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 the same as at the ordinary temperature; that is, with the exception of hydrogen, the product pv decreases to a minimum and then increases indefinitely.
The position of the minimum changes with the temperature. Thus for olefiant gas and carbonic acid gas (whose properties were also studied at these higher temperatures), the pressure at which the minimum occurs increases with the temperature, while in the case of nitrogen and marsh gas this critical pressure decreases as the temperature rises. Probably at some temperature higher than 100° olefiant gas and carbonic acid gas would begin to behave like nitrogen, and all would appear to tend more and more, as the temperature rises, to the condition of which hydrogen is the type. That is, the deviation from the Boylean law up to the minimum point would steadily decrease until finally the curve would cease to have a marked minimum. For any one gas, the higher the temperature the less the curvature at the minimum point; and the comparison of different gases seems to indicate that the curvature is greater for the more easily liquefiable gas. At sufficiently high temperatures the law of compressibility for all gases approximates to the relation
P( V - a) = constant, where P is the pressure, V the volume, and a a constant. Hydrogen follows this law very closely at the ordinary temperature of the air, as the straightness of its representative curve shows at a glance.
Amagat has further discussed by means of his results the law of dilatation of gases. This law is named Charles’s law, after the discoverer of it. Stated simply, it is that at constant pressure every gas expands by the same fraction of itself for a given rise from a given temperature. Charles did not publish his results; and it was not till fifteen years later, when Dalton and Gay-Lussac, working independently, rediscovered it, that the law became generally known.
Their results were published in 1801 and 1802 respectively; and it is upon the authority of the latter, who accidentally became acquainted with the fact, that the law is now named after Charles. The careful measurements of Magnus, Régnault, Jolly, and others have established that there is an appreciable difference in the coefficients of expansion for the different gases. The difference is slight for the so-called permanent gases—air, nitrogen, oxygen, hydrogen, and marsh gas; but for the more easily liquefiable gases it is quite marked. The mean coefficient of expansion for air between 0° C. and 100° C. and at the ordinary atmospheric pressure is Ό03665 per degree, and the value for any one of the gases just mentioned does not certainly differ from this by one-half per cent. This may be expressed by the formula v=KT, where K is a constant and T the temperature measured from absolute zero, which is 274° C. below the freezing point of water (see Heat). When T is constant, we have by Boyle’s law the product pro also constant. Hence we may combine the two laws in the form pι>=RT, where R is a constant. We thus see that, Boyle’s law being assumed to be true at all temperatures, Charles’s law, if true for any given pressure, is true for every other pressure. Further, if v is kept constant the rate of increase of log p with temperature will be expressed by the same number as the rate of increase of log v when p is kept constant. Experiment has fully verified this conclusion to as close an approximation as Boyle’s and Charles’s laws themselves are fulfilled. The rate of increase of log v with temperature, or, what is the same thing, the ratio of the rate of increase of the volume to the original volume, is given by the formula 1 ⅛ 1
v(fΓ'T ’
and this is the measure of the coefficient of expansion at temperature T. Hence the coefficient of expansion diminishes as the temperature rises, a conclusion also in accordance with experiment so long as we are dealing with gases which nearly obey Boyle’s and Charles’s laws.
We have seen, however, that even in the case of hydrogen the departure from Boyle’s law is very marked at the higher pressures; and therefore we cannot expect a closely numerical agreement between the results of experiment and the results of calculation from the above formula. Thus, it is not surprising that practically the coefficient of expansion should be affected by the pressure, as Amagat’s experiments clearly show,—although in the equation deduced above the pressure does not enter. In the following table given by Amagat, the second column contains the mean coefficients of expansion of hydrogen between 17° and 60° C. at the pressures given in the first column; and the third column contains the corresponding mean coefficients between 60° and 100" C. [table]
The temperature effect upon the coefficient of expansion, as shown by these numbers, is approximately that indicated above, viz., that at constant pressure the coefficient of expansion is inversely as the absolute temperature. A glance down each column shows at once the marked effect of pressure. In this steady decrease of the coefficient of expansion with increase of pressure, hydrogen stands alone amongst the substances discussed by Amagat. His conclusions are given in these words :—
1. The coefficient of expansion of gases increases with the pressure to a maximum, after which it decreases indefinitely.
