NOTES
3676
indices rii,Fi, Li, and Si all predict the same order of activity, SO they should all exhibit correlations with spin density of the type shown in Figure 5 - %me rnOlecules such as Phenanthrene do not show a regular intramolecular correlation, but in this case it is the hyperfine coupling constants which follow the reactivities.
Conclusions I n this paper we have shomn that the hyperfine coupling constants are good quantitative indices of relative chemical reactivity rates at various sites in a polycyc~ic hydrocarbon molecule. Various conventional molecular orbital reactivity indices are found to be unsatisfactory in this respect.
NOTES
Exact Geometrical Parameters for Pendular Ring Fluid
by James C. l\Ielrose and George C. Wallick Mobil Oil Corporation, Dallas, Texas 75221 (Receired April 5 , 1967)
That a fluid can be confined in the region of the contact point between two spherical solid particles by means of “capillary action’’ is well known. Such fluid is often said to be in a pendular ring configuration, and the problem of specifying the volume, meniscus area, and pressure deficiency is a classical problem in the theory of capillarity. The shape of the meniscus separating the pendular ring fluid from the surrounding fluid (vapor or liquid) is described by an equation due to Laplace,’**which relates the pressure deficiency to the A uid-fluid interfacial tension. From this equation it follows that, if gravitational distortion can be ignored, the meniscus must assume the form of a surface of constant mean c ~ r v a t u r e . ~ For many cases of interest it can be supposed that the two spheres are identical in size and that the properties of both solid-fluid interfaces are uniform a t all points in the region in which the meniscus meets the solid surface. It follows from the latter specification, together with Young’s equation14that the three-phase line of contact is characterized by a uniform contact angle. When this is so, the boundary conditions for Laplace’s equation have cylindrical symmetry. This establishes the fact that the meniscus will also take the form of a surface of revolution- Among surfaces of only the sphere, cylinder, unduloid, catenoid, and The Journal of Physical Chemistry
nodoid are also surfaces of constant mean curvature.jrs In the case of pendular ring fluid, the appropriate configuration of the meniscus is that of the nodoid. Since the two spheres are identical in size, the section of the nodoid involved is symmetrical with respect to the contact plane. The availability of these classical resuits7s8makes it possible to assess both the mathematical nature and the importance of various approximations which have been used in obtaining numerical solutions to the nodoid problem. Recently, AIayer and Stoweg have treated the problem by assuming the meniscus profile to be a circular arc. They then relate the curvature to differential changes in the volume, meniscus area, and area of contact between the solid and the confined fluid. The toroidal meniscus configuration was also assumed by Roselo and by earlier authors. In this previous work, it was recognized that the torus surface is not (1) J. C. Maxwell, “Capillary Action,” in Encyclopedia Britannica, 9th ed, 1875; “Scientific Papers,” Vol. 11, Cambridge University Press, London, 1890, p 541. (2) The classical hydrostatic and thermodynamic theories have been reconciled and extended by F. P. Buff, “The Theory of Capillarity,” in “Handbuch der Physik,” Vol. 10, Springer-Verlag, Berlin, 1960. (3) J. A. F. Plateau, “Statique exp6rimentale et thBorique des liquides,” Vol. 1, Gauthier-Villars, Paris, 1873, p 6. (4) T. Young, Phil. Trans. Roy. SOC.London, 95, 65 (1805). (5) J. A. F. Plateau, ref 3, p 131. (6) A lucid review is given by D. W.Thompson, “On Growth and Form,” J. T. Bonner, Ed., Cambridge University Press, London, 1961, pp 53-61. (7) H. Bouasse, “CapillaritB, phenomhnes superficiels,” Delagrave, Paris, 1924, pp 49-66. (8) G. Bakker, “Kapillaritat und Oberflachenspannung,” in “Handbuch der Experimentalphysik,” Vol. VI, Akademische Verlagsgesellschaft, Leipsig, 1928, pp 120-124. (9) R. P. Mayer and R. A. Stowe, J . Phys. Chem., 70, 3867 (1966). (io) w. Rose, J . Phys., 2 9 , 687 (1958).
