Virial coefficient, 105, 108 Virial expansion, 104... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

When the virial expansion (truncated after the third term) is substituted into Eq. (6), we obtain... 

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... 

Coefficients in the virial expansion of the osmotic pressure as a power series in the concentration c (Chap. XII et seq.). 

Coefficients in the alternative virial expansion of the osmotic pressure (see Eqs. VII-13 and XII-76). 

Finite concentrations are treated in terms of a virial expansion... 

As for the case of membrane osmometry, non-ideality is accounted for by a virial expansion (Equation (28)). 

Finite concentrations are accounted for by a virial expansion of both D and s-1... 

For ionic defects the individual terms in the formal virial expansions diverge just as they do in ionic solution theory. The essence of the Mayer theory is a formal diagram classification followed by summation to yield new expansions in which individual terms are finite. The recent book by Friedman25 contains excellent discussions of the solution theory. We give here only an outline emphasizing the points at which defect and solution theories diverge. Fuller treatment can be found in Ref. 4. 

Meeron60 62 first pointed out how the terms in S(Jt> in the solution theory can be arranged in a form much more compact than that above, which is of the form of a virial expansion in which the coefficients involve the Debye Hiickel potential of average force rather than the unscreened potential. Similar manipulations can be made in the present case, but we shall omit the details, which are very simple, and quote only the final result. It is found using Meeron s form of S that the activity coefficient of defect number s can be written... 

If the gas of charges (plasma) is sufficiently dilute, we could hope a priori that its equation of state would be described by the virial expansion ... 

We shall not dwell any further on the applications of Eq. (148) and its experimental verification (see Refs. 11 and 8, and references quoted in the latter). We just wish to end this section with a remark which will be relevant later the result (148) could have been obtained formally if we had taken a virial expansion of the type (115) limited to the second order but calculated with an effective potential ... 

Finally, we study the structure of the generalized Boltzmann operator. It can be expressed in terms of the transport operator, which allows one to obtain the virial expansion of the generalized Boltzmann equation. The remarkable point here is that the generalized Boltzmann operator can be expressed in terms of non-connected contributions to the transport operator. This happens for the correction proportional to c3 (c = concentration) and for the following terms in the virial expansion of the generalized Boltzmann operator. 

Let us mention first the work of Stecki who expanded Bogolubov s results in a series in A28 and who with Taylor showed that this expansion is identical to all orders in A with the generalized Boltzmann operator (85).29 Since the method is rather different from the virial expansions which we present here, we give in Appendix A.III the major thoughts of this general work valid for any concentration. 

Another remarkable point is the appearance in [Q(t0)Yfirst time when n = 4 (we cannot have two 6LW) with no particle in common if we do not have at least four particles), but also exist to higher orders in the concentration. Their evaluation necessitates some delicate mathematical manipulations (application of the factorization theorem) but the extension of this technique to the higher-order terms of the virial expansion does not seem to pose any new problem. 

Added in Proof.] We do not here discuss the logarithmic singularities which occur in the virial expansion and have recently been reported by I. Oppenheim and K. Kawasaki [Phys. Rev. 139A, 1763 (1965)]. 

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