32 SUGDEN THE VARIATION OF SURFACE TENSION VI.-The Variation of Surface Tension with Tern-perature and ~ome Belated Functions. By SAMUEL SUGDEN. IT is well known that a linear relationship between surface tension and temperature can only be used over small ranges of temperature. In 1894 van der Waals (2. physikal. Chem. 13 716) gave two formuh connecting surface tension and temperature which involve the critical constants. . . . . . . y = K,O,V,-S(l - m)B (1) = K28&p,3(1 - m)B (2) . . . . . . Here O, V, and p are the critical temperature volume and pressure m is the reduced temperature and K, K, and B are universal const ants. Ferguson (Phil. Hag. 1916 [vi] 31,37) has suggested the formula . . . . . . . . y = yo(l - bt)’& (3) where n varies from substance to substance.This variation is not large the values of n for non-associated liquids ranging from 1-192 to 1.248 and Ferguson states that the mean value 1-21 gives satisfactory results in most cases. More recently Macleod (Tram. Faraday Soc. 1923 19 38) has shown that where C is a constant for a given liquid over a large range of tem-perature. Here D is the density of the liquid and d that of the vapour. In a paper recently read before the Faraday Society (2nd July 1923) Fekguson has shown by eliminating 1 - m between (3) and Katayama’s modification of the Ramsay-Shields equation namely, . . . . . . . . . y = C(D - 4 4 (4) . . . . . . . y(&&Jt= A8,(1- m) ( 5 ) (Sci. Rep. Tohoku Imp. Univ. 1916,4 373) that Macleod’s relation results if n = 1-20.Hence if Macleod’s relation holds Ferguson’s equation reduces to that of van der Waals and the universal constant B has the value 1.20. This leads us to inquire whether it is necessary to have a variable exponent to reproduce the experimental figures with sufficient accuracy or whether a formula of the van der Waals type is adequate. We may rewrite (1) . . . . . . . . y = y,(l - m)’’‘20 where yo = K10cVc-3 = K,B,Jp,% (6) ( 7 ) . . . . WITH TEMPERATURE AND SOME RELATED FUNCTIONS. 33 By applying (ti) to the experimental data we can test the constancy of B and then from the values of yo and the critical constants see if K l and K are universal constants. The most important measurements of surface tension up to the neighbourhood of the critical temperature are those contained in the classical paper of Ramsay and Shields (Phil.Trans. 1893, 184 647). It is however now certain that the surface tensions obtained by these workers are too low because of an inadequate correction for the capillary rise in the wider tube. Fortunately, sufficient experimental data are recorded in this paper to enable one to calculate the necessary correction. Details of the calculation of this correction are given in an appendix to the present paper but the method employed may be described briefly here. In the first place measurements were made of the surface tension of benzene and chlorobenzene up to the boiling point by the method of maximum bubble pressure (this volume p. 27). To these figures a formula of type (6) was fitted and the surface tensions a t temperatures a few degrees above the boiling point were obtained by extrapolation.From these and the capillary rise observed by Ramsay and Shields the radius of the wide tube was calculated using the theory of the method of capillary rise worked out by the author two years ago (T. 1921, 119 1483). Once this radius is determined the corrected values of the surface tension can be calculated. To save labour the corrected surface tensions have been calculated at 30" intervals instead of the 10" intervals given by Ramsay and Shields. Further it should be noted that these corrected surface tensions are calculated from the observed values of the capillary rise whilst Ramsay and Shields's figures are deduced from a smoothed value. It seemed preferable to allow the figures to be burdened with the experimental error rather than to use an arbitrarily smoothed curve since the main object of the work was to