MOLECULAR ORBITALTHEORY OF ELECTRON DONOR-ACCEPTOR COMPLEXES
1 - p s v . - ppvp =
(The estimation is better for n
~ , g nL,g 6 4a -up,3[g(3n"/"
Appendix 111 The Behavior of L,g - nL,g and of Lg - nL,g for Large n. From the definitions (12) and (13), we have 48 Leg - nL,g = g-[(uCp3 3
+ up,)3 - ucp3]-
> 90 and LY < 3.)
+ 3n1l8+ 1) -
g is finite. Thus, for sufficiently large n, we get
48 - 3 From Appendix I1 for
3 and n = 90, we have
(This behavior is already achieved for n of the order of 200, and for smaller n if CY < 3). I n the same manner, for s a c i e n t l y large n we have
Log - nLpg + -nL,g
Molecular Orbital Theory of Electron DonorAcceptor Complexes. I. A Simple Semiempirical Treatment
by R. L. Flurry, Jr. Department of Chemistry, Louisiana State University in New Orleans, New Orleans, Louisiana YO122 (Received February 26, 1966)
A semiempirical linear combination of molecular orbitals description of electron donolacceptor complexes is presented. Using this, an empirical estimate of the interaction integral between the systems participating in EDA complex formation is made. This empirical PDA is used to predict stabilization energies and excitation energies for several complexes. The agreement with experiment is satisfactory. A brief comparison with the conventional treatments of such complexes is made.
I. Introduction Much literature on electron donor-acceptor (EDA) comdexes has arisen in the Dast few years. both from experimental and theoreticel points "of view. Most Of the papers with have been based on a valence bond (VB)t.ype of formalism.' Dewar's molecular orbital (MO) -treatment of treatment* comp1exes2 is a perturbation Murrel13 has presented a very sophisticated MO per-
turbation treatment of EDA interactions. The delocalisation method of Fukui and co-workers4 is an LCAO (1) R. s. Mulliken, J. Am. C h . SOC.,72, 600 (1950); J . Phys. Chem., 56,801 (1952),and subsequent papers. (2) M. J. S. Dewar and A. R. Lepley, J. Am. Chem. SOC.,83, 4560 (1961). (3) J. N. Murrell, ibid., 81,5037 (1959); J. N. Murrell and J. Tanaka, Mol. Phys., 7, 363 (1963-1964). (4) K. Fukui, A. Imamura, T. Yonezawa, and C. Nagata, BuU. chm. soc.Japan, 34, 1076 (1961); 35,33 (1962).
Volume 69,Number 6 June 1966
R. L. FLURRY, JR.
method which employs the atomic orbitals of both the donor and acceptor and then utilizes perturbation theory to estimate their interactions. The present research is part of an attempt to develop a simple MO theory of EDA complexes, utilizing orbitals of the entire complex as the basis, and to apply it not only to a calculation of charge-transfer (CT) spectra but also to predictions of stability and of orientation in EDA complexes. This paper presents a simple, semiempirical, oneelectron treatment of EDA complexes and applies it to benzene and methyl-substituted benzenes as donors and to chloranil, tetracyanoethylene, l13,5-trinitrobenzene, p-benzoquinone, and iodine as the acceptors. Nagakura and Tanaka have used a similar method to discuss the spectral effects of intramolecular electron migration in substituted aromatic systems.6 The present work differs from this in several important respects and, in addition, appears to be the f i s t attempt to apply such a treatment to intermolecular complexes and to calculating stabilities.
11. Method Considerations involving (HOMO)D and (LEMO)A. The ground (N) state wave function of the complex is expressed as a linear combination of molecular orbitals (LCMO) of the donor and acceptor where 4~ is the wave function of the donor and +A that of the 4 N = @D
acceptor, and a and 6 are mixing parameters (with the normalizing restriction that a2 b2 = 1, assuming formal neglect of overlap). The electronic energy of the ground state may be expressed as in eq. 2 (using
(2) Dirac notation). For the present, the form of the Hamiltonian will remain undefined. In addition to the energy arising from the interaction of the donor and acceptor orbitals, there will be an additional potential energy interaction owing to the attraction of the partial charges that result from a partial transfer of charge in the ground state of the complex. This potential may actually be more important in the stabilization of the complex than the direct orbital interaction.6 In the ground state, the amount of charge transferred will be equal to b2 in this approximation. Let us d e h e -Ves as the potential resulting if a complete charge transfer occurs between the components of the complex a t their equilibrium separation. The total energy of the ground state of the complex will then be The Journal of Physical Chemistry
EN = a2(h]HopIh) b 2 ( 4 ~ l H o p J ~ ~ ) 2ab($DlHopI4A)
or, making the defhitions
(+A /HOP l 4 A )
~ B D A= ( ~ D I H O ~ I ~ A )
the total energy of the ground state may be expressed as
EN = a2D 4- b2A
+ 2abjh~- b2Ve,
The excited state wave function is expressed as in eq. 6. In this approximation b and a have the same @E
values as in 4 ~ .(If overlap were included, this would not be the case.) The total energy of the excited state is thus expressed as eq. 7 (a2is the amount of charge transferred in this state).