2. This maximum occurs at the pressure for which at constant temperature the productiv is a minimum, that is, the pressure at which the gas follows for the instant Boyle’s law.
3. With increasing temperature this maximum becomes less and less sensible, finally disappearing with the minimum characteristic of the compressibility curve.
Thus, as hydrogen does not show this minimum characteristic, its coefficient of expansion has no maximum value. Possibly at lower temperatures hydrogen may, however, possess these characteristics.
It thus appears that the simple gaseous laws established by Boyle and Charles are most nearly fulfilled by those gases which are difficult to liquefy, and are better fulfilled by all the higher the temperature is. When a gas is near its point of liquefaction the density increases more rapidly than the pressure, or in other words the volume diminishes more quickly than Boyle’s law requires. When the point of liquefaction is actually reached, the slightest increase of pressure condenses the whole of the gas into a liquid; and in this state the alteration of volume is very small even for a large increase of pressure.
The transition from the gaseous to the liquid state is conveniently studied by the help of isothermal lines, which may be generally defined as curves showing the relation between two mutually dependent variables for given constant temperatures. Such variables are the pressure and volume of a mass of gas. Let the numbers representing the volumes be measured from a chosen origin along a horizontal axis, and the numbers representing the pressures similarly along a vertical axis passing through the same origin. If we consider a mass of gas at a given temperature, for any volume that can be named there will be a definite pressure corresponding, and vice versa. Hence the point whose coordinates arc the corresponding volume and pressure is completely determined if either coordinate is given. The temperature always being kept constant, let now the volume change continuously. The pressure will also alter according to a definite law; and the point whose coordinates are at any instant the corresponding volume and pressure will trace out a curve. This curve is an isothermal curve, or simply an isotherm. If Boyle’s law were fulfilled, the equation to the isotherm for any given temperature would be of the form pv ≈ constant.
The isotherm would be a rectangular hyperbola, whose asymptotes are the coordinate axes. For any gas not near its point of liquefaction the isotherm will not deviate greatly from the hyperbolic form.
Let now the pressure be kept constant, and the gas raised somewhat in temperature. The volume of course increases, and the corresponding point on the diagram moves off the original isotherm. Through this point in its new position we can draw a second isotherm corresponding to the new temperature. And thus the whole field may be mapped out by a series of isotherms, each one of which corresponds to a definite temperature. The higher the temperature the farther does the isotherm lie from the origin. Such a mapped out diagram or chart shows at a glance the relations between the volume, pressure, and temperature of a given mass of gas, so that if any two of these are given the third can be found at once.
So long as the substance is in the gaseous form, the isotherm remains approximately hyperbolic; but at the pressure at which liquefaction takes place a marked change occurs in the form of the curve. For greater definiteness consider the case of a gramme of steam at 100° C. and at a pressure somewhat below one atmosphere. As the pressure is increased, the volume diminishes appreciably faster than Boyle’s law requires, but still in such a way as to give an approximately hyperbolic form to the isotherm. .When the pressure reaches one atmosphere, however, any further increase is accompanied by the liquefaction of the whole; that is, the volume suddenly diminishes from 1647‘5 cubic centimetres to 1 cubic centimetre. Between these extremes of volume, the isotherm is a straight line parallel to the horizontal axis. The pressure remains constant until the whole of the gas is liquefied. In other words, the pressure of a gas in presence of its liquid does not alter provided the temperature is kept constant. This is a partial statement of the more general law that the temperature of the liquid surface alone determines the maximum pressure which its vapour or gas in contact therewith can exert (see Heat). After the whole has been liquefied, any increase of pressure is accompanied by a very minute diminution of volume. Hence the isotherm rises abruptly from the point whose coordinates are 1 cubic centimetre and 1 atmosphere, becoming nearly but not quite vertical. Thus, the isotherm for water-substance at 100° consists of three parts :—an approximately hyperbolic portion for pressures less than one atmosphere, the substance being then wholly gaseous; a horizontal portion, corresponding to the state in which the substance is partly liquid partly gaseous; and a nearly vertical portion for pressures higher than one atmosphere, the substance being then wholly liquid. If we trace out the isotherm for some higher temperature, say 150° C., we obtain the same general characteristics. The straight line portion, however, is not so long, for two reasons :—the steam must be reduced to a smaller volume before liquefaction begins; and the volume of the liquid when condensed is greater. The pressure corresponding to the transition state is in this case 4 ’ 7 atmospheres, and