By analogy, the virial expansion of the bulk molecular property X (such as the dielectric polarization) is written [Pg.292]

The first term of a virial expansion [296] of the correlation function is [Pg.103]

Coefficients in the virial expansion of the osmotic pressure as a power series in the concentration c (Chap. XII et seq.). [Pg.640]

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 [Pg.532]

Coefficients in the alternative virial expansion of the osmotic pressure (see Eqs. VII-13 and XII-76). [Pg.649]

Vinylidene dicyamide, 155 Virial coefficient, 105, 108 Virial expansion, 104 [Pg.412]

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 [Pg.233]

Finite concentrations are accounted for by a virial expansion of both D and s-1 [Pg.236]

Finite concentrations are treated in terms of a virial expansion [Pg.213]

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 [Pg.450]

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). [Pg.234]

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)]. [Pg.382]

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. [Pg.423]

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 [Pg.187]

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. [Pg.337]

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

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. [Pg.358]

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 [Pg.529]

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.) [Pg.104]

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 [Pg.2518]

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 [Pg.194]

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. [Pg.104]

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 [Pg.104]

Another remarkable point is the appearance in [Q(t0)Y

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