A Formula for the Vicosity of Liquids

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PROC. N. A. S.

"spin faster," i.e., increasing the velocity of the circulation, decreasing the time required for a given mass to complete the cycle. In either case he has increased the energy turn-over per unit of time. Whether, in this he has been unconsciously fulfilling one of those laws of nature according to which certain quantities tend toward a maximum, is a question well deserving of our attention.

Papers from the Department of Biometry and Vital Statistics, School of Hygiene and Public Health, Johns Hopkins University, No. 44. 1Lotka, A. J., Physic. Rev., 24, 1912 (235-238); J. Washington Acad. Sci., 2, 1912 (2, 49, 66); Science Progress, 55, 1920 (406-417); Proc. Am. Acad. Arts Sci., 55, 1920 (237-153); these PROCEEDINGS, Sci. 6, 1910 (410-415). 2Proc. Amer. Acad., loc cit., p. 142. 3See for example Picard, Trait,6 d'Analyse; H. Bateman, Differential Equations, 1918, p. 245. ,Spencer, First Principles, Chapter XXII; Winiarskie, "Essai sur la Mcanique Sociale," Revue Philosophique, 44, 1900 (113). 6 At the time of reading proof this project is partially realized. A discussion of the applicability of the Le Chatelier principle to systems of the general character here considered will appear in a forthcoming issue of the Proceedings of the American Academy of Arts and Sciences Lotka, A. J., London, Phil. Mag., Aug., 1911, p. 353. *

A FORMULA FOR THE VISCOSITY OF LIQUIDS1 By H. B. PHILLIPS DuPARTMENT OP MATHEMATICS, MASSACEUSET INSTITUTS OF TICMoi,OoY Communicated by A. G. Webster, March 22, 1921

1. In this paper I obtain for the viscosity of a liquid the formula

nNh J X2M (X-v -(1) where N is the number of molecules in a mol, h is Planck's constant, M is the molecular weight of the liquid in the gas phase, v its volume per gram, and n an integer. The quantity a is the co-volume as used in the equation of state of Keyes' A, RT (2) P v-6 (v-i)2 In all the cases to which I have applied the formula, n = 6 and so (1) tak-es the form

(v-5) = 3Nh/M

(3)

It is to be noted that 3N/M is the number of translational degrees of freedom of the molecules in the volume v of the liquid. 2. To prove equation (1), let x, y, z be rectangular coordinates and suppose the liquid to flow parallel to the x-axis in such a way that, u0 being

VOL,. 7, 1921

MA THEMATICS: H. B. PHILLIPS

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the velocity at the point (x, y, z),

(4) buo /bz = 1. Let the components of velocity of a molecule parallel to the x and x axes be u, w respectively. When a molecule of mass m moves in the positive direction of the s-axis across the xy-plane, the x-component of momentum transferred across that plane is mu. When it crosses in the negative direction the transfer is -mu. Under these conditions the viscosity v of the liquid is defined as the total x-component of momentum transferred across unit area of the xy-plane in unit time.2 To find the viscosity we therefore find the number of molecules of each velocity crossing unit area and multiply by the average momentum transferred by each. Let f (w) dw be the number of molecules per cubic centimeter with components of velocity parallel to the s-axis between w and w + dw. If the molecules did not influence each other, the number of these that would cross unit area of the xy-plane in unit time would be (5) wf(w) dw. Because of the interference of molecules with each other, this must be replaced by (6) v-5wf (w) dw.

v-

To see this we note that if the molecules moved independently, the zcomponent of momentum transferred across unit area of the xy-plane in both directions in unit time would be the pressure RT/v. (7) From equation (2), in the real,liquid this is replaced by (8) RT/(v-5) Now the mass of the molecule and the temperature (average kinetic energy) are the same in both cases. Hence the increased momentum (8) compared with (7), canionlybedueto an increasein the numberof moleculescrossing in unit time. Since the same law of distribution of velocities (Maxwell's Law) is assumed in both cases, the ratio in which the numbers are increased must be the same v/(v-8) for all velocities. Thus the number of molecules with velocity components between w and w + dw crossing unit area of the xy-plane in unit time is given by equation (6). The velocity u of a given molecule parallel to the x-axis will not always be equal to the average velocity u0 of the liquid at that point. There will, however, be a series of instants (which we may call collisions) when u will be equal to u0. I assume that the interval between two such instants will be a half period (interval between libration limits)3 in the sense of the quantum theory, and that the molecules can be treated as moving with constant velocity between consecutive collisions. Suppose then a molecule reaching the xy-plane has been moving a time, t, since its last collision.

