PNEUMATICS is that department of hydrodynamics which treats of the properties of gases as distinct from liquids. Under Hydromechanics will be found a general discussion of the subject as a branch of mathematical physics; here we shall limit our attention mainly to the experimental aspect.
The gaseous fluid with which we have chiefly to do is our atmosphere. Though practically invisible, it appeals in its properties to other of our senses, so that the evidences of its presence are manifold. Thus we feel it in its motion as wind, and observe the dynamical effects of this motion in the quiver of the leaf or the momentum of the frigate under weigh. It offers resistance to the passage of bodies through it, destroying their motion and transforming their energy—as is betrayed to our hearing in the whiz of the rifle bullet, to our sight in the flash of the meteor. In its general physical properties air has much in common with other gases. It is advisable therefore first to establish these general properties, and then consider the characteristic features of the several gases....
Matter is conveniently studied under the two great divisions of solids and fluids. The practically obvious distinction between these may be stated in dynamical language thus: —solids can sustain a longitudinal pressure without being supported by a lateral pressure; fluids cannot. Hence any region of space enclosed by a rigid boundary can be easily filled with a fluid, which then takes the form of the bounding surface at every point of it. But here we distinguish between fluids according as they are gases or liquids. The gas will always completely fill the region, however small the quantity put in. Remove any portion and the remainder will expand so as to fill the whole space again. On the other hand it requires a definite quantity of liquid to fill the region. Remove any portion and a part of the space will be left unoccupied by liquid. Part of the liquid surface is then otherwise conditioned than by the form of the wall or bounding surface of the region; and if the portion of the wall not in contact with the liquid is removed the form and quantity of the liquid are in no way affected. Hence a liquid can be kept in an open vessel; a gas cannot so be.
The mutual action between any two portions of matter is called the stress between them. This stress has two aspects, according as its effect or tendency is considered with reference to the one or the other body. Thus between the earth and moon there is a stress which is an attraction. The one aspect is the force which attracts the moon to the earth; the other is the force which attracts the earth to the moon. According to Newton’s third law of motion these are equal and opposite. Similarly the repulsive stress between the like poles of two magnets has its two aspects, which are equal but oppositely directed forces. In the case of a mass hanging by a cord, the stress is a tension at every point of the cord. At any given point this tension has two equal and opposite aspects, one of which is the weight of the mass and the portion of the cord below the given point. Finally, the stress between any body and the horizontal table on which it rests is a two-faced pressure, being downwards as regards the table, upwards as regards the body. The total pressure upon the table over the whole surface of contact is clearly the weight of the body. If the total pressure is supposed to be uniformly distributed, the measure of the pressure on unit surface is the quotient of this weight by the area of the surface. When we speak of pressure at a point, it is this pressure on unit surface that is meant. When the pressure varies from point to point over a surface, the pressure at any point is defined to be the limit of the ratio of the total pressure over any small element of surface around that point to the area of the element as the element is diminished indefinitely.
The stress which exists between the contiguous portions of a fluid is of the nature of a pressure. The ideal or perfect fluid is a substance in which this stress between contiguous portions is always perpendicular to the common interface. In other words there is no stress tangential to the interface-at any point. Hence if the contiguous portions are at relative rest, or have a relative motion parallel to the interface, neither state can be affected by the mutual stress. This condition is perfectly fulfilled in the case of any known fluid in equilibrium; but for a fluid in motion it is not even approximately fulfilled. For, any visible relative motion set up among the parts of a fluid rapidly decays in virtue of viscosity, which even for the subtlest gases is quite appreciable in its effects.
In a fluid at rest, then, the pressure over any surface which we may imagine to be drawn is perpendicular (or normal) to the surface at every point of it; and from this it follows at once, as has been proved in Hydromechanics, vol. xii. p. 439, that the pressure at any point of a fluid at rest has the same value in all possible directions.
