EXCHANGE "University of Cbtcaoo OX THE MOTION OF A SPHERE OF Oil THROUGH CARBON DIOXIDE AND AN EXACT DETERMINATION OF THE COEFFICIENT OF VISCOSITY OF THAT GAS BY THE OIL DORP METHOD LEO JOSEPH LASSALLK A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY, DEPARTMENT OF PHYSICS CHICAC.O 1917 tlbe Tllnlversitg of Cbtcago ON THE MOTION OF A SPHERE OF OIL THROUGH CARBON DIOXIDE AND AN EXACT DETERMINATION OF THE COEFFICIENT OF VISCOSITY OF THAT GAS BY THE OIL DORP METHOD BY LEO JOSEPH LASSALLE A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY, DEPARTMENT OF PHYSICS CHICAGO 1917 of, ON THE MOTION OF A SPHERE OF OIL THROUGH CAR- BON DIOXIDE AND A DETERMINATION OF THE COEFFICIENT OF VISCOSITY OF THAT GAS BY THE OIL DROP METHOD. 458684 [Reprinted Jrom the PHYSICAL REVIEW, N.S.. Vol. XVII, No. 3, March, 1921.] ON THE MOTION OF A SPHERE OF OIL THROUGH CAR- BON DIOXIDE AND A DETERMINATION OF THE COEFFICIENT OF VISCOSITY OF THAT GAS BY THE OIL DROP METHOD. BY LEO JOSEPH LASSALLE. SYNOPSIS. i I. Coefficient of Viscosity of CO z by the Oil-drop Method. The value of e being known, through work in air, to an accuracy of about .1 per cent., the oil-drop method as developed by Millikan has been used in COz for the determination of the coefficient of vicsosity of that gas with the result at 23 C. 77 = 1.490 X io~ 4 . II. Coefficient of Slip between CO?, and Oil. The value of the constant A in Millikan's equation e\ = e(i + AZ/o) 3/2 is found in COa to be 0.8249 as against Millikan's value in air 0.864. III. Variability of the Constant A for Values of I/a Greater than .5. Precisely as in Millikan's work in air, A was found to be constant only up to I/a = .5, beyond which it kept increasing as far as it was followed, viz., up to I/a = 12. INTRODUCTION. THE behavior of oil drops falling in air has been reported upon by R. A. Millikan. 1 Every precaution possible was taken to assure the highest degree of accuracy. The value of e obtained was shown to be accurate to within 0.2 of one per cent. The law of fall for oil drops in hydrogen was investigated by R. A. Millikan, W. H. Barber, and Y. Ishida. 2 The same precautions as those taken for air were observed. The work here described was undertaken at the suggestion of Dr. Millikan in order to find the correction factor to Stoke 's law for carbon dioxide, and the coefficient of viscosity of that gas by a new method. The value of e? lz is directly proportional to the coefficient of viscosity, 77, of the gas used. As is shown later, this relationship furnishes an elegant method of determining 77 for any gas, and is used in this paper to obtain the viscosity coefficient for CO 2 at 23 C. The apparatus used was the same one used by R. A. Millikan. 3 The same precautions were taken to give as high a degree of accuracy as 1 "On the Elementary Electrical Charge and The Avagadro Constant," PHYSICAL REVIEW, N.S., Vol. II., No. 2, Aug., 1913, p. 109. 2 "The Law of Fall of a Droplet Through Hydrogen," PHYSICAL REVIEW, Series 2, 5, - P- 334- PHYS. REV., N.S., Vol. II., No. 2, Aug., 1913, p. 109. MOTION OF A SPHERE OF OIL. 355 possible. All of the time observations were taken with a Gaertner recording chronograph. This instrument records time to one one- hundredth of a second. It was adjusted so as never to be in error by more than two hundred ths of a second per minute. The following method described by Langmuir was used in generating the CO 2 . Using a Kipp generator, CaCO 3 + 2HNO 3 = CO 2 + Ca(NO 3 ) 2 + H 2 O + HNO 3 vapor + H 2 O vapor. The gas is bubbled through NaHCO 3 , which takes up acid vapors but no CO 2 ; then it passes through P 2 O 5 tubes which take up