MEASUREMENT OF SURFACE TENSION BY
THE
PENDANT DROPTECHNIQUE
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
1933
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.
1.1,2
point ( X , Z ) , 4 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
I n this equation, P is the radius of curvature a t the
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.
The shape of a liquid drop being acted upon solely by gravitational and surface energy forces is given by eq.
(1) P. S. Laplace, “Mecanique Celeste,” supplement to the 10th
Volume 69, Number 6 June 1966
1934
where g is the gravitational constant, u is the effective density of the liquid drop (ie., the density of the liquid in the drop minus the density of the surrounding medium), y is the surface or interfacial tension, and b is the radius of curvature a t the origin. The unit of length used in eq. 1 is b, so that X,2, and p are nondimensional. The sign for p follows the convention of Bashforth and Adams,2 being negative for a pendant drop and positive for a sessile drop. Equation 1may be recast in the form of a differential equation by setting 1/P equal to d4/ds, where s is the linear distance along the drop profile. The resulting equation is not directly soluble a t the present time. However, Bashforth and Adams, using numerical integration techniques, solved the differential equation. Andreas, Hauser, and Tuckera suggested the use of the pendant drop technique for measurement of surface tension, but their method differed from earlier experimentalists in that they made measurements of the drop at two selected planes, measuring De, the diameter a t the equatorial plane, and D,,the diameter at the plane at a distance Defrom the tip of the drop. This obviated the necessity for determining such quantities as the position of the point of inflexion of the drop profile, which is very difficult to obtain with any degree of accuracy. On the other hand, the method of Andreas, et al., made the solution of the fundamental equation for several values of p imperative, since the calculation of surface tension from the experimental data obtained required the use of a correction factor 1/H, which is a function of the ratio D,/D,, or S. Andreas, et al., were rather dubious of the ma thematical methods available for solution of eq. 1, and preferred to construct the table of 1/H us. S by empirical evaluation of 1/H using liquids of known surface tension. Fordham4 and Xiederhauser and Bartells independently solved the fundamental equation in order to obtain a theoretically sound table of 1/H vs. S. They both used numerical integration techniques, similar to or identical with the method of Bashforth and Adams. Niederhauser and Bartells checked the consistency of their table by measuring the surface tension of water, using drops having S values from 0.68 to 0.99. They obtained the same value for surface tension with any size drops>and also showed that there was a systematic error in the table of Andreas, et al., amounting to as much as 1.47,. Their table, and a short discussion of the pendant drop technique, is also published in Adamson’s book on surface chemistry.6 Staicopolus7solved the fundamental equation by use of first-order approximations for the derivatives, with a The Journal of Physical ChEmiStTy
CLYDEE. STAUFFER
very small interval of integration. High-speed digital computers made such a procedure feasible. He used positive values for p in order to apply the results to sessile drops, and expressed the resulting factors as polynomial coeacients rather than as a table of factors. The table of 1/H vs. S published by Fordham and Niederhauser and Bartell covers a range of S values from 0.66 to 1.00. During the course of work in this laboratory, values for S less than 0.66 were often obtained. It is always a nuisance to change the size of the pendant drop in order to bring S above 0.66, and in some instances it was preferable not to disturb the droplet. Thus, a definite need for an extended table of 1/H vs. S was felt, and the present work was undertaken to fulfill that need.
