# APPENDIX TWO AN INTEGRAL ARISING IN THE DERIVATION OF THE KIRKWOOD G FACTOR IN THE M.S.M

In document Some topics in statistical physics (Page 135-141)

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

3

q(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).

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