test whether formula (6) was capable of representing the experimental figures with sufficient accuracy.A further check on the values for benzene chlorobenzene and carbon tetrachloride was obtained from the observations of Walden and Swinne (2. physikul. Chem. 1912 79 700). These workers employed the method of capillary rise and constructed an apparatus with one tube of large diameter (2-3 cm.) so that the correction for capillary rise in this tube was very small and could be calculated accurately. They deduced the radius of the capillary from observ-ations on benzene using Ramsay and Shields's figures for the surface tension of this substance and found a radius of 0.1930 mm.If this figure is recalculated using accurate values for the surface VOL. cxxv. 34 SUGDEN THE VARIATION OF SURFACE TENSION tension of benzene a much higher value is obtained as is shown below. The radius of the capillary is given by the equation Benzene. h D-d Y . r Temp. cm . gm./c.c. dynes/cm. mm. 20" 3.374 0.8787 28-88 0-1985 30 3.237 0.8680 27.59 0*2000 40 3.134 0.8569 26.31 0-1996 50 3.015 0.8455 25-03 0.1999 Similarly from the accurate values of Richards and Carver (J. Amer. Chem. Soc. 1921 43 827) for the surface tension of toluene and carbon tetrachloride two more values of r can be obtained, namely toluene r = 0.1991 mm. carbon tetrachloride r = 0.1995 mm.Taking the mean figure 0.1994 cm. the figures of Walden and Swinne can be corrected by multiplying by 0.1994/0.1930 = 1.0335. This simple method of correction can be adopted in this case because the correction for the wide tube is so small and mould not be altered appreciably if one employed the corrected surface tension instead of the old figure to determine the magnitude of the capillary rise in this tube. It is interesting to note that Walden and Swinne made a direct determination of the radius of the ca,pillary by means of a mercury thread. They found a value of 0.1978 mm. which however they disregarded adopting the figure 0.1930 in all their calculations. Having obtained the corrected figures for the surface tension, the next step was to test equation (6) This was done by plotting ryi against the temperature when as is seen in Fig.1 the points for any one liquid were found to lie on a straight line. From the intercepts of this line on the axes of the graph t.he constants of (6) were calculated and the formula was then used to predict the surf ace tensions. As will be seen from the table on pages 37,38 the observed figures are reproduced with considerable accuracy in the case of the six normal liquids benzene chlorobenzene diethyl ether carbon tetra-chloride methyl formate and ethyl acetate. The greatest differ-ences between observed and calculated figures are found with chlorobenzene but these dserences are irregularly distributed and could not be eliminated by giving to n another value than 1-20.In the column headed "observer," R. & S. stands for Ramsay and Shields W. & S. for Walden and Swinne R. & C. for Richards and Carver and S. for Sugden. The three associated liquids do not obey this law as can b WITH TEMPERATURE AND SOME RELATED FUNCTIONS. 35 seen in Fig. 1 where methyl and ethyl alcohols and acetic acid give lines of marked curvature. The next test is to compare the values of the critical temperature deduced from the surface tension measurements with the figures obtained by direct observation. These are shown in the table below ; the critical temperatures are those given by Young (Proc. Roy. SOC. Dublin 1910 12 374). FIG. 1. -+t 100 200 300 10 5 + t I. Benzene. II. Chlorobenzene. 111. Diethyl ether. IV. Carbon tetrachloride.V. Methyl formute. VI. Ethyl acetate. VII. Methyl alcohol +. VIII. Ethyt alcohol 0. IX. Acetic acid. Critical temperature. Substance. Calc. Obs. Diff. Benzene ..................... 257" 288.5" - 1.5" Chlorobenzene ............... 358 359.2 - 1.2 Diethyl ether ............... 193 193.8 - 0.8 Methyl formate ............ 212 214 -2 Ethyl acetate ............... 