NOTE ON THE MEASUREMENT O F SURFACE TENSIONS BY ROBERT B. ELDER
,
By the use of Jaeger's Method, Fergusonl has obtained some results bearing on the question of the effect of adsorbed gases on the surface tension of water, recently discussed by Mr. S. S. Bhatnagara2 Ferguson measured the surface tension of water in contact with air and with COz a t 15" C and obtained 73.88 and 73.04 dynes/cm respectively as the tension of these interfaces. Bhatnagar's results are 73.1 and 73.00 for the same interfaces a t the same temperatures. Bhatnagar also gives results on other gases besides air and COZ in contact with water, and E'erguson gives results on other liquids besides water in contact with air and with COz. There is reason to question the accuracy of Bhatnagar's results, however, as his method involves certain errors, which are partially compensating and evidently for th'is reason yield results in close agreement with those obtained by other methods. Bhatnagar's method, briefly, is as follows: A segment of a sphere of known radius is suspended from a Jolly balance spring. A dish containing the liquid whose surface tension is to be measured is slowly raised till contact is made between the liquid surface and the sphere. The dish is then lowered till this contact is broken, the segment of the sphere being pulled from the liquid surface by the spring, and the distance the dish is moved vertically from the making to the breaking of this' contact is taken as the amount the balance spring is extended. Knowing the constant of the spring the downward force exerted on the sphere by the liquid can then be found. The mistake in this lies in the assumption that the sphere and dish are in the same relative positions at the making and a t the breaking of the contact. This might be
* Phil. Mag., (6) 28, 403 (1914). * Jour. Phys. Chem., 24, 716 (1920).
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true in a very special case, but in general is not true as will be shown. Ferguson’s formula, which Rhatnagar uses in interpreting the experimental results described is, .-
and Bhatnagar explains that, in this formula “M = weight required to detach the circular disk from the surface of water.” This is a mistake, as this equation represents the condition of equilibrium when the sphere is in contact with the liquid, and its lowest point is in a line with the level portion of the liquid surface-(it being assumed that the angle of contact between the sphere and liquid is zero). A careful analysis of Ferguson’s derivation and use of this formula will show that it applies only when relative positions of the sphere and the liquid surface are as stated. The following quotation2 describes his application of this formula : “In order to apply equation (1) to determine surface tensions, a large hollow sphere of glass was taken and its external radius measured by means of a spherometer. No appreciable variation in the radius could be detected over the portion which was subsequently in contact with the liquid under examination. The sphere, after having been cleansed by alternate washings with caustic soda solution and distilled water, was dried and fastened by a thread about 20 cm in length to the underside of the pan of a balance standing on a high shelf, the thread passing through holes in the baseboard of the balance of the shelf. On a lower shelf, underneath the sphere, stood a small table whose height could be adjusted by Ferguson: Phil. Mag., (6) 26, 925 (1913) R = Radius of sphere, p = Density of the liquid
T,
a 2
= - , in which Pi?
constant.
I,.
c., page 931.
T = surface tension in dynes/cm and g is the gravity
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means of a screw which table carried a carefully cleaned porcelain basin containing the liquid under examination. “The sphere having been first counterpoised, the balance was left perfectly free and the height of the table adjusted until the liquid just touched the vertex of the sphere. This point was sharply defined by the sudden swing of the balancepointer as the sphere touched the liquid and was pulled down. (In order to obtain consistent results, great care must he taken that the balance is quite at rest and the pointer exactly over the zero at the instant when the sphere and liquid make contact.) Weights were then added till the pointer was brought back to the zero, the mass of these weights being the M of equation (l).” In a later paper Ferguson’ deals with the general case of the sphere in contact with liquid, and shows that the maximum downward force exerted on the sphere is developed when the lowest point of the sphere is at a distance, d l = a2
(AR - k) ,
from the horizontal surface of the liquid.