the range of volumes is from 384 ’ 4 cubic centimetres in the gaseous state to 1Ό38 cubic centimetres in the liquid state. It thus appears that the positions of the two points of abrupt change on an isotherm draw nearer the higher the temperature, coming together finally when the temperature has reached a certain critical value. In other words, at and above a certain temperature a liquid and its vapour cannot co-exist. This temperature for water-substance is very high, somewhere about the point of fusion of zinc, and is therefore difficult to measure. Dr Andrews, however, in his classical researches on carbonic acid gas, to which we owe most of what is said above, has discussed the whole subject in a very complete manner. This substance, at a temperature Of 13 o, l C., begins to liquefy at a pressure of 47 atmospheres. During the process of liquefaction there is a perfectly visible liquid surface separating the two coexisting states of the substance; and the isotherm has a corresponding straight line portion. At a temperature of 21 o, 5 C. liquefaction occurs at a pressure of 60 atmospheres. The horizontal portion of the isotherm, which marks the co-existence of the gaseous and liquid states, is considerably shorter than at the former temperature. The isotherm for 31° *l C., however, has no such rectilinear characteristic; and at this and higher temperatures the substance is never during the whole compression in two distinct conditions at once. It is impossible to say when the dense gaseous condition passes into the light liquid condition. The two states are absolutely continuous. The critical temperature—that is, the temperature below which there is a distinct separation between the liquid and the gas—is fixed by Dr Andrews at 30 o, 92 C. for carbonic acid gas. Above this temperature it is impossible to obtain a free liquid surface in a closed vessel. This conclusion had already been arrived at by Faraday in 1826, when he considered himself entitled to state that above a certain temperature no amount of pressure will produce the phenomenon known as condensation.
Andrews’s results also give the true explanation of the observations made by Cagniard-Latour in 1822 upon the effect of high temperature on liquids enclosed in glass tubes which they nearly filled. He found that at a certain temperature the free liquid surface disappeared, and the tube became filled with a substance of perfectly uniform appearance throughout. He concluded that the whole had become gaseous. In reality he had reached the critical temperature at which the liquid and gaseous conditions pass continuously the one into the other. The following are Cagniard-Latour’s estimated values for the temperature and pressure of various substances at the critical point [table]
Avenarius and Drion have studied the critical temperatures of other substances, such as sulphuric acid, aceton, and carbon tetrachloride. The substances, however, which can be so studied are comparatively few, since the greater number of those which are liquid under ordinary conditions have their critical temperatures very high, while the majority of those which are gaseous have theirs very low.
The necessity for a very low temperature long prevented the obtaining in a liquid form of the standard gases— hydrogen, oxygen, nitrogen, &c.—which were accordingly distinguished by the name permanent gases. Faraday proved that these could not be liquefied at a temperature of -110° C., even when subjected to a pressure of 27 atmospheres. Natterer likewise failed to reduce these gases to the liquid state, even at a pressure of 3000 atmospheres. His means for reducing the temperature were not satisfactory. In 1877 Cailletet and Pictet, working independently, first successfully effected their approximate liquefaction. The former compressed each of the gases oxygen, nitrogen, and carbonic oxide to 300 atmospheres in a glass tube, which was cooled to - 29° C. When the gas was allowed to escape, it did so in the form of a cloud, condensing for the moment to the liquid state under the influence of the extreme cold produced by the rapid expansion of the gas. Pictet in a similar way obtained an issuing stream of liquid oxygen. Von Wroblewski and Olzewski have more recently obtained oxygen, nitrogen, and carbonic oxide in a more evident liquid state. They used Cailletet’s form of apparatus, and cooled the gas by means of the evaporation of liquid ethylene. Under this extreme cold they observed these substances forming a well-defined liquid in the bottom of the tube. The following table gives the results of five different observations at slightly different temperatures. [table]
At slightly higher temperatures, the pressure necessary for the liquefaction increased very rapidly. Nitrogen and carbonic oxide were not so easily reduced, remaining still gaseous at - 136° C., and under a pressure of 150 atmospheres. By a sudden diminution of the pressure to 50 atmospheres there was obtained under the influence of the reduced temperature a rapidly evaporating liquid. The critical point of oxygen has been experimentally fixed by Von Wroblewski at - 113’ C. and 50 atmospheres pressure. With the data given by Amagat’s researches, Sarrau has calculated from a formula of Clausius’s the following values of the critical temperature and pressure for oxygen, nitrogen, hydrogen. [table]
It will be observed that Von Wroblewski’s observed values for oxygen are in remarkably close agreement with the calculated values given here. Apparently oxygen is just at the limit fixed by Faraday.
