Liquid Column in a Capillary

FIGURE 10.12. Column of a reactive fluid inside a capillary tube. 

At point A, the reaction has proceeded for a duration t = L/V. To estimate we resort to a simple rate equation 

In this equation, /c is a kinetic constant and c is the concentration of the reagent in the liquid. The factor (1 — ) describes the decrease in the number of available surface sites sustaining the reaction. For a surface continuously exposed during a time t, the solution is 

Typically, the chemical reactions involved are relatively slow. The time constant r is a few minutes at least. By contrast, the time t it takes for a drop to pass through is rather short (of the order of O.I s). Since i C r, we may simplify equation (10.41) to 

Our description of the fundamental principles is now nearly complete. We write a relation (at point A) between force and velocity, analogous to equation (10.37)  

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Capillarity — (a) as a branch of science, it concerns the thermodynamics of surfaces and - interfaces. It is of utmost importance for - electrochemistry, e.g., treating the electrode solution interface (- electrode, - solution), and it extends to several other branches of physics, chemistry, and technical sciences [i]. The thermodynamic theory of capillarity goes back to the work of Gibbs, (b) In a practical sense capillarity means the rise or fall of a liquid column in a capillary caused by the interplay of gravity and -> interfacial tension and also phenomena like capillary condensation [ii]. 

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... 

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). 

Geometry factors of concentric cylinder fixtures (ISO 3219) Average height of the liquid column in a capillary viscosimeter... 

It can be shown that r < h3, where h is the height of the mercury column, by considering the force of gravity and liquid flow in a capillary. So Ic h and l hx/2. This means that there is a minimum detection limit with current sampling this is around 10 7m (see Chapter 10 on pulse techniques). It is therefore important not to have too high a column of mercury. 

In Figure 4.7, the pressure at the liquid level and at point a just above the meniscus is atmospheric. The height of the column of liquid, h, will be such that the pressure at point c in the column is also atmospheric, due to the balanced effects of hydrostatic and capillary pressures in the liquid column. The pressure at point b, just below the meniscus, will be less than atmospheric by an amount (2ylr).kt equilibrium, AP is also equal to the hydrostatic pressure drop in the liquid column in the capillary. Thus,... 

Figure 4.7 Concave liquid meniscus in a capillary tube during a liquid surface tension measurement 6 is the angle of contact between the liquid and the capillary wall c is the point at the liquid level, a is the point just above, and b is just below, the meniscus level and h shows the height of the liquid column. The crown of the concave meniscus is the liquid between the top and the lower end of the meniscus. The term cp is the inclination angle, where (cp = 90° - 6) from plane geometry.

It follo vs from (17.13) that the smaller the radius of the capillary, the higher (or lower) the liquid can rise (or descend). Thus, for water, S = 73 mN/m, and in glass capillary with a = 0.1 mm we have H = 0.15 m. On the other hand, measuring the height of a raising (or descending) liquid column in the capillary, it is possible to determine the value of the surface tension coefficient with high accuracy. 

The rise or fall of a liquid interface in a capillary can be easily calculated by writing the balance of forces on a cylindrical column of height H (see Fig. 5). The water at the bottom of the tube, leveled with the free surface of the hquid, is at atmospheric pressure. The hydrostatic pressure drop just below the meniscus is balanced by the vertical component of surface tension at the wall. Therefore, 2nRycos9 = pginR H). Hence, the capillary rise is given by... 

The most common case is penetration under the action of gravity which affects the shape of the meniscus and the height of rise in the capillary. In a capillary-rise situation, AP is equal to the hydrostatic pressure drop in the liquid column in the capillary, as follows ... 

The curvature of the surface of the phase boundary creates a number of important so-called capillary phenomena. A large curvature of the surface of the phase boundary is characteristic of systems that contain small particles (highly dispersed systems). The result is, for example, an increased pressure (so-called Laplace pressure) inside the droplet of a certain radius due to the change in its size (radius). Another consequence is that the liquid substance in a capillary tube, with one end submerged in a container of Hquid, rises so high that the hydrostatic pressure of its column is equal to the Laplace pressure.In other words, adhesion forces between the fluid and the solid inner wall pulls the liquid column until there is a sufficient mass of liquid for gravitational forces to overcome these intermolecular forces. 

Now, when a liquid rises in a capillary tube and its diameter is sufficiently small, one knows that the surface atop the raised column does not differ appreciably from a concave half-sphere, whose diameter is consequently equal to that of the tube. Let us recall, moreover, part of the reasoning by which one arrives, in the theory of capillary action, at the law which relates the height of the column raised to the diameter of the tube. Let us suppose a very thin channel from the low point of the hemispherical surface... 