The pressures at two contiguous points in a fluid may either differ or not. If they differ, the change of pressure must be balanced by some extraneous force acting on the fluid in the direction in which the pressure increases. Any direction in which no such force acts must be a direction in which there is no change of pressure; otherwise, equilibrium will be destroyed. Suppose now the resultant force at every point in a fluid at rest to be given. In directions at right angles to the force at any given point the pressure will not vary. Hence we can pass to an infinite number of contiguous points at which the pressure is the same as at the given point. By making each ol these in turn the starting-point, we can pass on to another set of points, and so gradually trace out within the fluid a surface at every point of which the pressure is the same. Such a surface is called a surface of equal pressure, or briefly a level surface; and we see from the mode of its construction that it is at every point of it perpendicular to the resultant force at that point.
Imagine any two contiguous level surfaces to be drawn, at every point of the one of which the pressure is p, at every point of the other p + δp. Consider the equilibrium of a small column of average density ρ, bounded at its ends by these surfaces. Let A be the area of each end, and δr the length of the column or perpendicular distance between the level surfaces. If R is the average resultant force per unit mass acting on the column, then we have, for equilibrium of the column, ASp=ASr. pβ , or δ72 = Rpδr.
or the rate of increase of the pressure at any point per unit of length at right angles to the level surface is equal to the resultant force per unit of volume at that point.
If the applied forces belong to a conservative system, for which V is the potential (see Mechanics), we may write the equation in the form
δp = — pδV.
Hence over any equipotential surface, for which δV = 0, p is constant, and is therefore a function of V. Consequently p also is a function of V. For a fluid in equilibrium, therefore, and under the influence of a conservative system of forces, the pressure and density are constant over every equipotential surface, that is, over every surface cutting the lines of force at right angles.
Now in the case of gases, to which our attention is at present confined, the density (temperature remaining constant) varies with every change of pressure; in mathematical language p is a function of p. Thus, before we can solve the equation of equilibrium for a gas, we must be able to express this function mathematically; in other words, we must know the exact relation between the density of a gas and the pressure to which it is subject. This problem, which can only be settled by experiment, was solved for the case of air within a certain range of pressures by Robert Boyle (1662). Before discussing his results and the later results of other investigators, we shall first consider the general properties of our atmosphere as recognized before Boyle’s day.
It is evident that, for a fluid situated as our atmosphere is, the pressure must diminish as we ascend. The equipotential surfaces and consequently the surfaces of equal pressure and of equal density will be approximately spheres concentric with the earth. At any point there will be a definite atmospheric pressure, which is equal numerically to the weight of the superincumbent vertical column of air of unit cross-section. The effect of this pressure, as exemplified in the action of the common suction-pump, seems to have been first truly recognized by Galileo, who showed that the maximum depth from which water can be pumped is equal to the height of the water column which would exert at its base a pressure equal to the atmospheric pressure. As an experimental verification, he suggested filling with water a long pipe closed at the upper end, and immersing it with its lower and open end in a reservoir of the same liquid. The liquid surface in the pipe would, if the pipe were long enough, stand at a definite height, which would be the same for all longer lengths of pipe. The practical difficulty of constructing a long enough tube (33 feet at least) prevented the experiment being really made till many years later.
Torricelli, however, in 1642, by substituting mercury for water, produced the experiment on a manageable scale. As mercury is denser than water in the ratio of about 13.6:1, the mercury column necessary to balance by its weight the atmospheric pressure will be less than the water column in the inverse ratio, or a little under 30 inches. Torricelli’s experiment is exhibited in every mercurial barometer (see Barometer and Meteorology). By this experiment he not only gave the complete experimental verification of Galileo’s views relating to atmospheric pressure, but prα vided a ready means of measuring that pressure.