EE = b2D
+ a2A - 2abpDA - a2Ves
The energy of the charge-transfer (E + N) transition is simply EEminus EN.
UCT = (b2 - a 2 ) ( D- A
(8) If the degree of charge transfer in the N state of the complex and the value of &A are sufficiently small, eq. 8 can be approximated by equation 8a, a form which is frequently used for interpreting charge-transfer spectra. Ves)
U C T ~ I - E A + C (84 The relationships among D, A , WN, EN, WE, and EEare shown schematically in Figure 1. From a consideration of Figure 1, it is obvious that a resonance stabilization of the N state of the complex will be accompanied by a corresponding destabilization of the E state. If this were the only effect operative, the E + N transition would be of greater energy than the difference in the ionization potential of the donor and the electron affinity of the acceptor. This, of course, is not the case experimentally; hence, the electrostatic terms must be considered. If, on the other hand, there were no transfer of charge in the N state of the complex, as would be the case if there were no resonance stabilization, the slope of a plot of the chargetransfer transition vs. the ionization potential of the donor for a given acceptor would be unity. This, (6) 8.Nagakura and J. Tanaka, J. C h .Phys., 22,236 (1964). (6) J. N. Mumll, Quart. Rm. (London), 191 (1961).
MOLECULAR ORBITAL THEORY OF ELECTRON DONOR-ACCEPTOR COMPLEXES
ene (TCNE) with the methyl-substituted benzenes1° us. ionization potentials calculated by an SCMO (Figure 2) gives values of 0.89 for a,
Figure 1. The relationships between the highest occupied orbital of the donor (D), the lowest unoccupied orbital of the acceptor (A), the ground and excited state electronic energies ( W Nand W E )and the total ground and excited state energies (ENand EE)for EDA complexes.
again, is not true experimentally. It is therefore evident that both a resonance effect and an electrostatic effect must be operative. This treatment, of necessity, neglects solvation energies. If a suitable method for estimating the relative solvation energies of the N and E states could be found, these could be included in a manner similar to the electrostatic effect.
0.45 for b, and -0.63 e.v. for PDA. Table I lists the values obtained for all of the acceptors used. Just as in the Huckel treatment of 7-electron systems, the relative values of the calculated properties within a series will be independent of the absolute value of P D A ; however, these values of PDA are probably of the right magnitude. Brown has estimated that “bonds” between donors and acceptors in complexes usually have energies of 0.1-0.6P.1a P is generally accepted to have a value near -3 e.v. It is of interest to consider the significance of the slope of the plot obtained from eq. 8. The smaller the slope is, the greater is the amount of charge transferred in the ground state of the complex. For all of the acceptors, except p-quinone and trinitrobenzene, the apparent degree of charge transfer parallels the electron affinity of the acceptor. The data for pquinone are limited to those compounds which would
111. Calculations For the acceptors tetracyanoethylene, 1,3,5-trinitrobenzene, chloranil, and p-quinone, the distance between the donor and acceptor is assumed to be near that of the chloranil-hexamethylbenzene complex (3.65 B.).7 If the charges which arise are assumed to be evenly distributed around the donor and the acceptor, a value of 3.95 e.v. can be estimated for V e 8 . For iodine, a value of 2.91 e.v. can be estimated for V,, if the complex is assumed to have axial symmetry with a separation between the donor and the iodine molecule of 3.36 A., the experimental value for the bromine complex.* If the assumption is made that, for a given series of similar donors with the same acceptor, the degree of charge transfer is constant and PDA is constant, a plot of A E C T us. D (which, according Koopmaans’ theory, is the negative of the ionization potential of the donor) should give a straight line, the slope of which can be used with the normalizing condition to estimate a and b.9 If A (the negative of the electron affinity) and Ve8are known, the intercept of the line will give an estimate of PDA. Plotting the chargetransfer bands for the complexes of tetracyanoethyl-
8.5 I.P. (e.v.1
Figure 2. A graph of the CT transitionlo us. ionization potentials11 for TCNE-methylbenzene complexes.