The behaviour of a gas under varying pressure is a phenomenon of great practical importance, and gives a valuable method for measuring pressures (see Manometer).
A modification of the ordinary mercury manometer is used for measuring volumes, and is especially valuable in estimating the densities of substances which cannot be put into water, such for example as liquids or powders. The closed end of the manometer tube, which by means of a stop-cock may be opened to the air at will, is fitted to a flask with which it may be put into connexion when required. Two fiducial marks are then made upon it— one at the position where the mercury surfaces in the two limbs of the tube are co-level, and the other somewhat higher at a convenient spot. Between the two marks the tube expands into a bulb, thereby increasing the intervening volume and minimizing the effect of any slight error in bringing the mercury surface to the higher mark. Let the volume of the flask and tube down to the higher mark be V, and the volume of the rest of the tube down to the lower mark V. At first the volume of air in the flask and tube is V + V, at the atmospheric pressure P. Now pour mercury into the open end till the liquid surface in the closed end reaches the higher mark. The air has been compressed to volume v ; and the corresponding pressure, as measured by the balanced mercury column, is jt> + P. Hence
(p + P)u=P(V + v).
Let now a volume x be placed in the flask, and let the same succession of operations be made. Then, if p' be the increase of pressure necessary to bring the mercury surface from the lower to the higher fiducial mark, we have
(y + P)(v-x)-P(V+v-a!).
As V can be readily determined by gauging the tube, and as P, p, p' are all known, the quantities v and x can be at once found from these equations.
In many experiments on the properties of gases it is necessary to have an efficient and rapid means for altering the density. Instruments for this purpose are called air-pumps, and their function may be either to rarefy or condense the air—usually the former.
Otto Von Guericke of Magdeburg constructed the first air-pump about the year 1652. It was simply a spherical glass vessel opening below by means of a stop-cock and narrow nozzle into the cylinder of an “exhausting syringe,” which inclined upwards from the extremity of the nozzle. The cylinder, in which a well-fitting piston worked, was provided at its lower end with two valves. One of these opened from the nozzle into the cylinder, the other from the cylinder into the outside air. During the down-stroke of the piston the former was pressed home, so that no air entered the nozzle and vessel, while the latter was forced open by the air which so escaped from the cylinder. During the return-stroke the latter was kept closed in virtue of the partial vacuum formed within the cylinder, while at the same time the former was forced open by the pressure of the denser air in the vessel and nozzle. Thus, at every complete stroke of the piston, the air in the vessel or receiver was diminished by that fraction of itself which is expressed by the ratio of the volume of the available cylindrical space above the outward opening valve to the whole volume of receiver, nozzle, and cylinder.
Boyle, on hearing of Von Guericke’s success in applying the expansive properties of air as a means to its rarefaction, constructed a machine essentially the same as Von Guericke’s, of which no description had then been published. Boyle, however, made the exhausting cylinder a continuation of the nozzle, which was thus considerably reduced in size, and worked the piston by means of a wheel and racket work. He also employed a transparent glass receiver with removable cover, so that the operator could observe what was going on inside, and more easily alter the contents. Other modifications and improvements followed rapidly; and in the carrying out of these Boyle was greatly aided by Hooke, who conceived the happy idea of using two syringes opening into the same duct from the receiver. This improvement has recently been credited to Papin. In this form the pistons are worked by the same toothed wheel, and are so adjusted that the one rises as the other falls. This arrangement not only doubles the rate of exhaustion per stroke, but vastly increases the ease of working. In the single-barrelled form the piston is drawn back against a pressure of air which is greater the more complete the exhaustion is within; but in the double-barrelled form the downward pressures upon the two pistons to a certain extent counteract each other, producing opposite rotational effects on the toothed wheel and driving handle. Boyle also early adopted the flat plate, on which could be set receivers of various shapesand sizes. The junction of the plate and receiver he made tolerably air-tight by covering the plate with wet leather and having the receiver rim ground flat. According to Gerland, the invention of the plate is due to Huygens, who constructed the first air-pump so provided in 1661, shortly after a visit to London, where his interest in the subject was awakened by Boyle. To Huygens, probably in conjunction with Papin, is also due the applica tion of the mercury manometer to measure the pressure in the receiver—an indispensable equipment in all serviceable air-pumps. The form of the piston plugs and valves received the special attention of Papin, who by his refined and detailed improvements did much to increase the efficiency of the apparatus.