249 250.1 - 1.1 Carbon tetrachloride ...... 2 80 253.1 - 3.1 It will be seen that the predicted critical temperatures are usually a degree or two below the observed value but in general there is 0 36 SUGDEN THE VARIATION OF SURFACE TENSION good agreement although not so good as that obtained by Ferguson (Zoc. cit.). This however is only to be expected as his formula allows n to be varied slightly and should give a closer fit.The next point which arises is the relationship between the critical constants and yo which van der Waals deduced from the theory of oorresponding states. In the table below the critical data are due to Young (Zoc. c k ) and K and K are the constants in equations (1) and (2). Van der Waals's Constants. Substance. yo. 8,. atm. gm./c.c. C.C. K1. K,. Benzene ... ..... ...... 70-26 561-5 47.89 0.3405 256.1 5.047 0-646 Chlorobenzene . . . . . . $0.33 673.2 44-64 0.3654 307.8 4-763 0-638 Diethyl ether . . ... . . . . 55-96 466.8 35.61 0-2625 281.9 5.155 0.667 Carbon tetrachloride 66.27 556.2 44-98 0.5576 276.1 5.171 0.637 Methyl formate . . . . . . 75.85 487 59.25 0.3489 172.0 4.818 0.635 Ethyl acetate ...... 65-24 523.1 38.00 0.3077 286.0 5.414 0.716 It will be seen that K and K are roughly constant for these six substances but vary by several units per cent.from one sub-stance to another. Finally there remains for consideration the relation between surface tension and density discovered by Macleod (Zoc. cit.). The last column in the table on pages 37,38 gives the value of $/(D - d). This quantity in the case of the six normal liquids is remarkably constant up to within about 40" of the critical temperature where the surface tensions and densities are difficult to determine with accuracy. For the three associated liquids this function increases slowly with increasing temperature. Pe de v, E x P E R I M E N T A L. Details of the measurements on benzene have been given in the preceding communication (this vol.p. 27). Measurements of the surface tension of chlorobenzeiie were made by the same methods and details of these measurements are given below. PuriJication.-A good commercial specimen was fractionally distilled and the specimen used boiled steadily at 131" (corr.) a t 750 mm. Apparatus.-Two instruments were used No. 1 for which r2 = 0.163 em. and A = 0-003736 and No. 3 for which r = 0.159 em. and A = 0.003333. Densities are due to Young (Zoc. cit.). Temp. dynes/cm.2. gm. /c.c. 9. dyncs/cm. 12" 9070 1.115 1.014 34-36 80 7860 1-073 1-016 29.83 App. 1 93 6573 1-025 1.018 24.99 81 7 708 1.039 1.01 6 26.10 App. 3 123 6377 0.991 1.018 21-63 P D-d WITH TEMPERBTURJZ AND SOME RELATED FUNCTIONS. 37 Summary. (1) The surface tension measurements of Ramsay and Shields and of Walden and Swinne have been corrected.(2) For six non-associated liquids it has been shown that the variation of surface tension with temperature is represented accur-ately by the formula y = yo(l - rn)l'ao where m is the reduced temperature and yo a constant. (3) The relations between yo and the critical constants predicted by van der Waals have been shown to hold approximately. (4) Macleod's relationship between surface tension and density is found to hold accurately for non-associated liquids up to about 40" below the critical t,emperature. Temp. 13.5' 20 20-5 21 32-5 34.5 39 41.5 54.8 61 72 90 130 150 1 so 210 340 2 70 280 12 18-7 24.1 41 50 52.2 62.1 81 93 123 150 180 210 2 40 2 i 0 300 320 333 0 bservor.S. R. & C. TV & s. S. w. & s. S. S. w. a s. w. & s. S. S. R. & S. Y9 9 ) ? ? ? 9 9 9 37 S. w. 82 s. 9 9 ?? S. w. & s. S. 9 9 7 9 It. % 8. 9 9 9 ? ? ? ? ? ? ¶ ? ? 2 ) y obs. 29-72 28-88 28.91 28-74 27-30 26.98 26.36 26.08 24-28 23.61 22-1 5 20.13 16.42 13.01 9.56 6.46 3-47 1-05 0-36 11. 34.36 33-35 32-55 30.78 29.83 29.38 28-20 26-10 24.99 21-63 18.56 1540 12.16 9.30 6-43 4-05 2-39 1.63 TABLE. y cnlc. Diff. I. Benzene. 