Page:Encyclopædia Britannica, Ninth Edition, v. 5.djvu/78

66 CAPILLARY ACTION two surfaces. Since in the case of thin films the outer and inner surfaces are approximately equal, we shall con sider the area of the film as representing either of them, and shall use the symbol T to denote the energy of unit of area of the film, both surfaces being taken together. If T is the energy of a single surface of the liquid, T the energy of the film is 2T . When by means of a tube we blow air into the inside of the bubble we increase its volume and therefore its surface, and at the same time we do work in forcing air into it, and thus increase the energy of the bubble. That the bubble has energy may be shown by leaving the end of the tube open. The bubble will contract, forcing the air out, and the current of air blown through the tube may be made to deflect the flame of a candle. If the bubble is in the form of a sphere of radius r this material surface will have an area S = 47ir 2 ....... (1). If T be the energy corresponding to unit of area of the film the surface-energy of the whole bubble will be ST = 47rr 2 T ....... (2). The increment of this energy corresponding to an increase of the radius from r to r + dr is therefore (3). Now this increase of energy was obtained by forcing in air at a pr-essure greater than the atmospheric pressure, and thus increasing the volume of the bubble. Let II be the atmospheric pressure and 11+;? the pressure of the air within the bubble. The volume of the sphere is V=|irr>. (4), and the increment of volume is (5). Now if we suppose a quantity of air already at the pressure II -f p, the work done in forcing it into the bubble is pdV. Hence the equation of work and energy is pdV = Tds (6), or or ?) 9T /Q ? - zl r ( 8 )- This, therefore, is the excess of the pressure of the air within the bubble over that of the external air, and it is due to the action of the inner and outer surfaces of the bubble. We may conceive this pressure to arise from the tendency which the bubble has to contract, or in other words from the surface-tension of the bubble. If to increase the area of the surface requires the expenditure of work, the surface must resist extension, and if the bubble in contracting can do work, the surface must tend to contract. The surface must therefore act like a sheet of india-rubber when extended both in length and breadth, that is, it must exert surface-tension. The tension of the sheet of india-rubber, however, depends on the extent to which it is stretched, and may be different in different directions, whereas the tension of the surface of a liquid remains the same however much the film is extended, and the tension at any point is the same in all directions. The intensity of this surface-tension is measured by the stress which it exerts across a line of unit length. Let us measure it in the case of the spherical soap-bubble by con sidering the stress exerted by one hemisphere of the bubble on the other, across the circumference of a great circle. This stress is balanced by the pressure p acting over the area of the same great circle : it is therefore equal to irr^p. To determine the intensity of the surface-tension we have to divide this quantity by the length of the line across which it acts, which is in this case the circumference of a great circle 2-Trr. Dividing 7r? i2 /> by this length we obtain pr as the value of the intensity of the surface- tension, and it is plain from equation 8 that this is equal to T. Hence the numerical value of the intensity of the surface-tension is equal to the numerical value of the surface-energy per unit of surface. We must remember that since the film has two surfaces the surface-tension of the film is double the tension of the surface of the liquid of which it is formed. To determine the relation between the surface-tension and the pressure which balances it when the form of the surface is not spherical, let us consider the following case : Let fig. 8. represent a section through the axis Cc of a soap-bubble in the form of a figure of revolution bounded by two circular disks AB and ab, and having the meridian section APa. Let PQ be an imaginary section normal to the axis. Let the radius of this section PR be y, and let PT, the tangent at P, make an angle a with the axis. Let us consider the stresses which are exerted across this imaginary section by the lower part on the upper part. If the internal pressure exceeds the external pressure by p, there is in the first place a force iryp acting upwards arising from the pressure p over the area of the sec tion. In the next place, there is the sur face-tension acting downwards, but at an angle a with the vertical, across the circular section of the bubble itself, whose circumference is 2-n-y, and the downward force is therefore 2-n-yT cos. a. Now these forces are balanced by the external force which acts on the disk ACB, which we may call F. Hence equating the forces which act on the portion included between ACB and PRQ ir f-2) - ZTryT cos. a=-F (Q). If we make CR = z, and suppose z to vary, the shape of the bubble of course remaining the same, the values of y and of a will change, but the other quantities will be con stant. In studying these variations we may if we please take as our independent variable the length s of the meridian section AP reckoned from A. Differentiating equation 9 with respect to s we obtain, after dividing by 2rr as a common factor Now The radius of curvature of the meridian section is The radius of curvature of a normal section of the surface at right angles to the meridian section is equal to the part of the normal cut off by the axis, which is ?/ J| _p>q-___^ /T0 2 pns n /" Hence dividing equation 10 by y sin. a, we find

This equation, which gives the pressure in terms of the