The important characteristics of an efficient air-pump are as follows. The piston must work smoothly and easily. The valves must act precisely, and be when closed absolutely air-tight. The plate on which the receivers rest must be smooth and plane, so that the ground edges of the receivers may be in close contact all round. This perfect fitting is beyond the powers of the best workmanship, so that it is necessary to press between the receiver and plate a thin layer of lard, which renders the junction air-tight. Somewhere in the duct leading from the receiver to the piston cylinders, a stop-cock must be fixed, so that it may be possible to shut off the receiver completely from these. Then a second stop-cock is required as a ready means for admitting air to the receiver, whenever the need should arise. A combination three-way stop-cock is a very usual form. And, finally, the apparatus should be provided with a pressure gauge—a mercury manometer communicating by means of a duct with the main duct and receiver.
The double-barrelled form of reciprocating air-pump, as finally employed by Boyle, is still much in use; but it is gradually being superseded by Bianchi’s, which has but one cylinder and piston. The piston is, however, doubleacting, as the cylinder communicates both above and below by suitable valves with the main duct which leads to the receiver. Hence, during both the up and down strokes of the piston, exhaustion is being effected, the gas which is in the diminishing chamber being driven out through a suitable valve to the open air. The chief merit of Bianchi’s machine, however, as compared with the older form, lies in the mechanism by which the piston is driven. The end of the piston rod is attached by a crank to a rotating horizontal axle, which is in gearing with the axle of a flywheel 1 . The piston cylinder is capable of a reciprocating oscillatory motion about its lower end, which pivots on a horizontal axis. Thus, as the crank rotates, the piston rises and falls in the cylinder, and oscillates along with it from side to side. The driving power is applied to a handle fixed to the fly-wheel. This substitution of a continuous rotatory motion for a reciprocating motion greatly facilitates rapid exhaustion.
At every complete stroke of the piston the pressure of air in the receiyer is reduced by a definite fraction of itself, which depends upon the relative volumes of the receiver and piston cylinder. Hence the absolute change of pressure per stroke is smaller as the pressure is smaller; and the rate of exhaustion at very low pressures becomes practically inappreciable. There is, in fact, a practical limit to exhaustion, the particular value of which depends upon the special characteristics of the instrument. The best air-pumps of the type described above cannot reduce the pressure to less than what would balance one or two millimetres of mercury—that is, to what is technically called a pressure of one or two millimetres. To obtain a lower pressure or a higher vacuum, as it is commonly termed, requires the use of a different principle.
In a well-constructed barometer the region above the mercury contains no air. It is not an absolute vacuum, but is filled with vapour of mercury at a very low pressure —according tp Régnault, Ό372 mm. at 20° C., and Ό2 mm. at 0° C. The way in which the Torricellian vacuum has been applied to give a practical air-pump has been described in Mercurial Air-Pump. So long as mercury vapour is permitted to pass freely through the exhausting tubes, it is impossible to obtain by the mercurial air-pump pressures lower than the fortieth of a millimetre. If, however, there is placed somewhere in the duct leading from the mercury pump to the receiver a non-volatile substance which absorbs mercury vapour, the mercury vapour will be arrested. With similar precautions for the absorption of water vapour or other vapour which may be present, it is possible to obtain very low pressures indeed. Thus Crookes in his radiometer experiments obtained pressures as low as Ό0015 mm. or the ’ 2 millionth of an atmosphere. The form of pump used was of Geissler’s type. The water vapour was absorbed by phosphoric anhydride. Sulphur was used to absorb the mercury vapour; and on the further side metallic copper was placed to absorb any sulphur vapour which might tend to pass.