ExperimentaI Method. The method used was one of reiterated approximations, using a digital computer for all calculations. The values of X , 2, 4, and d4 are given a t the origin, and the values for the first three variables are estimated a t the point which is a distance ds along the drop profYe. A n estimated d4 is calculated (from eq. l),an adjusted value of 4 is obtained, new values of X and 2 are calculated at the point, a second estimate of d4 is calculated, and finally a second adjustment is made of 4, The calculation k, then advanced forward to the next point. The computer program was written so that any desired value of ds and b could be read into the computer at the beginning of the calculations, and with the option of beginning a t the origin or a t any point on the profile by supplying the appropriate values for X , 2,4 and d+. I n practice, ds was usually 0.001, while p was varied from -0.0125 to -0.2375 in increments of 0.0125, Also, the calculation was carried out a t other values of p for comparison with the results of Niederhauser and Bartell. The details of the calculations and the Fortran program are available upon request, In order to obtain X,,the radius of the drop a t the equatorial plane, X was plotted vs. s around the point where 4 had the value ~ / 2 . A smooth curve w w drawn through the points, and X,, the maximum value which X obtained, was read directly from the graph. (3) J. M. Andreas, E. A. Houser, and W. B. Tucker, J. Phz/s. Chem., 42, 1001 (1938). (4) S. Fordham, Proc. Roy. SOC.(London), A194, 1 (1948). (5) D. 0. Niederhauser and F. E. Bartell, “Report of ProgressFundamental Research on Occurrence and Recovery of Petroleum,
1948-1949,” American Petroleum Institute, Baltimore, Md., 1950, pp. 114-146. (6) A. W.Adamson, “Physical Chemistry of Surfaces,” Interscience Publishers, Inc., New York, N. Y., 1960. (7) D.N. Staicopolus, J. Colloid Sci., 17, 439 (1962).
MEASUREMENT OF SURFACE TENSION BY
THE
PENDANT DROPTECHNIQUE
1935
Table I : Tabulation of Calculated Data B
X.
X,
S
1/H
-0.0125 - 0.0250 - 0.0375 - 0.0500 - 0,0625 - 0.0750 - 0.0875 -0.1000 -0.1125 -0.1250 -0.1375 -0.1500 -0.1625 -0.1750 -0.1875 - 0.2000 -0.2125 - 0.2250 - 0,2375
1.00209724 1.00422263 I .00637708 1.00856146 1.01077669 1.01302373 I . 01530354 1.01761731 1.01996606 1.02235101 1.02477338 1.02723466 1.02973586 1.03227886 1.03486508 1.03749622 1.04017405 1.04290046 I . 04567752
0.19759259 0.26332724 0.31117094 0.35027590 0.38404412 0.41416856 0.44163167 0.46706015 0.49088254 0.51340630 0.53486074 0.55542181 0.57522944 0.59439493 0.61301086 0.63115402 0.64888967 0.66627366 0.68335468
0.19717906 0.26221998 0.30919915 0.34730248 0.37994952 0.40884389 0.43497501 0.45897426 0.48127341 0.50218200 0.52193075 0.54069613 0.55861844 0.57580849 0.59235824 0.60834344 0.62382798 0.63886601 0.65350423
19.88093704 9.91607923 6.58244559 4.91547255 3.91516035 3.24923111 2.77166126 2.41418769 2.13607278 1.91350647 1.73133711 1.57946287 1.45089161 1.34062679 1.24500550 1.16128006 1.08734921 1.02157857 0.96267757
X, was calculated by linear interpolation of X in terms of Z a t the point where Z = 2X,, Le., Z = De. The linearity of the relationship between Z and X in this limited region was verified by re-entering the calculations at successively smaller values of ds and obtaining values throughout this region. No discernible deviations from linearity were found in the eighth decimal place. For each value of p, the values X, and X, so obtained were used to calculate the values pertinent for our purposes, according to the equations
1/H
=
S = Xs/Xe -1//3(De)2 = -1/4p(Xe)2
(3)
(4)