NOTES
characterized by a constant mean curvature. Hence, several different approaches to the problem of computing an effective or average mean curvature were adopted. Kone of these approaches, however, corresponds precisely to the expression chosen by ,Mayer and Stowe. For example, Kruyer'l pointed out that, if the meniscus is assumed to be a hyperboloid of revolution, an excellent approximation to the nodoid curvature is obtained, particularly for small curvatures. Iczkowski12 has recently employed this approximation for the confined volume as well as for the curvature. It is the purpose of this note to draw attention to several exact mathematical treatments,13-16 to compare the numerical results so obtained with those reported by Mayer and Stowe, and to present a simple and useful relationship among meniscus area, curvature, and confined volume. The contact angle, e, is specified as the angle measured through the confined phase, and the filling angle, 9, is the angle between the axis of cylindrical symmetry and the line of meniscus contact. If R denotes the sphere radius, J the meniscus curvature, and V the confined volume, reduced parameters can be defined as JR/2 and V / R 3 . In Table I, values computed from the exact expressionsl6 are compared with those of Mayer and Stowe. For a contact angle of zero and for filling angles of 15" and larger, the assumption of the torus configuration leads to a positive error in the volume which is in excess of 1%. The error in the curvature is of the same order of magnitude but in the opposite direction. These errors decrease in magnitude appreciably as the contact angle increases. For 0 = 40°,the errors probably do not exceed 0.2% throughout the range of filling angles for which the curvature remains positive. The error in the curvature which results from IZruyer's approximation is of similar magnitude but in the opposite direction to the curvature error resulting from the 1Iayer and Stowe assumption. The approximation of a circular meniscus profile is seen to yield progressively less accurate results as the volume increases. This is to be expected since the nodoid is an anticlastic surface (the principal curvatures are opposite in sign). The principal radius of curvature which is negative in sign is represented by a normal drawn from the axis of revolution to the nodoid surface. This radius of curvature clearly will vary over the profile curve. Hence, a corresponding variation in the magnitude of the positive curvature mill occur. As the voIume increases, the mean curvature decreases, with the average negative curvature decreasing in magnitude less rapidly than the average positive curvature. It follows that the variation of the positive curvature over the profile must increase. I n the limit
3677
Table I : Approximate and Exact Nodoid Parameters +, deg
7 -
JR/2-
Exact
---lO~Y/R~Ref 9
Exact
Ref 9
0 0080 0.1121 0.5021 1.4128 3.0897 5.7739 9.699 13,091 23.177 31.193 43.979
0.01 0.11 0.51 1.44 3.16 5.95
0.0084 0.1231 0.s5770 1.6953 3.8647 7 ,313 13.107 21.138 31 546
0.01 0.12 0.58 1.70 3.89
0.0086 0.1311 0.6328 1.9129 4.482 8.946 12.354
0.01 0.132 0.635 1.916 4.486 8.960
e=o 5 10 15 20
25 30
35 40 45 50 55.64
5 10 15 20 25 30 35 40 44.77
126.229 29.983 12.485 6.467 3.729 2.267 1.3995 0.8460 0.4729 0.2108
126.125 29.89 12.40 6.395 3.665 2.21 ...
...
... ... ...
0.0000
e = 20" 114.15 25,925 10.22.5 4.95 2.61 ...
114.184 25.949 10.248 4.968 2.624 1.4050 0.7034 0.2704 0.0000
... ...
...
e 5 10 15 20 25 30 32.68
87.631 18.429 6.543 2.700 1 ,0688 0,2638 0.0000
... ... .
I
.
... ...
... ... ...
...
= 40""
87.62 18.423 6.5385 2.697 1.0665 0.263 ...
...
' Except for $ = So, values in columns 3 and 3 are t'aken from Table 111of ref 9.
as the mean curvature vanishes, the configuration of the meniscus is such that the profile is a catenary and the surface is a catenoid. The confined fluid volumes for this limiting case are also given in Table I. I n some physical situations involving pendular ring fluid, knowledge of the fluid interfacial area is required. Since the same mathematical functions (incomplete elliptic integrals) appear in the exact expression for area as in the expressions for curvature and volume,16 it is possible to obtain a relationship for the area in terms (11) S. Kruyer, Trans. Faraday Soc., 54, 1758 (1958). (12) R. P. Iczkowski, I n d . Eng. Chem. Fundamentals, 5 , 516 (1966). (13) R. A. Fisher, J . Agri. Sci., 16, 492 (1926). (14) L. V. Radushkevich, Iza. Akad. X a u k S S S R Otd. Khim. N a u k , 69, 1008 (1952). (15) E. C. Sewell and E. W. Watson, RILLEM BziZZ., New Series, No. 29, 125 (1965). (16) J. C. Melrose, A.I.Ch.E. J . , 12, 986 (1966).
