It may be shown that the constants denoting the surface energy and the surface tension are the same. To do this, consider a film of liquid bounded by the sides of a rectangle, one of which can be moved so as to increase or dimin ish the area of the film. Since both faces of the film act alike, we need consider only one of them. The force applied by the tension T in the film to the movable side, the length of which is represented by s, is Ts; and if the movable side moves in toward the opposite side through the distance r, the work done by the surface tension is Tsr. This is, therefore, the measure of the change in the energy of the film, and since sr is the change of area, T equals the energy per unit of area, or the surface energy.
The method of Gauss furnishes a proof that the contact-angle should be constant. If we consider that the potential energy of the system is a minimum, when the surface tensions which arise from die surface energies are in equi librium, this may be proved in the way already indicated; or a direct proof may be given.
The argument by which these conclusions have been reached fails if we take into consid eration the heat that must be introduced into the surface film to keep it at constant temperature during its enlargement. When this heat is taken into account it appears that the surface energy differs from the surface tension by a quantity equal to the product of the absolute temperature, and the rate at which the surface tension changes with the temperature. In all known cases the surface tension decreases as the tem perature rises and the surface energy is greater than the surface tension. The surface tension, while not equal exactly to the total surface energy, is equal to the so-called free energy of the surface. Since equilibrium depends on the free energy having a minimum value, the test of equilibrium based on the condition that the area of the liquid surface shall be a minimum, consistent with the boundary conditions, is not impaired by this modified statement.
A very interesting set of verifications of the theories of capillarity was devised by Plateau. In order to be able to examine a liquid taking shape under its surface tension only, he pre pared a mixture of alcohol and water having the same density as olive oil, in which the oil could be suspended. A mass of oil freely float ing in this mixture, assumed a spherical form. This form is manifestly that which would be produced by a tension acting uniformly in all parts of the surface; it is also that for which H t the internal pressure represented by— — 2 R R' is the same everywhere; and also that for which the surface, and consequently the potential energy, is a minimum. When the oil was sus
pended in a wire frame, it assumed various forms, depending on the shape of the frame and the quantity of oil, which were always such that the internal pressure, determined by Laplace's equation, was the same everywhere.
A similar set of verifications was afforded by the use of films of soapy water. Such films are so thin and light that their weight hardly distorts them at all, and the positions they as sume are due almost solely to the surface ten sion. Such a film, blown into a bubble, is spherical. When formed on a wire frame lying in a plane, the film is a plane. When the frame is twisted out of the plane, the surface of the film is the least that can be constructed with the edges of the frame as a boundary. It is one of the so-called minimal or ruled surfaces. Various films of this sort were examined by Plateau, and found to fulfil the geometrical conditions of the minimal surface.
Observers have ordinarily tested the theory by determining, from Laplace's equation, the various forms and dimensions of liquid sur faces, subject to various boundary conditions, and comparing the actual forms obtained by experiment with those deduced from the theory. For example, rough observations show that for any one liquid that wets glass, the heights to which it rises in various capillary tubes are inversely as the radii of the tubes, as the ele mentary theory declares they should be. More refined observations show that this statement is not strictly accurate, and a more complete theory leads to certain corrections of the state ment, to which the better observations conform. In a similar way, the rise of a liquid between parallel plates, the forms of large drops of mercury on a horizontal plate, or of large bubbles of air in a liquid under a horizontal plate, the force needed to lift a horizontal plate from the surface of a liquid which wets it, the maximum pressure exerted in a small bubble as it is enlarging in a liquid at the end of a tube, have all been used as means of testing the theory. Generally the observations are used in the appropriate formula to obtain a value for the surface tension 7', or for the constant a' = 2 T (Called Poisson's constant) and the g verification of the theory is found in the fact that the values of these quantities obtained by different methods are in good agreement with one another.