all H 2 O vapor. The HNO 3 was a 50 per cent, pure H 2 O solution through which com- mercial CO 2 had been bubbled for 5 hours before being put into the Kipp generator. The CaCO 3 (marble) was broken and boiled 4 hours in a very dilute solution of HNO 3 . The vessels into which the gas was intro- duced were evacuated to a pressure below 0.5 mm. before they were filled. They were then pumped down again to the same low pressure and refilled with CO 2 . This process was repeated three times before any observations were taken, thus assuring that not more than one part of air in (i.soo) 3 parts of CO 2 remained in the vessels. For a complete discussion of the apparatus used and the precautions taken to assure accuracy in the determinations of the constants that enter into the following results see paper by R. A. Millikan. 1 PART I. Determination of Coefficient of Viscosity of COz- Oil droplets were obtained for observation by aspirating oil with CO 2 gas under pressure. They were held for a sufficient length of time to observe the time of fall under gravity (/) on an average of eighteen times. The average number of changes of charge obtained for each drop was seven for the first thirty-six drops, which are the ones used in the deter- mination of the coefficient of viscosity and the correction factors "b" and "A." The results of the observations were such as to leave no uncertainty as to the greatest common divisor of [(i/tg) -f (i ///)], which will be represented hereafter by [(!//) + (iA/)]o, and which is the variable factor upon which the value of (ei 2/3 /i?) and (a) are dependent. It is significant that these thirty-six drops represent every drop observed where P.D. was constant to as much as 0.4 of one per cent, and where as many as three changes of charge were obtained. In the equation (. + .)(.)"' 1 R. A. Millikan, PHYS. REV., N.S., Vol. II., No. 2, Aug., 1913, pp. 109-143; also Phil. Mag., XXXIV., p. 13, 1917- 356 LEO JOSEPH LASSALLE. 771 is the coefficient of viscosity of CO2 at 23 C., g the acceleration due to gravity, 2 or in any other gas, say air. Since the value of e 213 in air is known to be 61.085 X io~ 8 E. S. units, with a probable error of less than o.i of one per cent., 1 then at i/pa = o, e^ 3 = e 2 ' 3 = 61.085 X io~ 8 ; .'. = 41.00 X io~ 4 1 This is the value obtained by reducing the number given by R. A. Millikan in Phil. Mag., XXXIV., p. 13, by .07 per cent, to allow for the change in the value of the coefficient viscosity of air from .0001824 to .00018227. LEO JOSEPH LASSALLE. gves 61.085 X io- 8 ill = - 7 = 1.490 X io~ 4 , 41.00 X io~ 4 which, because all the calculations were reduced to 23 C., gives the value of the coefficient of viscosity at 23 C. Wherever the temperature differed from 23 C. during an observation the correction factor to be applied to the value of 171 was introduced into the calculation of (0i 2/3 /T7i). Sutherland's equation was used and the variation of log 771 with temperature was obtained by plotting log 171 against temperature. With the exception of drops 5 and 15, which differ by about i C., the variation from 23 C. was always less than 0.5 C. By the oil drop method, then, the value of the coefficient of viscosity of CO 2 at 23 C. is found to be l?C0jat23-(7 = 1-490 X I0~ 4 . The accuracy of this result is dependent upon the accuracy with which the value of e 2/3 is known and the accuracy with which the inter- cept on the ei 2/3 /T7i axis for CO 2 is known. e 2/3 is known with an accuracy of about o.i of one per cent. The ei 2/3 /7ji intercept is known with a probable error of about 0.5 of one per cent. Therefore, the value of 771 given by these