The best test of a good vacuum is the electrical test. Disruptive discharge through a long tube filled with gas is possible only when the gas is very rare, but there is a point in the rarefaction of a gas at which the discharge passes most easily. In other words, the dielectric strength of a gas reaches a minimum as its pressure is diminished, and it is possible to obtain such a high vacuum that electric discharge will not take place through any considerable length of the exhausted space (see Electricity). In this connexion we may notice a simple but instructive experiment of Dewar’s. He carefully exhausted a vacuum tube, in which he had previously inserted a piece of carbon. In· the last stages of the exhaustion the carbon was strongly heated so as to drive off most of the gas which is always condensed on its surface. After the completion of the exhaustion, the tube was sealed up and the carbon allowed to cool. As it cooled, it condensed over its surface the greater portion of the small quantity of gas left in the vacuum tube. The vacuum was thus vastly improved, so that it was impossible to pass an electric spark between two platinum electrodes which had been previously fused into the glass. A gentle heat applied to the carbon, however, was sufficient to drive off from its surface enough of the occluded gas to raise the pressure to the point necessary for the passage of the electricity. As the carbon cooled again, the high vacuum was restored and the discharge ceased.
It is hardly necessary to do more than mention the other class of air-pumps. Let the essential valves in an exhausting pump have their actions reversed, and the apparatus will become a condensing pump. The condensing syringe is the usual form of such a pump; but, compared to the exhausting instrument, it has very limited applications. For the mere obtaining of high pressures hydraulic means are preferable, being at once more manageable and less dangerous. Besides, even moderately slow compression of a gas is accompanied by considerable rise of temperature, just as rarefaction of a gas is accompanied by appreciable lowering of temperature. In the former case work is done in compressing the gas against its own pressure, and this energy appears as heat which raises the temperature of the gas. In the latter case the gas in expanding draws upon its own energy and so cools.
The thermal properties of gases are treated as a branch of mathematical physics under Heat and Thermodynamics, where also will be found discussed such experimental details as bear intimately upon the theory. The article Atom contains a concise statement of the modern kinetic theory of gases; and in Constitution of Bodies and Diffusion certain more special aspects of the same theory are regarded. For the mathematical theory see Molecule. The principles embodied in these articles have been assumed i hroughout the present article.
The cooling of a gas by its own expansion may be observed in one of its effects during exhaustion in an ordinary receiver. Frequently a cloud of minute drops of water—a veritable fog—forms in the exhausted air. The reason simply is that the air has become cooled below its dew-point, or the temperature corresponding to the pressure of water vapour present. If the receiver is first filled with dry air and then exhausted, no cloud forms. Also if the air is carefully filtered through cotton wool, no cloud forms, even though the air be thoroughly saturated with water vapour. This latter fact was established by Mr Aitken in his beautiful series of experiments on dust, fogs, and clouds. It thus appears that the formation of fog and cloud depends not only upon the humidity of the air but also upon the amount of dust in the air. The little particles of dust are necessary as nuclei upon which the vapour can begin to condense. The more numerous the dust particles the finer are the drops which form on them. As these coalesce into larger drops and fall, they bring down the dust nuclei with them; and hence the tendency of rain is to clear the atmosphere, and make conditions less favourable for the formation of more cloud and rain. Thus rain, fog, and dew all require for their formation a free solid surface, colder than the temperature of saturation, on which to condense. In a dustless atmosphere no cloud can ever form.
It has been already pointed out that when a fluid is in motion it can no longer be regarded as even approximately possessing the properties of the ideal perfect fluid. The postulate that the stress between contiguous portions is perpendicular to the common interface cannot be for a moment admitted. A few examples will make this clear. Thus, if a vessel filled with a liquid be set in rotation, the liquid will soon be found to be rotating with the vessel; and if the motion of the vessel be stopped, the motion of the liquid inside will gradually subside. These phenomena show the existence of a stress tangential to the fluid elements, whereby the relative motion of the different parts of the system is gradually destroyed, until the vessel and its contents behave as one solid body. Again, a fluid stream flowing along a tube or canal moves fastest in the middle, slowest at the bounding walls, and with all possible intermediate speeds at intermediate places. This retardation, due in the first instance to the action of the walls upon the fluid in direct contact therewith, and then to the friction between the successive contiguous sheets of fluid, plainly implies the existence of tangential stress. The action of the wind in causing waves on the smooth surface of a sheet of water is a further illustration.