(7) T. T. Harding and S. C. Wallwork, Acta Cryst., 6, 791 (1953); 8, 787 (1955). (8) 0.Hassel, Mol. Phys., 1, 241 (1958). (9) a and b will not, in fact, be constant for the series but can later be derived, using a simple interaction matrix (eq. 9)between the donor and acceptor, once a value for @DA has been determined. (10) A. R. Lepley, J. Am. Chem. Soc., 86, 2545 (1964). (11) R.L.Flurry, Jr., and P. G. Lykos, ibid., 85,1033 (1963); R.L. Flurry, Jr., and J. G. Jones, Abstracts, 144th National Meeting of the American Chemical Society, Los Angeles, Calif., April 1963. (12) The average deviation of the calculated and available experimental values for the photoionization potentials is 0.13%. The maximum deviation (pseudocumene) is 0.85%. (13) R. D.Brown, J. Chem. Soc., 2232 (1959).
Volume 69, Number 6 June 1966
R. L. FLURRY, JR.
Table I: Parameter Values for the Various Acceptors Electron affinity, e.v.
1,3,5-Trinitrobenzene p-Quinone Chloranil Tetracyanoethylene Iodine See ref. 14.
0.70" l.llb 1.35" 1.60" 2.0OaSd
3.95 3.95 3.95 3.95 2.91
Slope (es. 8)
0.71 0.34' 0.77 0.60 0.55
-0.81 -1.40' -0.51
only 0.15 e.v. in the calculated spread of the transition energies. The magnitude of the benzene-TCNE transition varied by 1.59 e.v. from 4.67 to 3.08 e.v. The calculated log K/K" for hexamethylbenzene varied by a factor of 16.5 from 0.09 to 1.49. The estimated values of PDA can be used with the variational principle to set up a secular determinant (eq. 9) to determine the electronic energies of the
* Midpoint of range reported by M. E. Peover,
Trans. Faraday Soc., 58, 1656 (1962). These values are probably not very valid owing to the small amount of data available for evaluating them. Value for the axis of the iodine molecule perpendicular to the plane of the acceptor.
be expected to form the weakest complexes in the series; consequently, no definitive conclusions can be drawn from them. On the other hand, the data for trinitrobenzene, while encompassing only a few members, present examples a t both extremes of the series; consequently, the observed trends should be valid. Briegleb14 noted this anomaly in the relationship between the amount of charge transferred in trinitrobenzene complexes and its reported electron affinity. The large apparent degree of charge transfer in the ground state of the trinitrobenzene complexes makes it tempting to choose a value of about 1.42 e.v. for the electron affinity of trinitrobenzene, a value slightly larger than that of chloranil. It is interesting but probably not significant that this is near the experimental value of 1.6 e.v. for nitrogen dioxide.15 Using 1.42 e.v. as the electron affinity, the value obtained for PDA is -1.16 e.v. The uncorrected calculated values for the charge-transfer transition and for the relative stabilities are improved by this change. With the experimental value of the electron a h i t y , the calculated charge-transfer energies range from 4.81 e.v. for benzene to 3.32 e.v. for hexamethylbenzene as compared with an experimental range of 4.36 to 3.14 e.v. With an electron affinity of 1.42 e.v., the calculated range is 4.37 to 2.98 e.v. The calculated log K/Ko values, using the experimental electron affinity, range from 0 to 0.63. Using the estimated electron affinity, the range is from 0 to 1.56. The experimental range is 0 to 1.49. Variation Of the "' term has very little effect On the relative values of the calculated charge-transfer transition'* 'I does, have a effect on both the absolute value of the calculated transition and the spread in the calculated equilibrium constants. varying "' for the TCNE For over the range of 3.0 to 5.0 e.v. gave a variation of The Journal of Physical Chemistry
resulting complex orbitals and the coefficients a and b for eq. 1, using values of the ionization potential and electron afKnity for -D and - A . The lowest root of this determinant and the calculated value of b can be used with the V , to determine the stabilization energy of the complex. The complex will be more stable than the isolated donor and acceptor if it has a total energy lower than the sum of the total energies of the isolated donor and acceptor; i.e.
