But when the liquid wets one and not the other, the surface is lowered on the inside of the wetted one and raised on the inside of the other so that in each case the pressure on the inside. is greater than on the outside and the two are urged apart.

269. Soap Films. Some most interesting illustrations of surface tension are found in the phenomena of soap films. When a loop of thread is laid on a soap film formed in a wire ring and the film is broken inside the loop, the latter will be drawn into an exact circle, for it is pulled equally in every direction by the contracting film. And this circular loop may be moved

FIG. 148.-Loop in soap film.

FIG. 149.-Cylindrical bubble.

from one part of the film to another without changing shape, showing that the tension does not depend on the width of the film.

If wire frames forming the outlines of cube, tetrahedron, or cylinder are dipped into a soap solution and then carefully withdrawn, symmetrical figures of great beauty are formed by the films.

By blowing a bubble between two rings and then drawing the rings apart until the sides are cylindrical, the ends will be seen to bulge out, showing that the air within is under pressure, and the radius of curvature of the spherical ends will be found to be twice that of the cylindrical sides.

270. Equilibrium of Cylindrical Film and Formation of Drops. If such a cylindrical film is short relative to its length, it is in stable equilibrium; but if it is longer than its circumference, it is unstable and will collapse at one end and bulge out at the other because by so doing the surface will become smaller.

This will result in its breaking into two bubbles, a large and a small one (Fig. 150), with a very small one between them; a result which is of interest, because it illustrates why a thin jet of water breaks up into drops. Imagine a thin cylindrical jet escaping from a vessel of water. It is under pressure sidewise from the contractile force of the surface, but since it is long enough to be unstable it yields, becoming first undulatory, as shown in the upper part of the figure, then finally breaking up

1

FIG. 150.-Breaking of unstable
cylindrical bubble.

FIG. 151.-Jet breaking
into drops.

into alternate big and little drops which are elongated when first separated and vibrate from this to the flattened form, finally settling down to spherical shape. These details were first made out by Savart; they may best be studied from photographs made when the stream is instantaneously illuminated by an electric spark.

Problems.

1. Find the diameter of a drop of water in which the pressure is twice the atmospheric pressure on its surface, taking surface tension of water as 74 dynes per cm.

2. Two flat glass plates, 10 x 10 cm., placed face to face in a vertical position and separated only by bits of tinfoil, have their lower edge immersed in water which rises and fills the space between the plates. Find how much less the average pressure is between the plates than on the outside and thence find the force with which they are pressed together.

3. In case of a soap bubble 5 cm. in diameter, how much greater is the pressure within the bubble than without? Take surface tension 70 dynes per cm. Give answer in dynes and also in grm.

per sq. cm.

4. A glass hydrometer having a stem 8 mm. in diameter floats in water. With what force due to surface tension of water wetting its stem is it pulled downward?

5. How much deeper will the hydrometer in the last problem sink than if it had floated in a liquid of the same density that did not rise on its stem? Ans. 3.8 mm.

KINETIC THEORY.

271. Kinetic Theory of Gases. It was shown by Daniel Bernouilli (1700-1782) that the pressure of a gas could be best explained as due to the impacts of its molecules against each other and the walls of the vessel. In recent years Clausius and Maxwell especially have developed this theory, showing that the characteristic properties of gases are in harmony with it, and it is now generally accepted as giving a true conception of their

structure.

In this theory it is assumed that the molecules of gas are constantly striking against each other or the walls of the vessel and rebounding. When two molecules approach each other at a certain distance they experience a repulsive force which increases as they come nearer until the approach is stopped by the force and they are repelled apart or rebound. The distance between their centers when they are nearest together and about to rebound is called the diameter of the molecule. The molecule on rebounding soon gets out of the influence of the other and then flies in a straight line until it meets another from which it rebounds, either directly or glancing off sidewise, changing both its own motion and that of the molecule against which it strikes, and so it continues its path zig-zagging about. The average distance that a molecule travels between two successive impacts is called its mean free path. The velocity of a particular molecule is doubtless changed at every impact not only in direction, but in amount, sometimes increased and sometimes diminished, but there is no loss of energy on the whole; whatever one molecule loses the one impacting against it gains. The average kinetic

energy of the molecule, and consequently its average velocity, remains unchanged unless energy is in some way communicated to the gas from outside.

272. Pressure of a Gas.-An expression for the pressure of a gas may be deduced in an elementary way by neglecting the size of the molecules and their impacts against each other and considering each molecule as rebounding only from the walls of the vessel. Imagine a cubical vessel one centimeter each way, and for simplicity conceive the whole number of molecules N contained in it to be divided into three equal groups, one group rebounding be- A tween the sides AD and BC and producing pressure against them, the other groups being directed against the other pairs of sides.

If V

FIG. 152.

is the velocity of a molecule, it will strike against the side BC once every time that it travels across the vessel and back again, a distance of 2 centimeters. The number of impacts per second of one molecule against the side BC will therefore be1⁄2. The momentum of the molecule before impact is mV toward the side, after impact it is mV in the opposite direction; the total change in momentum of the molecule in one impact is 2mV where m is the mass of the molecule. But V there are impacts per second, so that each molecule in rebound

2

ing from one side experiences a change of momentum per second

of 2mV X -=mV2 and since the whole number of mole

V
2

cules impacting against the side BC is

N
3'

the total change in

momentum produced by that side in one second is NmV2, and this is, therefore, the force against the side. But since the side. BC has unit area, the force against it equals the pressure, hence p=}NmV2.

where p represents the pressure of the gas.

(1)

Now, the product Nm is the total mass of gas in one cubic centimeter, or its density d, and hence:

Maxwell's Law.-It has been shown on mechanical grounds, by Maxwell and others, that when two masses of gas are at the same temperature, the average kinetic energy of a molecule of the one is equal to the average kinetic energy of a molecule of the other.

where m1 and m2 are the masses of the molecules of the two gases and V, and V, are their velocities.

1

2

Boyle's Law. According to the law just stated the kinetic energy of the molecules in a mass of gas is determined by its temperature, and hence V changes only when the temperature of the gas changes. Formula (2) above, then, is in agreement with Boyle's law and expresses the fact that the pressure of a gas is proportional to its density when the temperature is constant.

This formula may be used to calculate the average molecular velocities, giving as follows:

Mean Velocity of Molecules in Gases at 0° C.

Avogadro's Law.-When two different gases have the same pressure we have by equation (1)

}N ̧m1V ̧2=}N1⁄2m2V22.

2

1

(4)

If the two masses of gas are also at the same temperature, we have by (3)

1

{m ̧V12 = {m2V22,

and combining the two equations we find

N1 = N2 ;

(5)

that is, the number of molecules per cubic centimeter is the same in all gases at the same temperature and pressure. This is known as Avogadro's law, and was reached by him from purely chemical considerations.