29.74 +0.02 28.88 +O.OO 28.75 +0*01 27-11 3-0.13 26.44 +0.08 26-12 +0*04 24.43 +0*15 23.65 fO.04 22-26 +0*11 20.06 -0.08 16.45 +Om03 12.97 -0.04 9.64 f-0.08 6.48 +Om03 3.59 4-0.12 1.06 +0-01 0-37 +O.Ol 28.82 -0.09 27.27 -0.03 Chlorobsnzene.34.41 -0.15 33-42 f0.13 32.78 -0.07 30.79 3-0.01 29.75 -0.08 29.49 +0*11 28-36 f0.14 26.19 +0*09 24.83 -0.16 21.50 -0.13 18.57 +0.02 15-40 &O*OO 12-35 +0*19 9.41 +0*11 6.61 +0.18 4.01 -0*04 2-25. -0.14 1.4G -0.17 U - d . 0.8857 0.8787 0.8782 0.8777 0.8653 0.8631 0.8581 0.8553 0-5400 0-8330 0.5207 0.8006 0.7616 0.7166 0.6657 0.601 1 0-5137 0.3696 0.2305 1.115 1.108 1.102 1.083 1.073 1.070 1.059 1.039 1.025 0.991 0.9545 0.9122 0-8622 0.8054 0.7341 0,6442 0.5628 0*4'314 Yt D y d ' 2.637 2.638 2.641 2-638 2.642 2.641 2-641 2.642 2.642 2.647 2-643 2.646 2.643 2-650 2-641 2.651 2.657 2.739 3.352 2.166 2.173 2.172 2.174 2.178 2.176 2.176 2-1 76 2.182 2.1 76 2.174 2.172 2-166 2.168 2.169 2.202 2.210 2.29 38 Temp.20 50 80 110 140 170 185 20 21.1 33-0 45.0 90 120 150 180 210 240 2 70 50 80 110 140 170 200 90 120 150 180 210 240 20 70 100 130 160 190 220 230 20 80 110 140 170 200 230 130 160 190 220 260 280 SUGDEN THE VARJATION OF SURFACE TENSION Observer. R. & c. R. & S. Y 9 Y 9 Y 7 7 9 7 9 R. & C. w. & s. R. % S. 9 ) 7 ) 9 7 Y Y 9 ) I ¶ 9 ) R. 8% s. Y Y ? Y 9 9 Y Y Y Y R.& S. Y 9 9 ) Y Y Y Y Y Y R. & C. R. & S. 9 Y Y Y 9 ) 7 ) 9 ) Y Y R. & C. R. & S. I t Y > Y Y Y Y ? 9 R. 8z S. 9 ) Y 9 Y 9 Y 9 9 9 111. Diethyl ether. yobs. ycalc. Diff. 17.01 17-04 +Ow03 13.69 13.56 -0.13 10.25 10.23 -0.02 7-00 7.06 +Om06 4-00 4-12 3-0.12 1-42 1.51 +0-09 0.40 0.43 +0.03 IV. Carbon tetrachloride. 36.95 26.80 -0.15 26.71 26.66 -0.05 25.22 25.20 -0.02 23.63 23.63 ,tO.OO 18.5 1 18.40 -0.11 14.95 14-96 +O.Ol 11-46 11.66 +0*20 8-50 8.51 +O.Ol 5.67 5.55 -0.12 2-64 2.83 +Om19 0.49 0-55 +0*06 V. Methyl formate. 20.48 20.35 -0.13 1545 15.92 -0.03 11.77 11.66 -0.11 7.63 7.69 +0*06 3-98 4.03 +Om07 0.92 0.90 -0.02 VI. Ethyl acetate.15-74 15.67 -0.07 12.08 12.19 +Om11 8-85 8.87 +0.02 5-70 5-75 +0.05 2-94 2.90 -0.04 0.50 0.50 &O*OO VII. Methyl alcohol. 22.61 18.50 15-73 12.66 9.34 5-56 1.95 0.77 22.27 17-97 14.49 11-34 7.85 4.26 0.95 VIII. Ethyl alcohol. IX. Scctic acid. 17-05 14.21 11.45 8-71 5-70 2-83 D -d. 0.7109 0.6713 0.6286 0.5707 0.4936 0-3785 0.2698 1.593 1.591 1.567 1.540 1.4475 1.3740 1.2914 1-1945 1.0687 0.8980 0.5955 0.9251 0.8698 0-8048 0-7156 0-6081 0.4131 0.8065 0.7580 0-7005 0.6265 0-5232 0.3278 0-7910 0.7445 0-7100 0.6676 0-6141 0.5369 0.4036 0.3223 0.7888 0.7360 0.7008 0.6516 0.5920 0-5050 0.3415 0-9190 0.8729 0.8245 0.7643 0.6841 0.5685 Ya-D-d' 2.857 2.865 2.850 2-865 2.865 2.947 1429 1.428 1.429 1.432 1.433 1.431 1.425 1.429 1.444 1.419 1.405 2.300 2-297 2.301 2-322 2.319 2-37 2.884 2.470 2.450 2-462 2-467 2-50 2-56 2.757 2.787 2,804 2.826 2.846 2.866 2.928 2-905 2.755 2-797 2.785 2.817 2.828 2.845 2.891 2.211 2.224 2.231 2.248 2.258 2,28 WITH TEMPERATURE AND SOME RELATED FUNCTIONS.39 Appendix. Correction of Ramsay and Shields's Data. The apparatus used by Ramsay and Shields consisted of a capillary tube mounted concentrically in a wider tube. It is very difficult to deal witsh this case mathematically but a complete solution of the analogous problem of two communicating tubes has been given by the author (T.1921 119 1483). Since the correction to be calculated is small (about 5 per cent.) it would seem that the quantity required might be calculated in the following manner. Using accurate data for some standard liquid and the observed capillary rise recorded by Ramsay and Shields one can cal-culate the radius of an equivalent wide tube which if it were placed in communication with the capillary and not around it would have the same effect as R'amsay and Shields's concentric tube. Then by the aid of the tables in the paper referred to above the observ-ations of these workers can be recalculated as the radii of both tubes have been determined. The assumption made here is that an external tube which has the same effect as a concentric tube a t one particular surface tension has the same