EN < ED
or again, considering only the (HOM0)D and the (LEMO)A,there is a net stabilization if
+ b2A + 2abPDA - b2Ti,, < D
The stabilization energy is then Mstab
D - EN
If the entropies of formation of the coniplexes are constant, the relative stabilities of the complexes can be obtained byll
Owing to the crudeness of the method, the uncertainty of the parameter values, and the large solvent effects,l*--l* the best that can be expected is a linear relationship between the calculated and observed values. Numerical agreement can be obtained, however, if a least-squares fit is made of these values. A similar fit can be made for the energies of the chargetransfer transitions. Tables 11-VI list the uncorrected and the corrected calculated values, the experimental values of the charge-transfer band, and the (14) G. Briegleb, Angew. Chem. Intern. Ed. Engl., 3, 617 (1964). o. Pritohard, Chem. Rev,, 52, 529 (1953). (15) (16) G. Briegleb and J. Czekalla, 2. Elektrochem., 63, 6 (1959); Angew. Chem., 72,401 (1960).
(17) G. Briegleb, " Elektronen-Donator-Acceptor-Komplexe," Springer-Verlag, Berlin, 1961, p. 114 ff. (18) C. C. Thompson, Jr., and P. A. D. dehlaine, J . Am. Chem. SOC., 85, 3096 (1963).
MOLECULAR ORBITAL THEORY OF ELECTRON DONOR-ACCEPTOR COMPLEXES
Table IV: Spectra and Relative Stabilities of Complexes with Chloranil
Table I1 : Spectra and Relative Stabilities of Complexes with l13,5-Trinitrobenzene T---AEcT,
Benzene Toluene &Xylene m-Xylene p-Xylene Hemimellitene Pseudocumene Mesitylene Prehnitene Isodurene Durene Pentamethylbenzene Hexamethylbenzene
4.81 4.40 4.16 4.15 4.05 4.04 3.81 4.03 3.67 3.67 3.53
4.38 4.08 3.91 3.90 3.82 3.82 3.65 3.81 3.54 3.54 3.44
4.36* (0) -0.02 (Wb 4.08 0.14 0.30 0.25 0.23 0.51 . . . ... 3.90 0.24 0.51 0.58 0.28 0.61 0.28 0.62 . . . ... 0.38 0.84 . . . ... 0.29 0.63 0.45 0.99 ... 0.45 0.99 . . . 3.61 0.51 1.14 1.01
3.32 3.28 3.14
" Least-squares fit with observed data. G. Briegleb and J. Czekalla, 2.Elektrochem., 58, 249 (1854); 59, 184 (1955).
Table III : Spectra and Relative Stabilities of Complexes with p-Quinone
Benzene Toluene o Xy1ene m-Xy 1ene p-Xylene Hemimellitene Pseudocumene Mesitylene Prehnitene Isodurene Durene Pentamethylbenzene Hexamethylbenzene
YAECT, e . v . 7 Calcd. Cor." Obsd.
4.04 3.63 3.38 3.37 3.26 3.25 3.02 3.24 2.87 2.87 2.74
3.69 3.37 3.17 3.16 3.08 3.07 2.89 3.06 2.77 2.77 2.66
3.65' 3.40 3.22 3.18 3.03 2.95 3.03 2.79 2.76 2.64
0.07 0.12 0.12 0.14 0.14 0.19 0.14 0.23 0.23 0.26
-0.08 (0)' 0.23 0.22 0.43 0.54 0.44 0.45 0.54 0.47 0.55 . . . 0.77 0.53 0.56 0.54 0.92 0.95 0.92 0.92 1.07 1.00
2.63 2.58 2.58
2.51 2.49 2.46
Least-squares fit with observed data.
Benzene Toluene +Xylene rn-Xy lene p-Xylene Hemimellitene Pseudocumene Mesitylene Prehnitene Isodurene Durene Pentamethylbenzene Hexamethylbenzene
4.87 4.49 4.27 4.26 4.16 4.15 3.95 4.15 3.82 3.82 3.70 3.61 3.51
4.06 3.92 3.83 3.83 3.79 3.79 3.71 3.79 3.66 3.66 3.62 3.58 3.54
4.06b 3.93 3.85 3.81 3.87c
... ... ... ... ...
(0) 0.44 0.73 0.74 0.87 0.89 1.19 0.90 1.40 1.40 1.60 1.76 1.95
Least-squares fit with observed data. * A. Kuboyama and S. Nagakura, J . A m . Chem. Soc., 77, 2644 (1955). Omitted in least-squares fit.
relative stabilities for the complexes of the methylbenzenes and the various acceptors. Benzene is adapted as the standard for the stabilities in each case. The numerical agreement is satisfactory.
