CHAPTER XVII

PROPERTIES OF LIQUIDS

144. Equilibrium of a Liquid at Rest.-In the case of a liquid at rest under the influence of gravity the free surface must be horizontal. If it were inclined to the horizon, then the weight of a particle P (Fig. 112) of the liquid at the surface would have a component parallel to the surface of the liquid. Since the surface is everywhere at the same pressure, there would be nothing in the nature of a hydrostatic pressure to resist this force, and as the liquid itself would offer no resistance, the particle P would move along the surface, and hence the liquid would not be at rest.

P

FIG. 112

B

Эс E

Although a comparatively small surface of a liquid is for all practical purposes plane, it is not absolutely so, and when dealing with large surfaces, this departure from planeness becomes appreciable. The condition that the particle P (Fig. 112) should be at rest is that the line of action of the attraction of gravity on P should be normal to the surface at P. Hence the surface of a liquid is always normal to the radius of the earth at the point considered, and therefore forms part of a sphere having the earth's radius as radius. The fact that the surface of a liquid is always horizontal is made use of in the spirit-level. This consists essentially of a glass tube ABC (Fig. 113), which is slightly bent, and fitted, with the convex surface upwards, D in a frame DE. This tube is closed at either end, and is filled with alcohol1 except for a bubble of air B, which is left in. This bubble constitutes the only free surface of the liquid, and it always sets itself at the highest point of the curved tube. Hence, if the tube is fixed in the frame in such a way that when the lower surface of the frame is horizontal the highest point of the bent tube is opposite a fixed mark on the top of the tube, then, whenever the bubble is opposite this mark, the lower surface of the stand will be horizontal. If one end, say E, is tilted up, then the marked point of the tube is no longer the highest, and the bubble moves towards E. 1 Alcohol is used on account of its extreme mobility.

FIG. 113.

145. Level of Liquid Surface in Communicating Vessels.-Suppose a U-tube ABC (Fig. 114) has the same liquid in either limb, then the two surfaces A and B will be in the same horizontal plane. For if we consider a point D within the liquid, at a depth

A

h1 below the surface at A, and at a depth h2 below the surface at C, then the pressure at D must be the same, whether caused by the column AD or the column CD; otherwise the liquid would move towards the side on which the pressure was least. Hence

U

B

FIG. 114.

h1gp=h2gp,

where p is the density of the liquid.

By an exactly similar line of argument it can be shown that the pressure at any pair of points, one in either limb, is the same if these points lie in the same horizontal plane.

If, instead of having the same liquid in both limbs, one limb AB (Fig. 115) contains a liquid of less density than that in the other; then, if B is the surface separating the two liquids, from what has

been said in the previous paragraph, the pressure at a point D in the denser liquid in the same horizontal plane as в must be equal to the pressure at B. Hence the pressure exerted by the column AB of the one liquid must be equal to the pressure exerted by the column CD of the other liquid. So that, if h1 and h are the heights of these columns, and p1 and p are the densities of the liquids, we have

That is, the heights of the columns of the two liquids above the level of their common surface are to one another inversely as the densities of the liquids.

146. Density of Liquids.-In order to determine the density of a liquid, the mass of a known volume must be measured. If, however, we know the density of any given liquid, say water, then we can find the density of any other liquid by comparing the mass of any volume of the liquid with that of the same volume of the standard liquid.

The density of water has been determined by Macé de Lépinay with great accuracy, by a method depending on the principle of Archimedes. A cube of quartz was prepared and its volume obtained by measurement.

The planeness of the faces was tested, and the distance between the opposite faces measured by an optical method depending on the production of Newton's rings (see § 376). This cube was then placed on the pan of a very delicate balance, a small cage suspended by a fine platinum wire hanging from the under side of the same pan. This cage was completely immersed in the water of which the density was to be measured, and which was kept at a constant temperature, this temperature being read by means of a thermometer. The weights necessary to counterpoise the quartz block (in air) and the wire cage (immersed in the water) having been placed in the second pan of the balance, the quartz block was transferred to the cage, and the weights found which were now necessary to counterpoise. The difference between these weights represents the loss of weight of the block when immersed in water, and this, by the principle of Archimedes, is equal to the weight of a volume of water equal to that of the block. Hence, knowing the volume of the block, i.e. the volume of the water displaced, the density can be calculated. The object of having the wire cage, which was always immersed in the water, was to allow for the weight of the suspending fibre and that of the water it displaced; also, by this arrangement the effect of the surface of the liquid on the suspending wire due to capillarity (§ 157) was the same during both the weighings, and was therefore eliminated.

Since the volume of the quartz cube altered with the temperature, this had to be allowed for, so that a preliminary measurement of the coefficient of expansion of quartz was made.

The following table gives the density of water at different temperatures:

Knowing the density of water, A, at different temperatures, we can determine the volume of a solid which is insoluble in water, by weighing it first in air and then when immersed in water at a known temperature. If w1 is the weight in air and w, the weight in water, then the loss of weight, that is, the weight of water displaced, is w1-w, and this must be equal to AV, where Vis the volume of the solid. Thus

V=(w1— W2)/▲.

One method of comparing the density of a liquid with that of water is to determine the loss of weight of a solid, which is unacted upon by either liquid, when weighed first in water and then in the liquid. In this way the weights or masses of equal volumes of the liquid and of water are obtained. If m, is the loss of weight in the given liquid of density p, and ma is the loss of weight in water of density A, then

and, by taking the value of ▲ for the temperature of the experiment from the table given above, p can be calculated.

Another method in common use for determining the density of a liquid is to weigh a small bottle, called a specific gravity bottle or pyknometer, when full, first of water, then of the liquid. Two forms of pyknometer which are in common use are shown in Fig. 116. One consists of a small glass bottle A, fitted with a well-ground-in glass stopper B. This stopper is pierced by a fine hole. The bottle is com

above the mark F. Then, by touching the capillary C with a piece of blotting-paper, liquid is withdrawn till the surface comes exactly to the mark F.

Let w, be the weight of the empty pyknometer, and W2 and W3 the weight when full of the liquid and water respectively. Then ww, is the weight of water which fills the instrument. Hence if A is the density of the water at the temperature at which the pyknometer was filled, its volume V is given by

V=(w3-w1)/A.

The weight of a volume V of the given liquid is w2-w1. Hence the density p of the liquid is

The following table gives the density of some liquids :

147. Flotation.-Since when a body is immersed in a fluid it experiences an upward force, due to the pressure of the fluid, equal to the weight of the fluid displaced, it follows that if the density of the body is less than that of the fluid, the weight of the displaced fluid will be greater than the weight of the body, and hence the upward force acting on the body due to the fluid will be greater than the downward force due to gravity. If no other forces are acting on the body, it will therefore rise until the weight of the displaced fluid is exactly equal to that of the body. In the case of a body such as a balloon in the air, this will happen when it has risen to such a distance that the density of the air has become so much reduced that the weight displaced by the balloon is equal to its own weight. In the case of a solid immersed in a liquid, it will rise till, some of the solid having risen above the surface of the liquid, the weight of the volume of liquid displaced by the remainder, which is still submerged, is equal to the weight of the body.

In order that a body floating in a liquid may be in equilibrium, not only must the upward pressure due to the liquid be equal in magnitude to the weight of the body, but it must also act vertically upwards through the centre of gravity of the body. If we suppose the body removed and

M