The most obvious applications of the barometer are these :—(1) to measure the variation in time of atmospheric pressure at any one locality on the earth’s surface (the existence of this variation was discovered soon after the date of Torricelli’s experiment by Pascal, Descartes, Boyle, and others); (2) to measure the variation of atmospheric pressure with change of height above the earth’s surface (Descartes mentions this application in the Principia Philosophiae, 1644; but to Pascal is the honour due of having first carried the experiment into execution, 1647); and (3) to compare pressures at different localities which are on the same level (if the pressures are equal, the air is in equilibrium; if they are not, there must be flow of air from the place of higher pressure to that of lower—in other words, there must be wind, whose direction of motion depends on the relative position of the places, and whose intensity depends on the distance between the places and the difference of pressures). The first and last of these measurements are of the greatest importance in meteorology. The second is a valuable method for measuring attainable heights, and is intimately connected with the problem as to the relation between the pressure and density of the air. Thus it would be possible, by barometric observations at a series of points in the same vertical line, to obtain a knowledge of this relation—more and more truly approximate the closer and more numerous the points of observation taken. At best, however, such a method could give the law connecting density with pressure for those pressures only which are less than the normal atmospheric pressure. The problem is better solved otherwise.
Assuming Boyle’s law that the density of air is directly as the pressure, we can now integrate the equation of equilibrium Zp = - p SV, and put it in the form 2>=2V^ v ' κ .
where p 0 is the pressure at zero potential and K is the constant ratio of the pressure to the density.
For all attainable heights in our atmosphere we may assume the force of gravity to be the same. Hence we may write N=gh, where g is the force acting on unit mass at height h. If we put K = ÿH, the equation becomes jo=2? o e'* /H , where H is obviously the height of a fluid of uniform density p 0 ∕K which would give at its base the pressure p 0 —in other words, the height of the homogeneous atmosphere, as it is called. Its value is readily found, since it bears to the height of the mercurial barometric column the same ratio which the density of mercury bears to the density of the atmosphere at the sea-level. For dry air at 0° C. and with g taken as equal to 981 dynes ( = 32'2 poundals nearly), the value of H is 7'9887 × 10 5 centimetres, or 26,210 feet. Hence the formula giving the height above the sea-level in terms of the pressure may be written h = 7 '9887 × 10 5 × Nap. log. (p 0 ∣ p).
In practice this formula must be modified to suit regions where g is other than 981, and where the temperature is other than 0 β . The effect of the water-vapour present must also be taken into account, and the constants involved carefully tested by observation. The subject is treated in detail under Barometer.
In an appendix to the New Experiments, Physico-Mechanical, Ac., touching the, Spring of Air (1660), Robert Boyle states that the density of air is directly as the pressure. His apparatus and method of ^experiment are as follows. A U-shaped tube is taken, one of whose limbs is considerably longer than the other. The shorter limb is closed at the end; and the whole apparatus is set vertically with the open end pointing upwards. A small quantity of mercury fills the bend, so that at the beginning of the experiment the two mercury surfaces are at the same level. Hence the air confined in the shorter limb is subjected to a pressure along its lower surface equal to the atmospheric pressure,, or one atmosphere as it is commonly called. As the height of the air column in the closed tube is small, the pressure and density are practically the same throughout. Now let mercury be poured into the longer limb. The free mercury surface will be observed to rise in the shorter limb, so that the air confined there becomes compressed into smaller bulk. Since the mass of air has not altered, the density is obviously inversely as the bulk, and can therefore be easily measured. Again, the pressure to which the confined air is now subjected is equal to the pressure over that surface in the mercury in the open limb which is at the same level as the free mercury surface in the closed limb. But this pressure is clearly the sum of the atmospheric pressure and the pressure due to the superincumbent column of mercury, which latter can be readily expressed in atmospheres if the height of the barometer is known. In other words, divide the vertical distance between the two mercury surfaces by the height of the barometer column. The quotient added to unity gives the required pressure in atmospheres.