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MA THEMA TICS: H. B. PHILLIPS

P-POC. N. A. S.

From (4) its momentum parallel to the x-axis due to the motion of the liquid at that point will be m w t. From (6) the total x-component of momentum transferred in the positive direction aeross unit area of the xy-plane in unit time will then be m dw. There will be an equal transfer due to molecules moving in the negative direction. Hence

V-f w2tf(Ui)

2mW2 tf(w)dw.

(9)

The time t can be considered as the interval between collisions at times4 to, ti. Since the molecule is assumed to move with constant velocity between collisions, 2mw2t=2 mw2dt = nh, (10) where, according to Sommerfeld's theory, n is an integer. Also ff (w) dw=N/2M, (11) since it is the number of molecules per gram for which w is positive. Combining (9), (10), and (11), we get (1) which was to be proved. 3. Owing to lack of data on the equation of state, the only substances on which the formula can at present be tested are carbon dioxide, ether, and mercury. Using the values of N and h given by Birge,5 3Nh = 3 (6.0594) (6.5543) 10-4= .011914, and so equation (3) takes the form

i7(v-a)=.011914/M.

(12)

Values of 5 for carbon dioxide and ether,6 tables I and II, were supplied by Professor, Keyes. At low temperatures the measured viscosities and those calculated by equation (12) do not differ by more than the experimental error. In case of carbon dioxide the temperature 300 is too near the critical point (t=310) for satisfactory use of the equation of state. Above 100 the calculated viscosity of ether is too large, the difference increasing with the temperature. This may be due to the fact that ether is a complex of more than one type of molecule. The equation of state was determined on the assumption that each liquid molecule is formed by the combination of two gas molecules. This may be substantially true at 100 and not at 1000. To obtain values of v-a for mercury, I make use of the fact that monatomic substances seem to have constant co-volumes7 B. Since mercury is monatomic in the gas phase, I assume that a is constant or nearly constant in the liquid phase. Also the term RT/(v-5) in case of mercury is very large (more than 15000 atmospheres). Hence at atmospheric pressures we may neglect p and so write (2) in the form

MA THEMA TICS: H. B. PHILLIPS

VOL. 7, 1921

'

T a =c (V-i

where c is constant. Differentiating this equation with respect to v and then with respect to T, assumning a constant, we get

(V6) 2 (3 d + 2 T-d 2) T(V.-b) (dt-T (d)=of whence -dv IdV\ 2Td2¶v + id+ dt 3t2 dv 3 + Td2v

dv V-S=Tdt

j(13)

According to Callendar and Moss8

V I+ 12444 100 o 108 +2539 \100/ 810" (14) VO=1+1805553 \100/ 10 is the ratio of the volume of mercury at t° to its volume at 00. The values of v-a in table III are obtained by calculating the derivatives of v from (14) and substituting in (13). Below 1000 the difference of the calculated viscosities and those measured is not greater than the difference between the values obtained by different observers. At higher temperatures the calculated viscosities are consistently smaller than those measured. The observations, however, differ greatly, owing probably to oxidation of the mercury in contact with the air. The result may be to increase the viscosity by almost any amount. This work should be repeated with mercury of the greatest purity in a vacuum or in contact with some inactive gas. TABLE I

CARBON DIOXI DU

lo TEMP. PIR*UURI

09780

a

0.1978, (v-6) v

1

1-5

5

Sat.

1.113

.278

10 15 20 20 20 20 30 30 30

" " "

1.168 1.228 1.306 1.291 1.224 1.189 1.566 1.339 1.274

.317 .360 .419 .408 .358 .332 .627 .445 .395

59 at 72 83 72 90 104

,jslOS CAL. 974

0002708 ; lOS OBS.

OBSERVBR

925

WarburgandBabo9

854

852

751

784 712 697 771

646 664 756 816 432 608 686

823 458 643

733

Phillips"

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MA THEMATICS: H. B. PHILLIPS

PRoc. N. A. S.

TABLE II

ETHER

log a = .48959 TEMP.

9

t-I

00 10 20 30 30 30 40 50 60 70 80 90 100

1.3583 1.3797 1.4016

.0573 .0612 .0663 .0703 .0703 .0703 .0761 .0828 .0888 .0961 .1058 .1177 .1302

1.4247 1.4247 1.4247 1.4506 1.4784 1.5019 1.5309 1.5620 1.6000 1.6380

-

.50981,

s(v-)C.0001608

;xlOS

IXlO'

CAL.

