We w i s h to c a l c u l a t e the i n t e g r a l
Q OO
J r 2 h . ( r ) d r (A2.1)
w h e r e h A (r) is a f u n c t i o n o c c u r r i n g in W e r t h e i m ' s s o l u t i o n 21
of the m e a n s p h e r i c a l m o d e l for h a r d s p h e r e s (diameter a) at
d e n s i t y p, w i t h i m b e d d e d p o i n t d i p o l e s (dipole m o m e n t y) . F r o m e q u a t i o n (41) of W e r t h e i m h A (r) 2K(h (r,2Kp) - h ( r , - K p ) ) (A2.2) (2) s ’ - = 9 S (r,p) - 1 is t he s o l u t i o n of the h a r d w h e r e h g (r, p ) sphe r e PY e q u a t i o n for d e n s i t y p. K is a c o n s t a n t d e t e r m i n e d by W e r t h e i m ' s e q u a t i o n (51), 4tt . ß y 2 3 j - ( - ± —) ( p o d ) ö 3 q ( j ( pö 3 ) K) - q (- ~ ( pu 3) K) (A2.3) w h e r e q (x) (l+2x) ( 1 - x ) 4 (A2.4) C o n s i d e r the i n t e g r a l I. F r o m (A2.2), I = (l7'P.° ) 2K (I (2Kp) - I (-Kp) ) (A2.5) w h e r e I (p)
1
" h s (r , p) dr (A2.6)I r 2 (g 2 (r / p) -l) dr (A2.7)
62
Theile has shown
{ r 2 (g^r,p)-1)dr
i + — -
4npö(q P 0 3)-1) (A2.8)
so that
I = i [q 1 (2. pa3K) + 2q 1 (- £ p a 3K) - 3] (A2.9)
Thus, if we define the variable
6 = I (pa3)K
(A2.10)then our integral is given by
I = i [q 1 (2C) + 2q 1 (-€) - 3] (A2.ll)
where £ satisfies (from (A2.8))
4 7T , 3 P 2 , 3 x
_ ( b ü _ ) ( p ö d )
3
a
3q(2£) - q(-£) (A2.12)
with the Wertheim restriction
§6.
REFERENCES
1. J .A . Barker, R.A. Fisher, R.O. Watts, "Molec.P h y s .", 2 1 , 657 (1971).
2. J.S. Rowlinson, "Liquids and Liquid Mixtures", 2nd edn. Butterworth (1969).
3. A . D . Buckingham, Disc . F a r a d . Soc . , 4_3, 205 (1967).
4. E .A . Mason and T.H. Spurling, "The Virial Equation of State" , Pergamon (1969).
5. J.R. Sweet and W.A. Steele, " J.Chem.Phys.", 4_7_, 3029 (1967) ; 5_0 , 668 (1968)
Y.D. Chen and W.A. Steele, " J .Chem. P h y s . " , 3_0, 1428 (1969) ; 5_2, 5284 (1970) .
6. R.O. Watts, "Molec. P h y s ." , 23_, 445 (1972). 7. R. Zwanzig, " J . C h e m . P h y s ." , 3_9 , 1714 (1963).
S.L. Brenner, D.A. McQuarrie, D. Olivares, "J .Chem.Phys.", 5_9 , 2596 (1973) .
K. Lakatos, "J. Stat.Phys. " , 2_, 121 (1970).
8. M. Born and H.S. Green, "P r o c .R o y .S o c .", A 1 8 8 , 10 (1946). 9. J.K. Percus, G.J. Yevick, "Phys.Rev.", 1 1 0 , 1, (1958). 10. J.M. Van Leeuwen, J. Groenweld, J. de Boer, "Physica",
2J5 , 792 (1959) .
11. J.L. Lebowitz, J.K. P e r c u s , "Phys.R e v .", 144 , 251 (1966). 12. L.S. Ornstein, F. Zernike, "Proc.Acad.S e i . (Amsterdam)",
17 , 793 (1914).
13. Y.D. Chen, W.A. Steele, "J.Chem. Phys. " , 54_, 703 (1971). 14. P.F. Morrison, PhD Thesis, C a l .Inst.Tech., (1972).
15. J.W. Perram, L.R. White, "Molec. P h y s ." , 28_, 527 (1974). 16. A. Ben Naim, " J . Chem. P h y s ." , 52_, 5531 (1970).
17. A. Ben Naim, "J .Chem. P h y s ." , 5_4 , 3682 (1971). A. Ben Naim, "Molec. P h y s ." , 24_, 705 (1972).
1 8
.
W .A . Steele, S.I. Sandler, "J .Chem.Phys. " , 61_, 1315 (1974) .D. Chandler, H.C. Andersen, " J . Chem. P h y s , 5J7, 1930 (1972) .
D. Chandler, " J . C h e m . Phys ." , 5_9 , 2742 (1973).
19. L.J. Lowden, D. Chandler, " J . C h e m . P h y s , 59_, 6587 (1973) 20. B. Fraser, R. Bearman, private communication.
21. M.S. Wertheim, " J .C h e m . P h y s , 55_, 4291 (1971). 22. L. Blum, "J .C h e m .P h y s .", 57, 1862 (1972); 58, 3295
(1973) .
23. G.N. Patey, J.P. Valleau, "Chem. P h y s . L e t t ." , 2\_, 297 (1
(1973) .