In the discussion of fluid motion, however, it is customary to consider first the properties of the ideal fluid in this respect. Under Hydromechanics, and especially under Atom, will be found the mathematical theory treated in full so far as the motion of a non-viscous fluid is concerned.
The possibility of setting up vortex-motion in a fluid depends upon its imperfection, upon the property of viscosity or fluid friction which is possessed by all known fluids. Some of the more obvious effects of this property have been already noticed. Its effects indeed are conspicuous wherever there is relative motion of the contiguous parts of a fluid. A current of air moving through a mass of air at rest soon loses its momentum; a solid body moving through still air likewise has its motion retarded. The loss of energy which a meteoric stone so suffers as it speeds through the atmosphere appears in the form of heat, which is sufficient to make the stone glow to incandescence or to the temperature of rapid combustion. The waves of rarefaction and condensation, which constitute sound physically, gradually decay in virtue of viscosity. It appears that the rate of this decay is quickest for the shortest waves; so that a sound after travelling through a long distance may lose its shriller constituents and so be modified and mellowed. It is viscosity also which supports the minute dust particles and cloud-forming drops of water in our atmosphere. These are bulk for bulk heavier than the atmosphere, and in tranquil air are slowly sinking. The slowness of their downward motion is directly due to the effect of fluid-friction.
The term fluid-friction is applied because of the similarity of its effects to the effects of friction between solid masses. Otherwise there need be no further resemblance. The true nature of friction between solid surfaces is not known; possibly, as suggested by Sir W. Thomson, it may be in great measure electrical. In the case of gases, however, the origin of friction is more apparent. Its laws can be deduced from the kinetic theory, and depend directly upon the principle of diffusion. According to the kinetic theory of gases, the molecules of a gas are in constant motion amongst themselves. Compared to their own dimensions, they are by no means closely packed, so that any individual molecule travels a comparatively large space between its encounters with other molecules. Any two contiguous regions are continually interchanging molecules. This diffusion of a gas into itself we have no means of measuring experimentally, as we cannot deal with the individual molecules. Suppose, however, that we have two contiguous layers of a gas flowing in parallel directions with different speeds. The general drift of molecules in the one layer is faster than in the other. But this difference will not prevent the individual molecules diffusing across the interface between the layers. Diffusion will go on freely. The result will be that the slower moving layer will on the whole gain momentum in the direction of its motion and the faster moving layer lose momentum. Thus, diffusion tends to the equalization of momentum between two contiguous regions, and the rate at which this takes place across unit area is the measure of the viscosity. Maxwell has proved that the viscosity so measured is independent of the density of the gas when the temperature is constant; whereas the relation between the viscosity and temperature depends upon the particular mode of action between the molecules when they approach each other.
The above definition of viscosity is not one which can be used in experimental determinations, since we cannot take account of the individual molecules of a gas. The coefficient of viscosity must be defined in terms of directly measurable quantities.
Maxwell has defined viscosity in these words :—the viscosity of a substance is measured by the tangential force on unit of area of either of two horizontal planes at the unit of distance apart, one of which is fixed, while the other moves with unit of velocity, the space between being filled with the viscous substance. This is the dynamical definition. When the effects of viscosity on the internal motions of a fluid itself are being considered it is often more convenient to use the kinematical definition. It is. given in terms of μ, the coefficient of viscosity, by the equation
M = pv, where p is the density of the substance, and v the kinematic viscosity.
The viscosity of fluids has been determined experimentally in three distinct ways—by flow of the fluid through tubes, by motion in the fluid of pendulums or vibrating disks, and by the oscillations of spheres filled with the fluid. The last was employed by Helmholtz and Von Piotrowski in their investigation of liquids, but it is not applicable to the case of gases. Experiments on the flow through capillary tubes have been carried out by Poiseuille for liquids, and by Graham, Meyer, 5 Springmühl, andPuluj β for gases. 'Γhis, the transpiration method, is the most effective for comparing viscosities, which are directly proportional to the times of transpiration of the respective gases. There is, however, a little uncertainty as to the effect of the capillary tube,—so that, for measuring absolutely the viscosity for any one gas, the method is not so trustworthy as the second method. Here we may use either pendulums swinging through small arcs or disks oscillating in their own plane under the action of torsion. In both the measurement depends upon the rate at which the amplitude of oscillation diminishes.
