The evolution of multicomponent systems at high pressures Part II.The Alder–Wainwright, high-density, gas–solid phase transition of the hard-sphere fluid

Abstract

The thermodynamic stability of the hard-sphere gas has been examined, using the formalism of scaled particle theory (SPT), and by applying explicitly the conditions of stability required by both the second and third laws of thermodynamics. The temperature and volume limits to the validity of SPT have also been examined. It is demonstrated that SPT predicts absolute limits to the stability of the fluid phase of the hard-sphere system, at all temperatures within its range of validity. Because SPT describes fluids equally well as dilute gases or dense liquids, the limits set upon the system stability by SPT must represent limits for the existence of the fluid phase and transition to the solid. The reduced density at the stability limits determined by SPT is shown to agree exactly with those of that estimated for the Alder–Wainwright, high-density gas–solid phase transition in a hard-sphere system, at a specific temperature, and closely over a range of more than 1000 K. The temperature dependence of the gas–solid phase stability limits has been examined over the range 0.01 K–10000 K. It is further shown that SPT describes correctly the variation of the entropy of a hard-core fluid at low temperatures, requiring its entropy to vanish as T→0 by undergoing a gas–solid phase transition at finite temperature and all pressures.

References

1. J. F. Kenney, Fluid Phase Equilib., 1998, 148, 21.
2. B. J. Alder and T. E. Wainwright, J. Chem. Phys., 1957, 27, 1208.
3. J. G. Kirkwood, J. Chem. Phys., 1935, 3, 300.
4. M. Born and H. S. Green, Proc. R. Soc. London, 1946, 188, 10.
5. H. Reiss, H. L. Frisch and J. L. Lebowitz, J. Chem. Phys., 1959, 31, 369.
6. H. Reiss, H. L. Frisch, E. Gelfand and J. L. Lebowitz, J. Chem. Phys., 1960, 32, 119.
7. H. Reiss, “Scaled particle theory in the statistical thermodynamics of fluids”, in Adv. Chem. Phys., ed. I. Prigogine, Interscience, New York, 1965, vol. 9.
8. T. De Donder, L'Affinity, Paris, 1936.
9. I. Prigogine and R. Defay, Chemical Thermodynamics, Longmans, London, 1954.
10. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations, John Wiley, New York, 1971.
11. I. Prigogine, Introduction to the Thermodynamics of Irreversible Processes, John Wiley, New York, 1967.
12. I. Prigogine and C. George, Proc. Nat. Acad. Sci., 1983, 80, 4590.
13. H. Reiss, H. L. Frisch and J. L. Lebowitz, “Mixtures of hard spheres”, in The Equilibrium Theory of Classical Fluids, ed. H. L. Frisch and J. L. Lebowitz, W. A. Benjamin, New York, 1964.
14. H. Reiss, J. Chem. Phys., 1967, 47, 186.
15. H. Reiss and D. M. Tully-Smith, J. Chem. Phys., 1971, 55, 1674.
16. H. Reiss and R. V. Casberg, J. Chem. Phys., 1974, 61, 1107.
17. H. Reiss, “Scaled particle theory of hard sphere fluids to 1976”, in Statistical Mechanics and Statistical Methods in Theory and Application, ed. U. Landman, Plenum, London, 1977.
18. D. M. Tully-Smith and H. Reiss, J. Chem. Phys., 1970, 33, 4015.
19. N. F. Carnahan and K. E. Starling, J. Chem. Phys., 1969, 51, 635.
20. H. F. Baker, Proc. London Math. Soc., 1902, 34, 347.
21. J. E. Campbell, Proc. London Math. Soc., 1898, 29, 14.
22. H. F. Baker, Proc. London Math. Soc., 1903, 2, 293.
23. H. F. Baker, Proc. London Math. Soc., 1904, 3, 24.
24. F. Hausdorff, Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math-Naturwiss., 1906, 58, 19.
25. B. J. Alder, W. G. Hoover and D. A. Young, J. Chem. Phys., 1968, 49, 3688.
26. U. K. Deiters and K. M. de Reuck, International Union of Pure and Applied Chemistry, 1997.
27. J. D. Bernal, Nature, 1959, 183, 141.
28. G. O. Jones and P. A. Walker, Proc. Phys. Soc. B., 1953, 69, 1348.
29. R. H. Fowler and E. A. Guggenheim, Statistical Mechanics, Cambridge University Press, Cambridge, 1939.
30. J. Timmermans and L. Deffet, Le Polymorphisme des Composes Organiques, Gauthier-Villars, Paris, 1939.
31. T. Boublik, C. K. Viga and M. Diaz-Pena, J. Chem. Phys., 1990, 93, 730.
32. W. G. Chapman, K. E. Gubbins, G. Jackson and M. Radosz, Ind. Eng. Chem. Res., 1990, 29, 1709.
33. I. Nezbeda, Mol. Phys., 1997, 90, 661.
34. P. Vimalchand, A. Thomas, I. G. Economou and M. D. Donohue, Fluid Phase Equilib, 1992, 73, 39.
35. Y. Zhou, S. W. Smith and C. Hall, Mol. Phys., 1995, 85, 1157.
36. U. K. Deiters and S. L. Randzio, Fluid Phase Equilib, 1995, 103, 199.
37. U. K. Deiters, Mol. Phys, 1992, 96, 539.
38. A. C. Mitus, H. Weber and D. Marx, Phys. Rev., 1997, E55, 6855.
39. W. G. Hoover, N. E. Hoover and K. Hansen, J. Chem. Phys., 1979, 70, 1837.
40. E. Helfand, H. L. Frisch and J. L. Lebowitz, J. Chem. Phys., 1961, 34, 1037.
41. L. Tonks, Phys. Rev., 1936, 50, 955.