observations should be accurate to about 0.5 of one per cent. Therefore *7co 2 at23'C = (1490 0.0080) X io~ 4 . Breitenbach 1 obtained at 15o(7 =1457 X I0~ 4 . Applying the Sutherland equation to his value there results *?C0 2 at23oC = 1494 X IO- 4 . If we assume that this investigator's value is too high for CO 2 in the same ratio that his value for air is too high, we get 1786 i?co 2 at23.c = (1494 X iQ" 4 ) = J 474 X io- 4 , a value I per cent, lower than the one obtained in this paper. Ernst Thomson, 2 using a vibrating disc, obtains a value of ij for CO 2 which is 0.0000004 higher than Breitenbach's. However, the main object of his investigation was to find the relationship between the 77 for 1 Breitenbach, Ann. d. Phys., 5, 1901. *"Ueber die innere Aeibung von Gasgemischen," Inaugural Dissertation, der Konigl. Christian-Albrechts Univ. zu Kiel. MOTION OF A SPHERE OF OIL. 359 two gases and the 77 for various mixtures of these two gases. He does not claim any high degree of accuracy for his value of the coefficient of viscosity of CO2. P. Phillips 1 gives a value of 77 which reduced to 23 C. by Sutherland's equation, becomes *7CO,at23C = 1-494 X I0~ 4 . He used the A. O. Rankin device, 2 which is a capillary tube method. The object of his investigation was to determine the variation of 77 with pressure, going to pressures of eighty atmospheres. No high degree of accuracy is claimed for the determination of the 77 at atmospheric pres- sure. The important point sought after was to obtain relative values as pressure varied. If, however, we take the mean of the values obtained by Breitenbach, Thomsen and Phillips, a value of 77 C 2at2 3<>c = 1-489 X io~ 4 is obtained. This value is different by less than o.i of one per cent, from the one here obtained. PART II. The A and b Correction Terms in CO 2. Table I. gives the values of ei z ' 3 /r]i corresponding to the various values of I / pa. ei 213 is obtained by multiplying ei 2l3 /iji by the value of 771 as found in part I. Fig. 2 gives the result of plotting e\- lz against i/pa. If, now, we write / T \ (7) and let y = e^ l3 \ y = e 213 , and x = (i/pa), we have y = y + byox. By differentiation, then, the above becomes / dy \ Slope of (y, x) curve b = ( ~, ~j~ y ] == ~~ . ~~ . (8) \ dx ) ^-intercept This gives a simple method of determining the correction factor &. 3 The last column in Table I. gives the values of e 213 calculated by sub- stituting in (7) and solving. The values of e follow directly from those of e 213 . It will be noticed that no value of e 213 varies from 61.085 by as much as 0.5 of one per cent. The value of b, obtained by substituting in (8) is _ Slope of y, x curve _ (75-37 61.085) y-intercept 600 X 61.085 1 P. Phillips, Roy. Soc. Proc., Ser. A, 87. pp. 48-61. s Roy. Soc. Proc., 1910, A, Vol. 83, p. 265. ' R. A. Millikan, PHYS. REV., II., p. 118, 1913, and XXXII., p. 381, 1911. 360 LEO JOSEPH LASSALLE. (SECOND [SERIES. O ; Ov'-i''* l ~*OvOr5OOOOONOO)OOOJ>-OiOOOrOOOI>.i-iOOOO M O\OOO\p'-;O|ppC'OO;'-ip'-oOi/ - joiOO) " -ipc < OiOoqO\O\r'>t--.-^^-i*-iOt-;pOJcoiOO i --;O^t-;- < =H'-HO c x ^1 OOO'OOOOOOOOOOOOOOOOOOOOOO I Tjnt^.ovPot^ IT l 1^ 1^ 1^ IT lOO' I' I' I O O p p p p p p p p p p o p p p p p p p p p p p p p p p X ooo'oo'ooo'ooc^oo'oooooooo'ooooo Tf O\ O w _^. ^ ' F rt> iOC3iOi^^jrrJV>wuuw""v^^-''.