IV. Comparison with Dative Bond and Perturbation Theory Treatments In an addendum to their 1962 review article,l9 Mulliken and Person compared the VB type of treatment with Dewar's MO treatment. In this section
'See ref. 3.
Table V : Spectra and Relative Stabilities of Complexes with Tetracyanoethylene
Benzene Toluene o-Xylene m-Xylene p-Xylene Hemimellitene Pseudocumene Mesitylene Prehnitene Isodurene Durene Pentamet hylbenzene Hexamethylbenzene
3.85 3.44 3.20 3.19 3.08 3.07 2.84 3.06 2.70 2.70 2.56
3.22 2.96 2.82 2.81 2.74 2.74 2.60 2.73 2.51 2.51 2.42
Log @/KO) oalcd.
3.19* 2.99 2.84 2.82 2.69 2.69 2.61 2.67
-0.17 0.29 0.60 0.61 0.76 0.77 1.11 0.79 1.34 1.34 1.56
0.12 0.19 0.20 0.23 0.23 0.32 0.24 0.38 0.38 0.43
2.46 2.36 2.38
2.34 2.29 2.27
0.28 0.54 0.48 0.58
... ... 0.94
a Least-squares fit with observed data. * See ref. 10. R. E. Merrifield and W. D. Phillips, J . Am. Chem. SOC., 80, 2778
the LCMO treatment will be compared with the VB treatment and with a simple perturbation theory treatment. To the present degree of approximation, the LCMO treatment of EDA complexes looks very similar to the traditional VB type of treatment,' and, in fact, the numerical treatment presented here can be carried (19) R. S. Mulliken and W. B. Person, Ann. Rev. Phys. Chem., 13, 107 (1962).
Volume 69, Number 6 June 1966
R. L. FLURRY, JR.
Table VI : Spectra and Relative Stabilities of Complexes vith Iodine -AEcT,
Benzene Toluene +Xylene m-Xylene p-Xy lene Hemimellitene Pseudocumene Mesitylene Prehnitene Isodurene Durene Pentamethylbenzene Hexamethylbenzene
4.82 4.44 4.22 4.21 4.11 4.09 3.89 4.09 3.75 3.75 3.63
4.31 4.07 3.93 3.93 3.87 3.86 3.73 3.85 3.65 3.65 3.57
4.24b 4.10 3.92 3.90 4.08
0.35 0.59 0.60 0.70 0.71 0.97 0.72 1.14 1.14 1.31
3.54 3.51 3.47 3.43 3.45 3.31
... ... 3.73
-0.02 0.19 0.32 0.33 0.39 0.40 0.55 0.41
(0)b 0.04 0.27 0.33 0.33
... ... 0.75
a Least-squares fit with observed data. L. J. Andrews and R. M. Keefer, J. Am. Chem. SOC.,74,4500 (1952).
out in the same manner in the VB formalism. The difference lies in the formal interpretation of the empirically determined parameters. In the present crude form, the prime advantage of the LCMO method is in the more correct conceptual interpretation of these parameters. In the simplest VI3 treatment, the ground-state wave function is expressed as in eq. 14, where 4o(DA) @JN =
represents a total wave function for the ground state of the donor and acceptor with no interaction between them. qh(D+-A-) is the dative structure in which a total charge has been transferred from the donor to the acceptor and which involves a bond between them. If we use the individual molecular orbitals of the donor and acceptor as the basis set for the VB treatment of the complex, we can draw an analogy with the VB treatment of heteronuclear diatomic molecules.2O The &(DA) corresponds to what, in the Heitler-London treatment, is an ionic state with both electrons on the same center (eq. 15). Equation 14 is then actually a +o(DA) =
+ (#JD/V(A)[email protected]
) A + (~A]V(D+)/#JA)
EN’ = D EE’ = ECT’= A - D
+ ( ~ A ] V ( D + ) /-~ A )
(4DI V(A) I4D)
4 ~ and , +A are as previously defined, and V(A) and V(D+) are the perturbing potentials. If either of the integrals ( 4 ~ l V ( D + ) l 4 ~or) ( h l V < A > / 4 ~can ) be assumed to be proportional to the ionization potential of the donor, the perturbation theory treatment can predict a non-unity slope for a plot of the charge-transfer transition energy vs. the ionization potential of the donor. The greatest difficulty in using the perturbation theory explicitly in semiempirical calculations is the evaluation of the perturbation terms. In Murrell’s treatment, the first-order approximations to the wwe function for the N and E states are as expressed in 4 ~ =‘ 4~ 4E’ =
configuration interaction equation which allows for the mixing of the pseudo-ionic +o(DA) configuration and the covalent @1(D+-A-) configuration. The inclusion of higher order terms in eq. 1 (allowance for “reverse dative structures”2 etc) amounts to the inclusion of additional configurations in the VB treatment. The Journal of Physical Chemistry