Volume 71. hTumbeT 11
October 1967
3678
NOTES
Denoting the area by a, this ex-
of these parameters. pression is
+ $)I + [sin (6 + IC.) ](sin +) (1 + cos Jf(3V/2) + rR2(1-
Metal Complexes of Methyliminodiacetic Acid
D = 2rR2 (sin2 +){ [cos (6
$)-I}
and Hydroxyethyliminodiacetic Acid1*
(1)
COS #)2)
This simple and useful result has not been explicitly pointed out in previous treatments of the nodoid problem. I n Table 11,numerical values for the reduced area, Q/R2,are given. These values were obtained by means of eq 1, using the corresponding reduced values of J and V from Table I. Table I1 : Nodoid Solutions for Meniscus Area $3
O l-DR /’
-
deg
e
5
0.0608 0.4515 1.4175 3.1309 5.7087 9.2252
10 15 20 25 30 32.68 35 40 44.77 45 50 55.64
00
... 13.723 19.221
...
e
= 200
0.0513 0.3887 1,2442 2.8013 5.2050 8.5693
...
e
400
0.0458 0.3527 1.1486 2.6306 4.9710 8.3220 10.579
12.984 18.519 24.895
25.721 33.209 42.819
In the limit of vanishing curvature and for the special case of zero contact angle, the meniscus area obeys an elementary relationship involving that portion of the total area of the two spheres which is in contact with the pendular ring fluid. It follows from eq 1 that the area of the meniscus, which is now a catenoid tangent to the spheres, is simply the harmonic mean of the contacted and noncontacted areas of the two spheres. Since this theorem holds for the catenoid (a minimal surface in mathematical terminology), it will necessarily fail for the surface of a torus. I n summary, it is concluded that the approximation which Mayer and Stowe adopt gives results having an accuracy which is entirely adequate for many purposes. However, it should be emphasized that the method they employ is not exact and may yield errors of unacceptable magnitude for certain problems involving capillaryheld fluids.
Acknowledgment. We are indebted to Dr. S.-T. Hwang for assistance in obtaining the results given in Table 11. Appreciation is also expressed to the Mobil Oil Corp. for permitting publication of this work. The J O U Tof~Physical Chemistry
Proton Magnetic Resonance Studies on Some
by G. H. Nancollas and A. C. Park‘b Chemistry Department, State University of Xew York at Buffalo, Buffalo, New York 14314 (Received M a y 36,1967)
In recent years, the metal complexing ability of ethylenediaminetetraacetic acid (EDTA) and its derivatives has been the subject of a large number of studies. The strong complexes formed with the normally weakly associating alkaline earth metal ions are of particular interest insofar as their structure in solution is concerned. Hoard and his c o - ~ o r k e r s ~ - ~ have established the crystal structures of a number of heavy metal ion complexes of EDTA, but the validity of the assumption that the aqueous species have geometries which are not essentially different from those in the crystalline state has not yet been established. It is therefore desirable to use as many physical methods as possible in order to determine the structures of the species in solution and the nature and strengths of the bonds formed. Thermodynamic data, while providing valuable information about the overall energy changes taking place in a reaction, can give no insight into the specific interactions involved within a single species. On the other hand, spectral changes, brought about by variations in electron distributions within a ligand due to interaction with different metal ions, can afford information of this nature. So far, infrared and nuclear magnetic resonance (nmr) spectroscopy have proved to be the techniques most useful in the study of complexes, and many workers5a6 have been involved in the interpretation of data obtained from these sources. I n the present work, proton magnetic resonance studies have been carried out on the alkaline earth metal ion complexes of N-methyliminodiacetic acid (MIDA) and hydroxyethyliminodiacetic acid (HEIDA) which represent simple analogs of the analytically im-
(1) (a) Supported in part by N.S.F. Grant No. GP-6042; (b) Postdoctoral Fellow, Chemistry Department, Cornel1 University, Ithaca, N. Y. (2) H. A. W-eakliem and J. L. Hoard, J. Am. Chem. SOC.,81, 549 (1959). (3) G. S. Smith and J. L. Hoard, ibid., 81, 556 (1959). (4) J. L. Hoard, M. Lind, and J. V. Silverton, ibid., 83, 2770 (1961). (5) D. T. Sawyer and J. E. Tackett, ibid., 85, 2390 (1963). (6) R . J. Day and C. N. Reilley, A n a l . Chem., 36, 1073 (1964).