> < ^>.v oHPOO'-!pfO^t|o)-H^p'^-;poqoiOcso;pO^p'-*'-io>'O I H OOCNO^HOIfOTtilOO ^ ^-H ^-H ^H *^ *"H ^~t t^ ^H ^-( *-H Ol Ol Ol Ol CN O4 Ol VOL. XVII.1 No. 3. J MOTION OF A SPHERE OF OIL. x ,_;,_; o *-< -< *-< o' M X >o >o c o" o M X c I oooooooo O O\ ' O i i o X o ^2 o o o o o o o o o o o o o'oo'oo'ooo i 4 3 Q t LEO JOSEPH LASSALLE. [SECOND [SERIES. or b = 0.0003898 (see Fig. 2). It should be noticed that the value of b is independent of any theory and is determined by the use of values which are directly measurable. If, however, it is desired to write (7) in the form ,2/3 _ (9) then A can be calculated in the same way that b was, provided the values of I/a are known. Using 77 = o.35O2ww/C, where the Boltzmann value of K is used so 76 TS 74 73 7* fl 70 69 6! 66 62] rtrf ^ 50 60 90 300 360 420 480 509 Fig. 2. as to give an A that can be compared with that for air obtained by R. A. Millikan in the paper previously referred to, the values of I [a can be calculated. From equations (7) and (9) it is evident that '--= a pa (10) At a constant temperature i/pl is a constant. Therefore, A = bB, which means that the (ei 273 , ijpa) curve becomes the (ei 2/3 , //a) curve simply by changing the scale of the abscissae. By calculating for a given drop and substituting in equation (10) the value of A/b = B is obtained. This is checked by making the calcu- VOL. XVII No. 3. MOTION OF A SPHERE OF OIL. 363 ation for several drops. In this way B was found to be equal to 2116. This gives for CO A = 0.8249. The value obtained by R. A. Millikan in his work in air was A = .864. The difference would seem to be slightly more than the experimental error involved in the determination of the slope of the line from which A is obtained. This error should scarcely exceed two or three per cent. There is then here a somewhat uncertain indication that the coefficient of slip between CO 2 and oil is a trifle less than that between air and oil. TABLE II. No. Temp. P. D. Volts. (Sec.) X (eV)- n aXlO 5 Cms. / Cms. Hg. i a ' -*" 35 22.70 3245.0 53.833 0.014186 3-9 10.63 16.70 631.6 0.2985 76.19 36 23.04 2604.8 38.036 0.00971 3-13 12.58 12.02 661.3 0.3125 77.01 37 23.15 2583.0 22.63 0.00771 ; 8-19 16.11 8.160 760.83 0.3595 79.33 38 23.15 2589.5 27.67 0.00920 5-13 14.22 7.638 921.09 0.4352 82.94 39 22.90 1288.5 9.32 0.00274 67-168 24.24 4.210 980.14 0.4632 84.63 40 22.87 1288.4 14.83 0.003817 18-66 18.59 4.588 1172.5 0.5540 90.38 41 23.38 1938.6 24.02 0.00735 7-13 14.57 5.566 1233.3 0.5828 91.05 42 22.85 1286.5 20.81 0.00460 8-56 15.60 5.279 1214.6 0.5739 91.50 43 22.87 1286.0 26.67 0.00560 | 7-87 13.45 5.365 1386.1 0.6550 96.06 44 22.90 1288.2 20.36 0.00514 10-44 15.15 4.410 1497.1 0.7074 99.18 45 23.35 1933.3 38.55 0.01073 3-12 10.97 5.910 1542.7 0.7290 100.05 46 22.86 1288.8 19.20 0.00522 11-35 15.37 4.110 1583.1 0.7480 102.14 47 22.80 650.3 7.01 0.00169 86-137 24.94 2.450 1636.3 0.7732 106.2 48 22.85 648.5 16.91 0.00298 26-99 15.37 3.356 1938.5 0.9160 115.9 49 22.74 1289.5 27.39 0.00772 5-33 11.99 3.860 2161.6 1.021 117.8 50 22.79 651.4 10.19 0.00254 29-153 19.22 2.290 2271.5 ,1.073 123.0 51 22.90 653.5 9.58 0.00267 38-129 19.32 2.055 2518.7 1.190 129.5 52 23.20 92.70 9.90 0.000391 240-376 18.91 1.970 2684.7 1.269 131.0 53 22.83 649.5 23.23 0.00530 12-60 11.42 2.638 3319.6 1.569 152.9 54 22.95 656.7 16.06 0.00550 12-44 12.80 1.990 3925.1 1.855 176.0 55 22.79 651.1 42.68 0.01510 2-6 6.58 2.390 6356.7 3.004 250.4 56 23.55 649.8 9.96 0.00809 12-31 13.15 1.060 7171.5 3.389 269.3 57 22.08 156.0 31.26 0.00364 26-77 7.29 1.840 7457.8 3.524 278.4 58 23.14 155.72 13.45 0.00275 37-159 10.59 1.100 8582.6 4.056 307.2 59 22.83 192.50 7.18 0.00117 122-225 14.60 0.798 8584.2 4.056 302.7 60 22.90 155.21 6.87 0.00241 j 64-122 13.84 0.719 10051. 4.750 352.2 61 24.46 157.00 28.32 0.00507 1 3-85 6.76 1.450 10205. 4.822 359.9 62 23.22 156.8 13.23 0.00469 17-27 8.94 0.736 12076. 5.706 438.8 63 22.82 652.2 82.18 0.05502 1-2 3.44 2.211 13147. 