Ferguson2 fuither states that the “pull is therefore a maximum when the vertex of the sphere is slightly below the level of the free horizontal surface of the liquid.” (The italics are the present writer’s.) This is an inaccurate statement of the case, as it will be seen that dl will be positive or negative depending on the relative values of R and a. If R is less than 3Ga the ex1 pression is negative, and hence d, is negative G 4aR i. e., the lowest point of the sphere is above the “free” horizontal surface of the liquid. For water a t 20” C, T = 72.8, T 72.8 p = 1.0, g = 981, and u 2 = - = - - 0.0742. Then a = P -R 981 -_40.0742 = 0.2724, and 3Ga = 9.8 approximately, so that with water and any sphere smaller than about 0.8 cm radius, the
-(
k)
Phil. Mag., (6) 28, 149 (1914). L. c., page 153.
Note on the Measurewent of Surface Tensions
561
maximum force is developed with the lowest point of the sphere above the level surface of the liquid, and not below, as Ferguson states. The sphere Ferguson used in this test was 7.321 cm radius, and that used by Bhatnagar was 5.007 cm radius. Substituting 5.007 for R in the above formula we find that the maximum forces would be developed on Bhatnagar’s sphere with the lowest point of the sphere at a distance (approximately) 0.0042 cm vertically above the horizontal portion of the liquid surface. It is obvious that Ferguson’s experiment with the sphere suspended from the balance arm would have been impossible if the maximum development of forces occurred when the lowest point of the sphere was below the level portion of the liquid surface. It would, in fact, be impossible under otherwise similar conditions, and using a sphere larger than 9.8 cm radius. Considering the case of a segment of a sphere suspended from the spring of a Jolly balance, as is the case in Bhatnagar’s experiment, it is evident t h a t the rupture between the sphere and the liquid in the dish will not take place before the forces acting on the sphere are a t their maximum. This maximum occurs when the first derivative of the forces pulling down becomes zero, but, since the force pulling the sphere up also varies differentially with the vertical motion of the sphere, rupture does not occur a t the point of maximum development of the forces, because at this point a slight upward movement of the sphere will decrease the force pulling up more than it decreases the force pulling down. In general, for a solid of any shape being pulled from a liquid surface, the position in which rupture occurs is fixed by the equality between the first derivative of the force pulling the solid up and that of the forces pulling it down, rather than by the maximum development of these forces. In the case of the gravity balance, the derivative of the force pulling up is zero, and the rupture occurs when the forces pulling down reach a maximum, that is, when their derivative becomes equal to zero. If the force pulling up varies with the differential vertical motion of the solid, the
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position of rupture is that position after the maximum development of forces is obtained in which the derivative of the forces pulling down equals that of the forces pulling up. I n Bhatnagar’s experiment the rupture between the liquid and the sphere comes after the maximum development of forces and when the lowest point of the sphere is above the level or “free” portion of the liquid surface. The level liquid surface is lower in the dish at the time of breaking the contact than at the time of making it, since some of the liquid is lifted up with the sphere. The amount of this lowering of the liquid surface in the dish would depend, among other things upon the size of the dish. These two errors, i. e., the lowering of the liquid surface in the dish and the lifting of the lowest point of the sphere above the level liquid surface a t the time rupture occurs, tend to compensate each other under the conditions of Bhatnagar’s experiment. In addition to these errors there is the misinterpretation of Ferguson’s formula, so that Bhatnagar’s results are of little value as to their absolute accuracy. Their relative values as compared with one another are doubtless correct. The lowering of the level liquid surface in the dish due to some of the liquid being lifted by its contact with the sphere would also be a factor for consideration in Ferguson’s experiment with the segment of the sphere suspended from the pan of the gravity balance, and Ferguson has apparently overlooked this. In this case, however, the area of the dish might have been so large that the error introduced by neglecting this factor would have been negligible, which is not likely to have been the case in Bhatnagar’s experiment. Idaho School of Mines Moscow, Idaho 1 For a mathematical discussion of the forces involved in the case of a wedge-shaped solid suspended from a Jolly balance spring, and a method of measuring surface tensions thereby, see Pamphlet No. l, of the Idaho Bureau of Mines and Geology-“Interfacial Tension Measurements and Some Applications to Flotation” by R. B. Elder, and for further discusRion of the subject and description of a n apparatus for accurately and rapidly measuring surface tension by this method, see “Surface-Energy and Adsorption in Flotation” by A . W. Fahrenwald: Min. and Sci. Press, 123, 127 (1921).