Already in 1830, an American scientist. Dr. Hough, had tried to arrive at the measurement of the pressure exerted either on a bubble of air contained in an indefinite liquid, or on the air contained in a soap bubble. He has a rather right idea of the cause of these pressures, however, he does not distinguish one from the other, and, to evaluate them, he starts, as I did, with the consideration of the hollow surfaces atop a column of the same liquid raised in a capillary tube but, although a clever observer, he did not know about the theory of capillary action also he arrives by reasoning whose error is palpable, at necessarily false values and a law. 

Fig. 4 illustrates the time-dependence of the length of top s water column in conical capillary of the dimensions R = 15 pm and lo =310 pm at temperature T = 22°C. Experimental data for the top s column are approximated by the formula (11). The value of A is selected under the requirement to ensure optimum correlation between experimental and theoretical data. It gives Ae =3,810 J. One can see that there is satisfactory correlation between experimental and theoretical dependencies. Moreover, the value Ae has the same order of magnitude as Hamaker constant Ah. But just Ah describes one of the main components of disjoining pressure IT [13]. It confirms the rightness of our physical arguments, described above, to explain the mechanism of two-side liquid penetration into dead-end capillaries. 

A sample to be examined by electrospray is passed as a solution in a solvent (made up separately or issuing from a liquid chromatographic column) through a capillary tube held at high electrical potential, so the solution emerges as a spray or mist of small droplets (i.e., it is nebulized). As the droplets evaporate, residual sample ions are extracted into a mass spectrometer for analysis. 

Since capillary tubing is involved in osmotic experiments, there are several points pertaining to this feature that should be noted. First, tubes that are carefully matched in diameter should be used so that no correction for surface tension effects need be considered. Next it should be appreciated that an equilibrium osmotic pressure can develop in a capillary tube with a minimum flow of solvent, and therefore the measured value of II applies to the solution as prepared. The pressure, of course, is independent of the cross-sectional area of the liquid column, but if too much solvent transfer were involved, then the effects of dilution would also have to be considered. Now let us examine the practical units that are used to express the concentration of solutions in these experiments. 

The height, h, of a column of liquid in a capillary tube can be estimated by using h = lylgdr, where y is the surface tension, d is the density of the liquid, g is the acceleration of free fall, and r is the radius of the tube. Which will rise higher in a tube that is 0.15 mm in diameter at 25°C, water or ethanol The density of water is 0.997 g-cm-3 and that of ethanol is 0.79 g-cm-3. See Table 5.3. 

With this idea in mind, the horizontal surface in Figure 6.3b can be taken as a reference level at which Ap = 0. Just under the meniscus in the capillary the pressure is less than it would be on the other side of the surface owing to the curvature of the surface. The fact that the pressure is less in the liquid in the capillary just under the curved surface than it is at the reference plane causes the liquid to rise in the capillary until the liquid column generates a compensating hydrostatic pressure. The capillary possesses an axis of symmetry therefore at the bottom of the meniscus the radius of curvature is the same in the two perpendicular planes that include the axis. If we identify this radius of curvature by b, then the Laplace equation applied to the meniscus is Ap = 2y/b. Equating this to the hydrostatic pressure gives... 

ELECTROCAPILLARITY. The surface tension between two conducting liquids in contact, such as mercury and a dilute acid, is sensihly altered when an electric current passes across Ihe interlace. As a result, when the contact is in a capillary tube, the pressure difference on the opposite sides of the meniscus is affected hy a current traversing the capillary column, to an extent dependent upon the direction of the current across the houndary. 

SFC has received attention as an alternative separation technique to liquid and gas chromatography. The coupling of SFC to plasma detectors has been studied because plasma source spectrometry meets a number of requirements for suitable detection. There have been two main approaches in designing interfaces. The first is the use of a restrictor tube in a heated cross-flow nebuliser. This was designed for packed columns. For a capillary system, a restrictor was introduced into the central channel of the ICP torch. The restrictor was heated to overcome the eluent freezing upon decompression as it left the restrictor. The interface and transfer lines were also heated to maintain supercritical conditions. Several speciation applications have been reported in which SFC-ICP-MS was used. These include alkyl tin compounds (Oudsema and Poole, 1992), chromium (Carey et al., 1994), lead and mercury (Carey et al., 1992), and arsenic (Kumar et al., 1995). Detection limits for trimethylarsine, triphenylarsine and triphenyl arsenic oxide were in the range of 0.4-5 pg. 

In the special case of a fluid rising in a capillary tube we have the following situation. The upward pull due to surface tension must balance a column of liquid with height, say h, and density p0. Since the tension is exerted along the contact of the liquid with circumference of the tube, then the total upward pull due to this tension is 7r Dcc where Dc is the diameter of the capillary and a the surface tension. This upward pull must equal the downward gravitational pull on the mass of... 

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