6.212 476.1 64 22.86 156.72 13.03 0.00571 14-18 8.41 0.841 14142. 6.683 502.7 65 22.80 158.22 36.08 0.01945 3-6 3.99 1.112 22524. 10.64 805.2 66 23.02 92.56 9.87 0.00708 14-27 7.21 0.540 25702. 12.14 904.4 67 22.93 89.97 23.59 0.01103 2-8 4.60 0.824 26552. 12.55 926.4 LEO JOSEPH LASSALLE. PART III. [SECOND [SERIES. Limits to the Validity of Millikan's Equation in CO 2. Observations were taken on drops at as large values of I/a as practicable with the method of obtaining drops that was used throughout these observations. It was found very difficult to get drops at pressure below one cm. of Hg. A more direct method of blowing the drops and one which will introduce less gas into the system is desirable in order to go to low pressures. At low pressures it is not possible to have such high potential differ- ences; also it is not possible to observe on drops of the same size as those used at pressures above, say 2 cm. These two factors tend to counterbalance each other as far as the difficulty of observing on drops with a small number of charges is concerned. If the number is not small the greatest common divisor of the series of speeds begins to be uncertain. However, no drop was used in these calculations where there was doubt as to this greatest common divisor. Table II. gives the values of the various factors entering into the goo /oao /zco MOD 1600 /goo -?ooo ,2400 Z400 600 Fig. 3. determinations of e^ 13 , i/pa and I/a for drops where the values of i/pa are greater than 600. Fig- 3 gives the relationship between e^ 13 and if pa, which is also that between e x 2 / 3 and I/a. It gives this relationship from i/pa = 600 to i/pa = 2,500. It will be seen that from 600 to about 1,100 the slope is the same as that for i/pa = o to i/pa = 600. But at about the value of 1,100 the slope clearly begins to change. This behavior is quite like that found by R. A. Millikan in his work with air. He found (Pnvs. VOL. XVII. No. 3. MOTION OF A SPHERE OF OIL. 365 REV., II., p. 138) that the linear relation between ei w and i/pa began to break down at about i/pa = 650, which corresponds to a value of I/a of about .5. Here the break comes at about i/pa = 1,100 which will be seen from Table II. to also correspond to a value of I /a = .5. J9foo S3SOO Fig. 4. Fig. 4 shows the graph of the relations contained in Table II. in the range I/pa = 2,500 up to if pa = 26,500, i.e., in the range I/a = I to I/a = 12. SUMMARY. I. Using the oil-drop method, the coefficient of viscosity, ij, of CO 2 at 23 C. was found to be 1.490 X io~ 4 . II. The correction factors b and A as given in the equations 2/3 _ and were determined for CO 2 and found to be b = 0.0003898, A = 0.8249. Applying the correction term to the various drops, 36 in number, values of e 213 were obtained, no one of which varies from 61.085 X io~ 8 by as much as 0.5 of one per cent. 366 LEO JOSEPH LASSALLE. III. The relationship between ei 2/3 and I/pa was found for values up to I / pa = 26,500. The value of A = 0.8249 holds until I/a = 0.50, approximately. The slope of the curve then increases. The values of I/a for CO%, at which the change in slope begins is the same as that found by Millikan in the case of air. It gives me pleasure to acknowledge the cheerful and able assistance of Dr. Y. Ishida and Mr. B. L. Steele. I am especially indebted to Dr. R. A. Millikan, who suggested the problem and who advised me throughout the course of the investigation. RYERSON PHYSICAL LABORATORY, UNIVERSITY OF CHICAGO, August 10, 1917. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. LD 21-100m-7,'33 Gaylord Bros. Makers Syracuse, N. Y. PAT. JAN. 21, 19C8 YD 02232 1 U-3 UNIVERSITY OF CALIFORNIA LIBRARY