G.N. Patey, J.P. Valleau, " J . Chem. P h y s , 6_1 , 534 (1974). 24. G.S. Rushbrooke, G. Stell, J.S. H o y e , "Molec.P h y s .",
26_, 1199 (1973) .
25. G.A. Pople, "P r o c .R o y .S o c .", A 2 2 1 , 498 (1954).
26. K.E. Gubbins and C.G. Gray, "Molec. P h y s ." , 2_3 , 187 (1972) 27. R. Zwanzig, " J . Chem. P h y s ." , 2_2, 1420 (1954).
28. J . A . Barker and D. Henderson, " J . Chem. P h y s ." , 47_, 2856 4714 (1967).
29. G. Stell, J.C. Rasaiah, H. N a r a n g , "Molec.P h y s .", 2 7 , 1393 (1974).
30. I.R. McDonald, "J.Phys.C", 1_, 1225 (1974).
31. L. Onsager, "Chem.Rev.", 1_3, 73 (1933).
32. G.S. Rushbrooke, "Trans. F a r a d . S o c ." , _36 , 1055 (1940). 33. I. Prigogine, "Molecular Theory of Solutions", North
Holland, Amsterdam (1957).
R. Balescu, "B ull. A c a d . B e l g .C 1. S c i ." , 4_1, 1242 (1955). 34. J.S. Rowlinson, "Molec. P h y s ." , 1_, 414 (1958).
35. D. Cook and J.S. Rowlinson, "P r o c .R o y .S o c .", A 2 1 9 , 405 (1953).
36. J.S. Rowlinson, " Trans . F a r a d . Soc , 5_0, 647 (1954); 51, 1317 (1955) .
37. G. Stell, J.C. Rasaiah, H. N a r a n g , "Molec.P h y s . " , 2 3 , 393 (1972).
38. L. V e r l e t , J.L. Weiss, "P h y s .R e v .A " , 5, 939 (1972). 39. S. Sung, D. Chandler, "J .Chem.Phys.", 56, 4989 (1972). 40. W.A. Steele, S.I. Sandler, " J . Chem. P h y s , 6_1, 1315
(1974).
41. A. Bellemans, " P h y s . R e v . L e t t ." , 2_1, 527 (1968).
42. K.C. Mo, K.E. Gubbins, "Chem. P h y s . L e t t ." , 21_, 144 (1974). 43. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, E. Teller,
A.H.J. Teller, "J .Chem.Phys. " , 7\_, 1087 (1953).
44. J. Veil lard-Baron, " J . Chem. P h y s ." , 56_, 4729 (1972). 45. G.A. Few, M. Rigby, "Chem. P h y s . L e t t , 20^, 433 (1973). 46. I. McDonald, K. Singer, "Chemistry in Britain", 9_, 54
(1973) .
47. J. Barojas, D. Levesque, B. Quentrec, "Phys.Rev.", 7_, 1092 (1973).
48. See references 23, 29, 30, 37.
49. A. Rahman, F.H. Stillinger, " J.Chem. P h y s ." , 5_5, 3336 (1971); 57, 1281 (1972); 60, 1545 (1974).
50. J .A . Barker, R.O. Watts, "Molec.P h y s ." , 26, 789 (1973) 51. E.P. Wigner, "Group Theory" , Academic Press NY (1959). 52. M. Abramowitz , I.A. Stegun, "Handbook of Mathematical
Functions", Dover (1965).
53. E.T. Whittaker, G.N. Watson, "A Course of Modern Analysis", Cambridge (1969).
54. W.H. Keesom, "Comm.Phys.Lab.Leiden", Supp. 24b, Section 6 (1912) (see also Ref.55, p.210).
55. J .0. Hirschfelder, C.F. Curtis, R.B. Bird, "Molecular Theory of Gases and Liquids", Wiley, NY (1954).
5 6. S.A. Rice, P. Gray, "The Statistical Mechanics of Simple Liquids", Interscience, NY (1965).
57. A .D . Buckingham, "Disc.F a r a d .S o c , £3, 205 (1967).
R.L. Armstrong, S.M. Blumenfeld, C.G. Gray, "Con.J.Phys. £6, 1331 (1968).
P.A. Egelstaff, D.J. Page, J.G. Powles, "Molec.Phys.", 2_0, 881 (1971).
V.F. Sears, "Can.J.P h y s .", ££, 1279 (1966); ibid, £5, 237 (1967).
58. W.R. Smith, D. Henderson, J.A. Barker, "J.Chem.Phys.", 53, 508 (1970).
59. K.E. Gubbins, W.R. Smith, M.K. Tham, E.W. Tiepel, "Molec.Phys.", 22, 1089 (1973).
F.P. Buff, F.M. Schindler, "J.Chem.Phys.", 29,1075 (1958) 60. J. Kirkwood, E. Boggs, "J .Chem.Phys.", 10, 394 (1942). 61. L. V e r l e t , J. J. Weis, "M o l e c .P h y s ." , 28, 665 (1974). 62. E. T h e i l e , "J.Chem.Phys .", 39, 474 (1963).
63. F. K o h l e r , J.W. P e r r a m , "Molec.Phys .", submitted 1974. 64. F. K o h l e r , J.W. Perram, L.R. White, "J .Chem.Phys.",
submitted 1974.
65. C.J.F. Böttcher, "Theory of Electric Polarization, Vol.I" 2nd e d n . , Elsevier (1973).