A list of the values so obtained is given in Table I. Interpolation was done using Newton’s method of interpolation with divided differences. The logarithm of l/H was used in this interpolation to decrease the extreme contributions of the large differences a t the small value of S. Differences up to and including the fourth one were used in the calculations. The resulting tabulation of 1/H us. S , in increments of 0.001 in S , is presented in Table 11. In any work such as this, the question of accuracy is of prime importance. The calculations were done on an IBM 1620 digital computer with 12-digit precision, or on an IBM 7094 computer with 16-digit precision. Calculations done on both machines a t one particular value of p gave results identical to the eighth decimal place. Results obtained from calculations with p equal to -0.20, -0.30, -0.375, -0.45, or -0.55 cor-
respond exactly with the numerical results published by Niederhauser and Bartell.5 The plot of X V S . s for the determination of X, was exact in the seventh decimal place, and the value of X, could be read to f1 in the eighth place. A difference table of X, us. /3 showed essentially zero thirddifferences. As mentioned above, the linearity of the plot of Z vs. X in the region of Z = 2Xe was checked by recalculating the values of X and 2 a t successively smaller values of ds to a h a 1 ds of 0.000001. The plot of the values so obtained showed no deviation from linearity, although it was of such a scale that differences of 1 2 in the eighth place would have been easily discernible. The difference table of Xs in terms of p did not readily go to zero a t the lower values of p. This probably means only that X, cannot be expressed as a simple polynomial function of p. The differencing, however, did not indicate the presence of mathematical errors in any particular term. The use of no divided differences higher than the fourth in the interpolation step resulted in a maximum error of 1 in the fifth place in the natural logarithm of 1/H, corresponding to less than 1 in the sixth place in the value of 1/H. The value of 1/H a t S = 0.670 agrees exactly with the value calculated by Niederhauser and Bartell,jg6but in the S range 0.660-0.670, the values of 1/H given here are slightly lower than those found by F ~ r d h a m . ~ To check the unbiased nature of Table 11, especially with respect to the table given by Xederhauser and Volume 69, Number 6
June 1965
CLYDEE. STAUFFER
1936
Table 11: Tabulation of I I H us. S S
0
1
2
3
4
5
6
7
8
9
0.30 0.31 0.32 0.33
7.09837 6.53998 6.03997 5.59082
7.03966 6.48748 5.99288 5.54845
6.98161 6.43556 5.94629 5.50651
6.92421 6.38421 5.90019 5.46501
6.86746 6.33341 5.85459 5.42393
6.81135 6.28317 5.80946 5.38327
6.75586 6.23347 5.76481 5.34303
6.70099 6.18431 5.72063 5.30320
6.64672 6.13567 5.67690 5.26377
6.59306 6.08756 5.63364 5.22474
0.34 0.35 0.36 0.37
5.18611 4.82029 4.48870 4.18771
5.14786 4.78564 4.45729 4.15916
5.11000 4.75134 4.42617 4.13087
5.07252 4.71737 4.39536 4.10285
5.03542 4.68374 4.36484 4.07509
4.99868 4.65043 4.33461 4,04759
4.96231 4,61745 4.30467 4.02034
4.92629 4.58479 4,27501 3,99334
4.89061 4.55245 4.24564 3.96660
4.85527 4,52042 4.21654 3.94010
0.38 0.39 0.40 0.41
3.91384 3.66427 3.43572 3.22582
3.88786 3.64051 3.41393 3.20576
3.86212 3.61696 3.39232 3.18587
3.83661 3.59362 3.37089 3,16614
3.81133 3.57047 3.34965 3.14657
3.78627 3.54752 3.32858 3.12717
3.76143 3.52478 3.30769 3.10794
3.73682 3.50223 3,28698 3.08886
3.71242 3.47987 3.26643 3.06994
3.68824 3.45770 3.24606 3.05118
0.42 0.43 0.44 0.45
3.03258 2.85479 2.69110 2.54005
3.01413 2.83781 2.67545 2.52559
2.99583 2.82097 2.65992 2.51124
2.97769 2.80426 2.64452 2.49700
2.95969 2.78769 2.62924 2.48287
2.94184 2.77125 2.61408 2.46885
2.92415 2,75496 2.59904 2.45494