Fourteen years after the date of the publication of Boyle’s results, Mariotte, working independently, discovered the same law, which is still widely known on the Continent as Mariotte’s law. He supplemented Boyle’s experiments by investigating the effect of pressures less than that of the atmosphere, and proved that the same law held at these diminished pressures. His method was essentially as follows. A barometer tube is filled in the ordinary way with mercury and fixed up as in the Torricellian experiment. A little air is then introduced at the lower end of the tube which is dipping in the reservoir of mercury. This air travels up the tube and fills the Torricellian vacuum at the top, thereby depressing to a slight extent the barometer column. The amount of depression divided by the true height of the barometer gives the pressure in atmospheres which acts upon the air in the tube. The tube, always kept truly vertical, dips in a reservoir of mercury sufficiently deep to admit of its complete immersion. For a certain position of the tube the free surfaces of mercury in the tube and reservoir are at the same level. For that position the confined air is at the atmospheric pressure; and for any higher position of the tube the pressure in the confined mass of air is less than the atmospheric pressure by the pressure due to the column of mercury between the free surfaces. Recent experiments by Kraevitch and Petersen (Journal of the Russian Chemical Society, vol. xvi.) seem to show that very rarefied air is very far from obeying Boyle’s law. At such low pressure, the condensation of the gas upon solid surfaces is an important factor.
For most ordinary purposes Boyle’s law—that, at constant temperature, the density of a gas varies directly as the pressure—may be assumed to be true, at least for moderate ranges of pressure; but the careful investigations of later experimenters, such as Oersted, Despretz, Dulong, Régnault, Andrews, Cailletet, and Amagat, have proved that the law is only approximate for every known gas, and that the deviation from correspondence with the law is different for each gas. The most recent investigations are those of Cailletet and Amagat, who have carried the results to much higher pressures than former experimenters employed. Both adopted in the first place a form of apparatus essentially the same as Boyle’s, only much longer. The gas was subjected to the pressure of a mercury column enclosed in a strong narrow steel tube; and, as oxygen acts vigorously upon mercury at high pressures, nitrogen was used. In this way Cailletet attained to a pressure of 182 metres of mercury, and Amagat to a pressure of nearly 330.
Having thus determined accurately the corresponding pressures and densities of nitrogen, Amagat proceeded to determine the relation for other gases by Pouillet’s differential method. That is, the pressure to which the new gas was subjected was made to act simultaneously upon a given mass of nitrogen, whose volume could be readily measured and pressure estimated. Oxygen, hydrogen, carbonic oxide, dry air, olefiant gas, and marsh gas were investigated in this way. The general results obtained by Amagat are exhibited in the subjoined chart taken from his paper. For all gases except hydrogen the product pv (pressure into volume), instead of being constant, as Boyle’s law would require, diminishes at first as the pressure is increased. At a certain pressure, however, different for each gas, the diminution ceases, and if the pressure is still further increased the product pv begins to increase also, and continues so to do to the greatest pressure used. In tho case of hydrogen the product increases from the very beginning.
On the diagram, absciss® represent pressures in metres of mercury, and the ordinates represent the deviations from the Boylean law. It will be observed that all the curves pass through the point on the pressure axis which represents a pressure of 24 metres of mercury. If π represents the product pv for any gas at this pressure, and √ the corresponding product for any other pressure, then we may write
τr∕ιr ’ = l + δ, where δ represents the deviation from Boyle’s law. All the curves except that for hydrogen show a well-marked minimum, at and near the pressure corresponding to which the particular gas obeys Boyle’s law. For the several gases these positions occur at the pressures as given in the following table :—
Nitrogen 50 m. I Carbonic acid 50 m.
Oxygen 100 m. Marsh gas 120 m.
Air 65 m. ∣ Olefiant gas 65 m.
For olefiant gas δ is so great, and varies so rapidly, that only portions of the curve are represented. The value of δ for its minimum point is - 1 '3, while the corresponding value for oxygen is -0Ό5. In these experiments the temperature of the gases varied between 18° and 22° C.
Amagat has extended his researches to higher temperatures up to 100° C. The general characters of the curves obtained for hydrogen, nitrogen, olefiant gas, and marsh gas remain...
