If we take a body of any shape and select a square-bounded element of the surface, the tensions in the plane of the diagram are equivalent to a normal force ads' and those acting on the 1 other two edges to (rds2X Hence per unit area Here and are the radii of curvature in two planes at right angles. Owing to Gauss's theorem of integral curvature the sum of two such curvatures at a point is a constant. The above the orem is therefore true whether the two rectangular planes con sidered are principal planes or otherwise. In the case of a sphere P = R Effect of Temperature.—The variation of surface tension with temperature was not taken into account in Laplace's theory. The forces between moleCules were taken, like gravitational forces, to be independent of temperature. The rise in a capil lary tube was supposed to vary merely because the density varied. Experiment has shown, however, that the tension itself in all cases diminishes with rise in tempera ture. That it must ultimately diminish to zero can be inferred from the fact that according to the usual conceptions as to the critical state a liquid and its vapour will at that point become identical with one another; the tension at the interface must then be zero. Eotvos introduced the conception of molecular surface tension. By means of some what doubtful reasoning he concluded that o (Mv) (where My is the molecular volume), plotted against temperature should give the same kind of curve for different substances, and he found in fact that it was representable by where T, is the critical temperature and K a constant. According to Ramsay and Shields K should be the same constant for all non-associated substances. In reality K varies betyeen 1.5 and 2.6 and it is necessary also to change to 6 where 8 is a small constant, so that the law is only a rough one. According to this equation the curve would be a straight line cutting the temperature axis a few degrees below the critical point.
In connection with this, it is important to remember that a is only the free energy per unit area and not the total energy per unit area (u). Lord Kelvin was the first to prove that the varia tion of a with temperature, involves that u must be greater than a ; or in other words, when an expansion of the surface takes place heat must be added to keep the temperature constant. The connection between these quantities he proved thermo dynamically to be u = a —T (vide THERMODYNAMICS applied dT to chemistry). Now, at the critical point, u (which is only the extra energy due to the existence of unit surface and not the total energy of the whole body) must also vanish. But if both u and a are then zero so also must T be zero.
Hence, if our conceptions of the critical state are correct the curve of a plotted against T instead of being a straight line must become horizontal at the critical point. In 1894, van der Waals showed that a formula of the type a= A was to be expected and gave the average value of n= 1.27 as deducible from the experimental values. In 1916 A. Ferguson (Phil. Mag., Jan. 1916), examining the data for 14 different organic substances showed that the best values of n ranged from 1.187 to 1.248. The advantageous character of this power equation is that it makes a, u, and all equal to zero at the critical point and it dT thus satisfies all that is required of it theoretically. Liquids,
however, which are believed to be associated do not follow the above simple law.
The surface energy, u, also falls to zero at the critical point, but according to a different law. For taking the equation da u = —T dT it follows that u= A — l]T). The curves and u are shown in fig. 5 for the case of benzene. The curve shows how nearly a straight line law satisfies the values of a until the critical point is closely approached. An interesting relation has been given by Prof. E. T. Whittaker between u and the latent heat (internal) of evaporation. The internal latent heat (Li) is obtained by subtracting the external work done from the ordinary u latent heat. The relation in question is that • 7 is nearly a reality depend. It is appropriate to summarize here the leading characteristics of this property.
The tension of a liquid across any straight line drawn on the surface is normal to that line and is the same for all directions of the line and is measured by the force across an element of the line divided by the length of that element.
For any given homogeneous liquid surface, as the surface which separates water from air or oil from water, the surface tension is the same at every point, of the surface and in every direc tion. In the case of mixtures, however, when circumstances de mand it (e.g., when the surface is inclined and the effects of a gravi tation field require consideration) the concentration at the surface may adjust itself from point to point of the surface in such a way as to produce a corresponding variation in the value of the surface tension sufficient to satisfy the conditions of equilibrium. This effect is very minute and in nearly every problem may be ignored.
When the surface is curved the effect of the surface tension is to make the pressure on the concave side greater than that on the convex side by the amount a where and C2 are the curvatures in mutually perpendicular normal planes.
The tension of the surface separating two liquids cannot be deduced by any reliable method from the tensions of the liquids when separately in contact with air. The experiments of C. G. M. Marangoni, van der Mensbrugghe, and Quincke have led to results which show that the common surface between two liquids has a tension always less than the difference of the tensions of the separate liquids. This is usually referred to as Marangoni's rule.
If three liquids meet along a line the interfaces at the corn mon edge must be parallel to the sides of a triangle proportional to the three tensions a12, otherwise equilibrium cannot obtain (Neumann's rule). This triangle cannot exist unless two of the interfacial tensions are greater than the third. Marangoni's experimental rule shows that the triangle is always imaginary. Hence three pure fluids (of which one may be a gas) accordingly cannot remain in contact. If a drop of oil stands in a lenticular form upon a surface of water, it is because the water-surface is already contaminated with a greasy film.