2.90659 2.73880 2.58412 2.44114
2.88918 2.72277 2.56932 2.42743
2.87192 2.70687 2,55463 2.41384
0.46 0.47 0.48 0.49
2.40034 2.27088 2.15074 2.03910
2.38695 2.25846 2.13921 2.02838
2.37366 2.24613 2.12776 2.01773
2.36047 2.23390 2,11640 2.00715
2.34738 2.22176 2,10511 1.99666
2,33439 2,20970 2.09391 1.98623
2.32150 2,19773 2.08279 1,97588
2.30870 2.18586 2.07175 1.96561
2.29600 2.17407 2.06079 1.95540
2.28339 2.16236 2.04991 1.94527
0.50 0.51 0.52 0.53
1.93321 1.83840 1.74808 1.66369
1.92522 1.82909 1.73938 1.65556
1,91530 1.81984 1.73074 1.64748
1.90545 1.81065 1.72216 1.63946
1.89567 1.80153 1.71364 1.63149
1.88596 1.79247 1.70517 1.62357
1.87632 1.78347 1.69676 1.61571
1.86674 1.77453 1.68841 1,60790
1.85723 1.76565 1.68012 1.60014
1.84778 1.75683 I.67188 1.59242
0.54 0.55 0.56 0.57
1.58477 1.51086 1.44158 1.37656
1.57716 1.50373 1.43489 1.37028
1.56960 1.49665 1.42825 1.36404
1.56209 1.48961 1.42164 1.35784
1.55462 1.48262 1.41508 1.35168
1.54721 1.47567 1.40856 1.34555
1.53985 1.46876 1.40208 1.33946
1.53253 1.46190 1.39564 1.33341
1.52526 1.45509 1.38924 1.32740
1.51804 1.44831 1.38288 1.32142
0.58 0.59 0.60 0.61
1,31549 1.25805 1.20399 9.15305
1.30958 1.25250 1.19875 1.14812
1.30372 1.24698 1.19356 1.14322
1.29788 1.24149 1.18839 1.13834
1.29209 1.23603 1.18325 1.13350
1.28633 1.23061 1.17814 1.12868
1.28060 1.22522 1.17306 1.23389
1.27491 1.21987 1.16801 1.11913
1.26926 1.21454 I.16300 1.11440
1.26364 1.20925 1.15801 1.10969
0.62 0.63 0.64 0.65 0.66
I . 10501 1.05967 1.01684 0.97635 0.93803
1.10036 1.05528 1.01269 0.97242 0.93431
1.09574 1.05091 1.00856 0.96851 0.93061
1.09114
1.08656 1.04225 1.00037 0.96077 0.92327
I.08202 1.03796 0.99631 0.95692 0.91964
1.07750 1.03368 0.99227 0.95310 0.91602
1.07300 1.02944 0.98826 0.94930 0.91242
1.06853 1.02522 0.98427 0.94552 0.90884
1.06409 1.02102 0.98029 0.94176 0.90528
I.04657 1.00446 0.96463 0.92693
Bartell, determinations of the interfacial tension between water and n-heptane were made. Three dsferent tips were used, and approximately 12 drops were formed with each tip, photographed, and measured to obtain drop dimensions. The water was tap distilled water which had been deionized with a mixed ion-exchange resin. The n-heptane was a commercial product, b.p. 98-99', which was treated three times with one-fifth of its volume of activated silica gel and was stored over silica gel. The n-heptane was held in a jacketed container held a t 26" and the water droplet was expressed from a micrometer syringe. The range of S values represented in each group of droplets, the The Journal of Physical Chemistry
average interfacial tension, and the standard deviation of each mean are presented in Table 111. The differences between these values were not significant at the 5% level by Student's t-test. Table 111: Interfacial Tension of Water and %-Heptane
range
Interfacial tension, dynes/cm.
0.540.57 0.64-0.66 0.70-0.72
50.11 49.43 50.34
S, values,
Std. dev. of the group
= t l 13 .
rrt0.85 2~0.79
Numbere of dropleta
11 11 12
MEASUREMENT OF SURFACE TENSION BY
THE
PENDANT DROPTECHNIQUE
Owing to the particular manner in which the drop measurements are made and also to the nature of the subsequent calculations, the experimental errors are not simply additive, but tend rather to combine in a pseudo-exponential fashion. I n addition, the magnitude of this large increase depends upon the shape of the drop, as exemplified by the ratio D,/De (= S). The purpose of this section is to explore the nature of the error propagation in the pendant drop method, and to determine the connection between the probable experimental error, the shape parameter S, and the actual experimental error of measurement.