The simple VB formalism is basically a two-electron theory while the simplest MO formalism is basically a one-electron theory; hence, the interpretation of the various parameters used here, particularly the use of the electron alKnity, is formally correct only in the MO scheme. In a simplification of the perturbation theory treatment of EDA complexes,a the highest occupied orbital of the donor may be taken as the unperturbed groundstate orbital of the complex, and the lowest empty orbital of the acceptor as the unperturbed excitedstate orbital. The perturbation of the ground state may be considered to be the potential of the neutral acceptor while the perturbation of the excited state will be the potential of the donor bearing a single positive charge. Since the first-order perturbation energy for a given state is just the perturbation function averaged over the corresponding unperturbed state of the system,21the energies of the N state, the E state, and the charge-transfer excitation will be, to first order, as expressed in eq. 16-18, respectively. D, A,
C ~ A
4 A -k d h
The coefficients c and d are obtained in the standard manner.21 Except for normalization, these functions (20) J. C. Slater, “Quantum Theory of Molecules and Solids,” McGraw-Hill Book Co., Inc., New York, N. Y., 1963,p. 142. (21) L. Pauline and E. B. Wilson, Jr., “Introduction to Quantum Mechanics,” McGraw-Hill Book Co., Ino., New York, N. Y., 1935, p. 159.
MEASUREMENT OF SURFACE TENSION BY
are very similar to those used in the present work; consequently, a second-order perturbation theory treatment of EDA complexes should give results similar to the LCMO treatment. Charge-transfer spectra have frequently been used to estimate ionization,potentials and electron affinities either by use of eq. 8a or simply by fitting known values of ionization potentials or electron affinities to a linear plot of observed CT spectra with a given acceptor or donor, and estimating values for the unknown substances from the plot. While this method has been useful, it can be seen from eq. 8 that it must be applied with caution. If, in the series chosen, either V,, or PDA varies significantly, the validity of the esthated values will be doubtful. Values of ionization potentials obtained in this manner are
more likely to be reliable than are electron afiities, owing to the fact that a series of similar compounds for which ionization potentials are known is more often available than is the case with electron affnities. The results of parameter variation in this work imply that it might be possible to obtain reliable estimates of the electron affinities of the acceptors by using these and the V,, values as variation parameters and minimizing the error in both the transition energies and the relative stabilities with respect to them.
Acknowledgment. The author wishes to express his appreciation to the Cancer Association of Greater New Orleans for financial aid which helped make this research possible and to the L.S.U.N.O. Computing Center for generous use of the IBM 1620 computer.
The Measurement of Surface Tension by the Pendant Drop Technique
by Clyde E. Stauffer The Procter and Gamble Company, Miami Valley Laboratories, Cincinnati, Ohio 46,299 (Received December 18, IQ64)
The Laplace equation describing the shape of a liquid drop being acted upon solely by gravitational and surface energy forces has been solved by a technique involving reiterated approximations, employing a high-speed digital computer for the purpose. This method of solution has been used to extend the table of 1/H as a function of S , quantities which are basic to the Andreas-Hauser-Tucker method of measuring surface tension by the pendant drop method, to lower values of S than are currently available. This extended table will make the pendant drop technique more widely usable. The statistical error inherent in the pendant drop technique has been evaluated and found to be greatly dependent upon the shape of the drop as expressed in the ratio S. More nearly spherical drops are inherently subject to greater imprecision in the measurement of their surface tension.
The shape of a liquid drop being acted upon solely by gravitational and surface energy forces is given by eq. 1.1,2
1 sin p+,=2+PZ C#I
I n this equation, P is the radius of curvature a t the
point ( X , Z ) , C#I is the angle made by the tangent at the point ( X , Z ) and the X coordinate axis, and p is the shape parameter, given by (1) P. S. Laplace, “Mecanique Celeste,” supplement to the 10th
book, Duprat, Paris, 1806. (2) F. Bashforth and J. C. Adams, “An Attempt to Test the Theories of Capillary Action,” University Press, Cambridge, England, 1883.
Volume 69, Number 6 June 1966