The capillary height method

The capillary height method. The approximate theory of this method was given in Chap. I, 12. It was shown that the curvature of the meniscus of the liquid in the tube determines the pressure immediately 

When the radius of the tube is appreciable, equation (2) requires correction, because the meniscus is no longer spherical. In Fig. 57, let b be the radius of curvature of the lowest point O of the meniscus (the two radii will be equal at this point, since the tube is cylindrical and the point is on the axis of revolution). The pressure immediately under the centre of the 

1 D is the density of the denser phase, and d that of the lighter, throughout this chapter. If the liquid is taken in contact with air, d is the density of air if in a vacuum, d is the density of the saturated vapour at the temperature of the experiment. In these cases d may often be neglocted without serious loss of accuracy. The equations also apply to interfaces between two liquids in which case d is the density of the lighter liquid. 

The symbol Dx denotes the density below the vertex of a fluid interface, and Dt that above the vertex. 

A few writers, including Rayleigh, have used another capillary constant, with the same symbol a, half that defined above this is, however, much the commonest. 

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There are static and dynamic methods. The static methods measure the tension of practically stationary surfaces which have been formed for an appreciable time, and depend on one of two principles. The most accurate depend on the pressure difference set up on the two sides of a curved surface possessing surface tension (Chap. I, 10), and are often only devices for the determination of hydrostatic pressure at a prescribed curvature of the liquid these include the capillary height method, with its numerous variants, the maximum bubble pressure method, the drop-weight method, and the method of sessile drops. The second principle, less accurate, but very often convenient because of its rapidity, is the formation of a film of the liquid and its extension by means of a support caused to adhere to the liquid temporarily methods in this class include the detachment of a ring or plate from the surface of any liquid, and the measurement of the tension of soap solutions by extending a film. 

Application to the capillary height method. Sugden1 has pointed out that, when the contact angle is zero, the ratio r/6 of the radius of the tube to the radius of the lowest point of the meniscus is the x/b of Bashforth and Adams s tables. By (6)... 

The above methods, with a wide reference tube from which to measure the height of rise, require much liquid. Sugden3 employed a U-tube with two limbs of different diameter, one about twice that of the other, comparing the height of rise in both. The corrections to the simple formula (1) were found from his tables, and the method appears capable of little less accuracy than the highest refinements of the capillary height method, if... 

Ferguson4 has also devised two ingenious methods for determining the surface tension of very small amounts (I or 2 cu. mm.) of liquid. The pressure required to force the meniscus, located at the end of a capillary tube, to a plane form, is measured it is equal to 2yjr for tubes so narrow that the meniscus would naturally be spherical. In one method the tube is vertical in the other it is horizontal, and in this modification the density of the liquid need not be determined. The contact angle must be zero, as in all variants of the capillary height method. 

Where great economy of material is desirable, Ferguson s methods, or Sentis s, ( 7) are perhaps the most economical Sugden s modification of the capillary-height method is also good and accurate. 

Harkins, W. D. and Brown, F. E., The determination of surface tension (free surface energy), and the weight of falling drops the surface tension of water and benzene by the capillary height method, J. Am. Chem. Soc., 41, 499-524 (1919). 

The Wilhelmy plate method [323,324], the sessile drop method [328,340], and the capillary height method [325-328] measure equilibrium surface tension, if sufficient time is allowed for the adsorption of surfactant molecules at the surface to attain the state of equilibrium. The Wilhelmy plate method measures the force exerted on a vertical plate partially immersed in the liquid (Fig. 9.20). If wetting of the plate is complete, the force, F, is proportional to the surface tension, y, and the circumference, L. of the plate ... 

The approximation that limits this analysis of capillary rise originates from neglecting the weight of the liquid in the crown of the curved meniscus. We see in Section 6.8b that the height of capillary rise can be related to surface tension without making this approximation, although the connection is somewhat unwieldy. A more detailed description of the experimental aspects of the capillary rise method can be obtained from advanced textbooks (e.g., Adamson 1990). 

As discussed in Section 4.4, when a solid capillary tube is inserted into a liquid, the liquid is generally raised (or rarely depressed) in this tube. In the capillary rise method, the height of a liquid column in a capillary tube above the level of the reference liquid contained in a large dish is measured. The container must be sufficiently large so that the reference liquid... 

While determining the liquid height, it is better to measure with a falling (or receding) meniscus, so that the liquid level is initially raised above its equilibrium value by a slight suction above the capillary tube, and then left to equilibrate. On the other hand, two-armed capillary tubes, connected with a cross tube above the liquid level, are also used to ensure that the pressure in both arms of the glass apparatus is the same. An interesting modification of the capillary rise method is to measure the pressure, AP, that is required to force the meniscus down until it is on the same level as the plane surface of liquid outside the capillary tube. This method is useful to compare the surface tension of water and its dilute solutions. 

From Eq. (10.2.4) it is evident that for small capillaries Hg can become relatively large. For example, for water with o- = 73mNmin a 0.1-mm-radius clean glass capillary the liquid will rise to an equilibrium height of about 0.15 m. The capillary rise method is one of the most accurate means for the measurement of surface tension. 

The time required to reach equilibrium is much reduced through the use of novel technology in commercially available automatic membrane osmometers. If, for example, the capillary height in the solution chamber increases because solvent permeates from the solvent chamber, this is immediately compensated by the application, via a servomechanism, of a pressure on the solution chamber, such that the capillary heights above solvent and solution remain the same. Since this method involves the transport of only very small amounts of liquid, equilibrium is reached after only 10-30 min. 

In the capillary rise method, the surface tension, 7, of a liquid can be determined from the height, h, of the liquid column in a capillary tube of radius r. If the liquid completely wets the tube (zero contact angle). 

FIGURE 6.10. In the capillary rise method of surface tension measurement, surface tension effects canse the wetting hquid to rise in the small capillary to a height that just balances the hydrodynamic force dne to gravity (a). For non-wetting liquids such as mercury, a depressing effect is observed (Z>). 

Some variants of the capillary rise method have been employed. In the differential capillary rise technique, two tubes of different diameters are used, and the difference in height between the two menisci is measured. In the inclined capillary rise scheme, the tube is inclined at a known angle so that the length Q of the tube occupied by the wetting fluid is greater than the height h of the rise in a vertical tube. This method is useful when the interfacial tension is low because under these conditions the vertical rise h is small and difficult to measure accurately. 

The capillary rise method is a classical example of category (2) for measuring liquid-vapor interfacial tension, in which the liquid height in a capillary tube is measured [27]. On the other hand, the geometrical shape of the... 

The capillary rise method was the earliest technique by which surface tension was measured and, indeed, was the technique by which the force itself was recognized. If a narrow tube of radius r is partially inserted into a liquid, the liquid rises up inside the tube to some equilibrium position as shown in Fig. 22. This occurs because the attractive interaction of the wetting liquid (aqueous solution) with the solid surface is stronger than that of the gas phase. Gravity opposes the rise, and the equilibrium height H corresponds to the minimum free energy of the system. The treatment is based on the Laplace equation that gives the pressure difference across a curved interface due to the surface or interfacial tension of the liquid [62]. Let us assume that we have a spherical bubble Of gas in a liquid... 

In the capillary rise method, surface tension can draw liquid up the tube in a phenomenon known as capillary action. The height of the liquid column is given by... 

FIGURE 3 Temperature dependence of the surface tension of p-anisaldazine. A Jaeger [6], maximum bubble pressure method (Tni = 180°C) B Ferguson and Kennedy [7], capillary height method (Tni = 180.5°C) C Krishnaswamy and Shashidhar [17], pendant drop method (Tni = 180°C). 

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