2e De
e = - X 100
(5)
If the individually measured value of De is in error, for example if it is too large by the amount 6, then the wrong plane 2, will be chosen for the measurement of D,. As a result, the value of D, will be, in this case, too small by the amount 2 6 A X / A Z (plus the experimental errors in determining 2, and measuring Os). These two errors will magnify each other in giving a calculated value of S which is smaller than the true value. From this a value of 1/H which is greater than the true figure is read from the table of correction factors and is inserted in the equation for the calculation of surface tension (eq. 6) along with the too large value of De. The combination of these two erroneously high Y = sAP(DJ2(1/H)
(‘)
figures gives a calculated value of interfacial tension
1937
4
I
20 2r
77:’’ 10-
I
I
Table IV : Probable Limits of Error in Surface Tension S
0.959 0.879 0.797 0.722 0.653 0.592 0.522 0.459 0.380
ZSY/PY*
70
2.130 2.264 2.893 3.873 5.166 6.773 9.393 12.855 19.791
which is high, but by a factor which is more than just the sum of the measuring errors in De and D,. The propagation of errors in this method has been evaluated by considering the error of an individual measurement, *e, to represent the 95% confidence Volumc 69, Number 6
June 1966
R. W. MATTHEWS AND D. F. SANGSTER
1938
limits =k2u (where u represents the standard deviation of the individual measurement). Treating all measurements statistically and combining them in a straightforward algebraic manner, an estimate of the probable limits of the error in the true value of surface tension, p,, is obtained. The variance of a single determination and calculation is fu,, so to include 95% of the cases the limits were taken to be =k2u,. The probable limits expressed as a percentage of true surface tension are given in Table IV. A per cent error of measurement, e, of 1% is assumed. For other per cent errors a simple
multiplication is all that is necessary. The details of the error analysis are available upon request. The values of 2u, are also plotted vs. S in Figure 2. From this plot the desirability of working at S ratios close to 1.0 is evident. Although this is not always feasible because of other factors, if one chooses (or is forced) to use more nearly spherical drops, of smaller S values, he should be aware of the loss of accuracy entailed thereby.8 (8) 0. L. Davies, “Statistical Methods in Research and Production,” Hafner, New York, N. Y., 1961.
Measurement by Benzoate Radiolytic Decarboxylation of Relative Rate
Constants for Hydroxyl Radical Reactions
by R. W. Matthews and D. F. Sangster Australian Atomic Energy Commission Research Establishment, Suthedand, Sydney, Australia (Received December $3,1964)
The decarboxylation of 14C carboxyl-labeled benzoate during the irradiation of a dilute, aerated aqueous solution by ionizing radiation has been studied. It was found that, in addition to the 14C02produced during the irradiation, there was a post-irradiation production of further carbon dioxide during the following few hours. From the kinetic salt effect and competitor studies, it is concluded that the hydroxyl radical is involved in the production of all the carbon dioxide found. Provided comparisons are made a t a set time, the system can be used to measure over-all reaction rates for solute-hydroxyl radical reactions relative to the benzoate-OH reaction rate. Values for 20 solutes are given and compared with some of the published values. The highest values found were 2.1 for ferrocyanide-OH and 2.4 for iodide-OH. Values found for the aromatic compounds studied lie between 0.39 for nitrobenzene and 1.6 for phenate. Where hydrogen abstraction is involved, lower values are found.
Introduction Products which have been identified from the radiolysis of dilute aerated aqueous solutions of benzoic acid Or benzoate salts are the three isomeric hydroxy acids, carbon dioxide from the carboxyl group,l,2 and a dialdehyde of unknown structure which is thought to be related to mucondialdehyde.S Post-irradiation The JOUTWEof Physical Chemistry
effects have been observed in irradiated aqueous solutions of b e n ~ e n e ,but ~ none have been reported for (1) H.Loebl, G. Stein, and J. Weiss, J . Chem. SOC.,405 (1951); W. A. Armstrong, B. A. Black, and D. W. Grant, J . Phys. Chem., 64, 1415 (1960); A. Sugimori and G. Tsuchihashi, B J ~ C . h m . sot. Japan, 33, 713 (1960); A. Sakumoto and G. Tsuchihashi, ibid., 34, 660,663 (1961). . . (2) ‘A. M. Domes, Australian J . Chem., 11, 154 (1958).
