﻿ ﻿Final Remarks - Modeling the Structure of Liquids: The Integral Equation Approach - Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 7.6. Final Remarks

Although the concepts as discussed in this chapter provide a direct “ab-initio” approach to liquid structure (and, if elaborated, to dynamics), the approach is practically troublesome. Apart from the fact that, in general, numerical solutions must be found, the extension to molecular fluids, as indicated, is not that straightforward. Starting with an EoS using the second virial coefficient, Song and Mason [25] have extended perturbation theory in a semi-empirical fashion, claiming an accuracy better than 1% for the LJ fluid. Their theory has also been applied to more realistic fluids. Nevertheless, since in most cases our interest lies more in molecular fluids than in atomic fluids, other approaches are called for.

Notes

1) We could write for but with keeping ρN−1(1) generalization to n > 2 is more obvious.

2) If we have ⟨exp(−βA)⟩ = Σn=0(n!)−1(−β)nAn⟩ and the function A itself can be expanded as A(−β) = Σn=1(k!)−1ak(−β)k, we generally have a1 = ⟨A⟩, a2 = ⟨A2⟩ − ⟨A2, a3 = ⟨A3⟩ − 3⟨A2⟩⟨A⟩ + 2⟨A3, . … The coefficients an are denoted as cumulants for which a general expression can be derived [26]. If the cumulant, e.g. a1, can be written as a product xy, the cumulant is zero if the factors x and y are statistically independent [27].

3) Fitting MC results of the coexistence curve by Lu et al. [28] showed that d = (a1T + b1)σ /(a2T + b2) with a1 = 0.56165k/ε, a2 = 0.60899k/ε, b1 = 0.9718 and b2 = 0.92868 describes d for a LJ fluid reasonably well.

4) See, e.g., for pyrazine, Ref. [24]. Experiments do show small but significant differences for some other, less rigid molecules.

References

1 Barker, J.A. and Henderson, D. (1976) Rev. Mod. Phys., 48, 587.

2 (a)Yvon, J. (1935) Actualities and Scientifique et Industrielles, nr. 203, Herman and Cie, Paris; (b)Born, M. and Green, H.S. (1946) Proc. R. Soc. Lond., A188, 10.

3 Kirkwood, J.G. and Boggs, E.M. (1942) J. Chem. Phys., 10, 394.

4 Kirkwood, J.G. (1935) J. Chem. Phys., 3, 300.

5 Bellemans, A. (1969) J. Chem. Phys., 50, 2784.

6 See McQuarrie (1973).

7 See Hansen and McDonald (2006).

8 See Friedman (1985).

9 Percus, J.K. and Yevick, G.J. (1958) Phys. Rev., 110, 1.

10 Broyles, J. (1962) J. Chem. Phys., 37, 2462.

11 See Watts and McGee (1976).

12 Fawcett, W.R. (2004) Liquids, Solutions and Interfaces, Oxford University Press, Oxford.

13 (a) Thiele, E. (1963) J. Chem. Phys., 39, 474; (b) Wertheim, M.S. (1963) Phys. Rev. Lett., 10, 321; (c)Baxter, R.J. (1968) Phys. Rev., 154, 170.

14 Smith, W.R. and Henderson, D. (1970) Mol. Phys., 19, 411.

15 Carnahan, N.F. and Starling, K.E. (1965) J. Chem. Phys., 51, 635.

16 Isihara, A. (1968) J. Phys., A1, 539; See also Friedman (1985).

17 (a) Barker, J.A. and Henderson, D. (1967) J. Chem. Phys., 47, 4714; (b) Barker, J.A. and Henderson, D. (1967) J. Chem. Phys., 47, 2856.

18 Henderson, D. and Barker, J.A. (1971) in Physical Chemistry, an Advanced Treatise, vol. 8A (eds H. Eyring, D. Henderson, and W. Jost), Academic Press, New York, p. 377.

19 Barker, J.A. and Henderson, D. (1972) Annu. Rev. Phys. Chem., 23, 439.

20 Weeks, J.D., Chandler, D., and Andersen, H.C. (1971) J. Chem. Phys., 54, 5237.

21 Andersen, H.C., Chandler, D., and Weeks, J.D. (1976) Adv. Chem. Phys., 34, 105.

22 Verlet, L. and Weis, J. (1972) Phys. Rev., A5, 939.

23 Barker, J.A. and Henderson, D. (1971) Acc. Chem. Res., 4, 303.

24 Bormans, B.J.M., de With, G., and Mijlhoff, F.C. (1977) J. Mol. Struct., 42, 121.

25 Song, Y. and Mason, E.A. (1989) J. Chem. Phys., 91, 7840.

26 Zwanzig, R.W. (1954) J. Chem. Phys., 22, 1420.

27 Kubo, R. (1962) J. Phys. Soc. Jpn, 17, 1100.

28 Lu, B.Q., Evans, R., and Telo da Gama, M.M. (1985) Mol. Phys., 55, 1319.

29 Watts, R.O. (1969) J. Chem. Phys., 50, 984.

30 Mikolaj, P.G. and Pings, C.J. (1967) J. Chem. Phys., 46, 1401.

31 Chandler, D.W., Weeks, J.D., and Andersen, H.C. (1983) Science, 220, 787.

Friedman, H.L. (1985) A course on statistical mechanics, Prentice-Hall, Englewood Cliffs, NJ.

Hansen, J.-P. and McDonald, I.R. (2006) Theory of Simple Liquids, 3rd edn, Academic, London (1st ed. 1976, 2nd edition 1986).

McQuarrie, D.A. (1973) Statistical Mechanics, Harper and Row, New York.

Watts, R.O. and McGee, I.J. (1976) Liquid State Chemical Physics, John Wiley & Sons, Inc., New York.

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