effect as the concentric tube at all surface tensions This assumption whilst perhaps not rigidly true must be so to a first approximation and since the total correction for the capillary rise in the wide tube is about 10 per cent at the lower temperatures and diminishes to zero a t the critical temperature it does not seem likely that any large error will result if this method is adopted.What may be termed " internal evidence " for the truth of this hypothesis may be found in the table on pages 37,38 where it will be seen that the corrected data obtained in this manner are in harmony with the values found by other observers. The standard liquids chosen for the determination of the radius of the wide tube were benzene and chlorobenzene.From a series of measurements by the method of maximum bubble pressure it was found that the surface tensions of these liquids could be repre-sented by the formulae Benzene y = 70.26 ( 1 -&y2 . . . Chlorobenzene y = 70-33 1 - ( &)'? . . . where 6 is the absolute temperature. for benzene at 90" y = 20-05 dynesjcm. D - d = 0.8006 whence a2 = 5.105 mm.2. with two communicating tubes From (9) it is found tha,t At this temperature, Now for an apparatua l / b - l / b 2 = H/a2 40 THE VARIATION OF SURFACE TENSION WITH TEMPERATURE ETC. where b and b are the radii of curvature at the lowest points of the liquid surface in the capillary and wide tubes respectively. To calculate l/bl we proceed as follows. In Ramsay and Shields's apparatus rl = 0-012935 cm.hence rJa = 0-05275. From the table in the a,uthor's paper referred to above this corresponds to r,/b = 0.9989 whence l / b = 77.22. The capillary rise for benzene found by Ramsay and Shields a t this temperature was H = 3.460 em. from which H/a2 = G7.78 and l/b2 = 9-44. By the aid of the table connecting r/a and rjb it was found that for this value of a2 1,'b = 9.50 when r2 = 0.099 cm. and l / b = 9-39 when r2 = O-lCO cm. whence by interpolation r2 = 0.0996 cm. Similar calculations were made for benzene a t 120" and for chlorobenzene a t 150" and 180" with the following results. Temp. ......... 90" 120" 150' 180" r2 crn ............. 0.100 0.096 0.096 0.097 Mean 0-097 cm. Using this value of r2 any of the observations of Ramsay and Shields may be recalculated by the method of successive approxim-ations detailed in the author's 1921 paper (Eoc.cit.). This would involve a very laborious series of calculations which can be avoided in the following manner. Benzene. Chlorobonzene. Since (12) (13) 2 H a2 = -Y -g(D - d ) - l / b l - l/&i * * ' we can write where . . . . . . . . y = H(D - d)+ Kow # is a function of rl r2 and a2 and for a given apparatus, where rl and r2 are fixed is a function of H only. Hence if + is calculat'ed for a few values of H intermediate values can be obtained by interpolation. The table below gives corresponding values of # and H for the range required. H ...... 0.10 0.15 0.20 0.40 0.70 1 -00 d ...... 6.828 6.839 6.881 6.881 7.108 7.158 ...... 4.0 5.0 OD ...... 7.280 7.321 H 1.50 2.00 3-0 Q) 7.208 7-233 7.259 1 . ~ 7 1 From these values a curve was drawn to give intermediate values of Q and the correct'ed surface tensions mere calculated by equation (13). Liquid. Temp. II. D-d. 9. y corr. y R. & S. Benzene ......... 90" 3.460 0-8006 7.365 20.13 19.16 ......... 240 0.945 0.5137 i-155 3.47 3.41 Chlorobenzene ... 150 2.68 0.9546 7-251 18-55 17.67 9 . ... 320 0.60 0.5638 7.084 2.39 5-35 - 0 A few examples are given below. ? STEREOISOMERISM AND LOCAL ANZSTHETIC ACTION ETC. 41 The last columii in the table gives the surface tension calculated by Ramsay and Shields. It will be seen that the corrected figures are about 1 dynelcm. higher at the lower temperatures the difference diminishing as the temperature rises. The authbr is indebted to the Research Fund Committee of the Chemical Society for a grant which has largely defrayed the cost of this investigation. BIRRBECK COLLEGE, UNIVERSITY OF LONDON. [Received October 23rd 1923.