OBS.

OBSERVER

2806 2860 Thorpe and Roger" 2626 2585 2425 2345 2287 2120 2287 2134 Pound12 2287 2150 Heydweiller'3 2113 1970 1942 1810 1811 1666 1673 1528 1519 1400 1366 1284 1235 1177 TABLE III MERcURY i(v-8) =.00005935

TEMP.

U-8

V-CAL.

-18.1

.0033282

.01784

.01836

.00006111

Koch'

.0035606

.01661 .01688

.00005914 .00006010 .00005817 .00005975 .00005898

PHUSS1

.0035606 .0036885 .0036897 .0037538 .0037557 .0037635 .0037661 .0037668 .0037687 .0037700 .0037717 .0038006 .0038165 .0038165

.01667 .01667 .01609 .01609 .01583 .01581 .01578 .01577 .01577 .01576 .01575 .01574 .01562 .01555 .01555

.0039967 .0040732 .0043682 .0045817 .0048225 .0048362 .0051589 .0053304 .0055456-

.01485 .01457 .01359 .01295 .01230 .01227 .01151 .01114 .01070

.01476 .01483 .01360

0 0 10 10.1

14.95 15.1 15.7 15.9 15.95 16.1 16.2 16.5 18.76 20 20 34.05 40 62.9 79.4

97.95 99

"BS.

.01577 .01620 .01571 .01575 .01570

.01583 .01581 .01563 .01563 .01572 .01558 .01547 .01579

.01299 .01263 .01227 .01171 .01167 .01092

-5)

OBSERVER

Koch Umani Koch

Benard16

.00005914 .00005909

.00005961 .00005954 .00005889 .00005892 .00005929 .00005921 .00005904 .00006026 .00005899 .00006041 .00005941 .00005952

" " PlUss " Schweidler Pliss Schweidler P1Uss

.00006091

Koch .00005934 .00006041 " .00006221 Pl1iss Koch .00006056 av. .00005968 'F. G. Keyes,"A New Equation of Continuity," Proc. Nat. Acad. Sci., 3, 1917 124 137.5 154

(323-330).

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'Jeans, Dynamical Theory of Gases, p. 247. 3Adams, "The Quantum Theory," Bull. Nat. Res. Council, 1, No. 5, Oct., 1920. 4 Jeans, hc. cit., p. 246. 6 Ithaca, Physic. Rev., 14, p. 368. 6WWm. A. Felsing (Thesis), Technology Press, June, 1918. 'Keyes, loc. cit. 8 Proc. R. Soc. London, 84A, p. 595. ' Leipzig, Ann. Physik, 17A, 1882 (418). 10 Proc. R. Soc. London, 87A, pp. 556-57. " Phil. Trans. R. Soc., 185A, p. 572. "2J. Chem. Soc. London, 99, p. 708. 1 "Ann. Physik, 59, p. 200. "Landolt Bornstein (1912). 16 Z. Anorg. Chem., 93, p. 18. 16Brillouin, Leqons sur la Viscosite', pp. 152-159. 7rHE ANGULAR DIAMETER OF ALPHA BOOTIS BY T HE INTERFEROMETER By F. G. PUASE MOUNT WILSON OBSZRVATORY, CARNZGII: INSTITUTION OF WASHINGTON Communicated by G. E. Hale, May 2, 1921

Since the measurement of the diameter of Betelgeuse' in December, 1920, observations for the determination of stellar diameters have been continued with the 20-foot interferometer attached to the upper end of the 100-inch Hooker reflector. Several definite results have been obtained, and others, less definite because of poor observing conditions, have been partially checked by comparison with stars whose diameters are known to be too small for measurement with this instrument. Let the visibility of the zero fringes for any particular separation of the mirrors be reduced by seeing which is poor as compared with that on a fine night; it is then hopeless to make final measures. If, however, the interferometer is turned to a neighboring check star and the fringes are seen, and then turned to the star under consideration and they do not appear, the observer is justified in saying that there is a definite decrease in visibility. The data thus far obtained are as follows: For a Tauri (Aldeberan) fringes of gradually decreasing visibility have been observed at 13, 14.5 and 19 feet. Further observations are necessary between the last two positions to determine whether the fringes vanish between these points or whether a longer beam will be necessary to obtain a measure of the diameter. If the fringes vanish around 16-18 feet, as observations under poor conditions lead one to suspect they will, then the fringes observed at 19 feet must lie beyond the point of disappearance, on the ascending branch of the visibility curve where it rises toward the second maximum.