Stokes, who first satisfactorily discussed the true nature of viscosity, tested the theory by a discussion of the pendulum experiments of Dubuat (1786), Bessel (1826), and Baity (1832). From Baily’s results he calculated ∙000104 (in metric units) as the coefficient of viscosity of air. Meyer similarly deduced from Bessel’s and Girault ’ s experiments the values ’ 000275 and Ό00384. It is not in the least surprising that these are all pretty wide of the true value, seeing that the experimenters had not the special problem of finding the viscosity before them.
Meyer, to whom we owe a very complete series of valuable memoirs on the subject, has more recently experimented with three different-sized pendulums. The values deduced for the viscosity were ’ 000232, Ό00233, and Ό00184. The last number, given by the shortest pendulum, Meyer considers to be the best.
Maxwell, Meyer, and Kundt and Warburg have experimented with oscillating disks. The methods of Maxwell and Meyer were so far similar that each used an arrangement of three horizontal circular disks, fixed centrally to the same vertical axis, and suspended by a torsion wire inside a receiver. The pressure and temperature of the air or gas inside could be adjusted to any desired values within certain limits. In Maxwell’s apparatus, which Meyer adopted in his later researches, the moving disks vibrated between parallel fixed disks, which were perforated in the centre so as to allow the vertical suspended axis to pass freely through them. Each disk thus oscillated in its own plane between two parallel fixed surfaces. After the disks were set in position, and the air in the receiver brought to the desired temperature and pressure, the suspended disks were set in oscillation. This was effected magnetically, a small magnet fixed to the end of the suspended axis being acted upon by an external magnet suitably adjusted. Each disk, in its oscillations, dragged after it the layer of air in immediate contact with it; and in virtue of viscosity this oscillation was trans mitted with diminishing amplitude from layer to layer until the fixed disks were reached. In thus setting and sustaining in motion a mass of gas, the disk was doing work; and, if left to itself and to the action of the torsion suspension, it oscillated with gradually diminishing range until it came to rest. The viscosity of the air was not the only retarding influence. The torsion wire had also a coefficient of viscosity; aud then there was a possible resistance due to the slipping of the fluid at the surfaces of the disks. These various effects were discriminated by suitable modifications. Thus by placing the oscillating disks in contact with each other, and setting two of the fixed disks at measured distances above and below, Maxwell reduced the number of surfaces in contact with the fluid, and so increased the relative importance of the effect due to the wire’s viscosity. Again, by diminishing the distances between the fixed and oscillating disks, he made the conditions more favourable to the effect (if any) due to the slipping. This latter effect was found to be so small as to be almost within the errors of observation; consequently Maxwell felt himself warranted in calculating the coefficient of viscosity on the assumption that there was no slipping.
Maxwell’s final result in metric (G. G. S.) units for the coefficient of viscosity of dry air is μ≈ ’ 0001878(l + ΌΟ3650), where θ is the temperature in degrees Centigrade. Meyer’s result is μ — ∙000190(l + · 00250).
Maxwell found the effect of pressure to be inappreciable down to a pressure of 12 mm., and thus verified the deduction from theory.
Kundt and Warburg, in their experiments, used only one disk, which oscillated under the influence of a bifilar suspension between two fixed disks. They earned the pressure down to as low as 0 ’ 6 mm. At 20 mm. pressure the viscosity was the same as at the atmospheric pressure; but at lower pressures a slight diminution began to show itself. According to Crookes’s later researches, this diminution becomes more and more marked at the higher exhaustions. The manner in which the viscosity then diminishes coincides remarkably with the manner in which the free path increases. It could not be expected that in such modified circumstances Maxwell’s law would continue to apply. When the gas becomes so far rarefied that the mean free path of a molecule is not small compared to the space in which the gas is confined, the motion of the molecules cannot be treated statistically. Hence the deductions from a theory based upon the statistic method will no longer hold good.
Maxwell, Kundt and Warburg, and Crookes investigated by the disk method the viscosities of other gases, the values for which are compared below with the transpiration times of the same gases through capillary tubes. Maxwell also found that damp air, at 100 mm. pressure, and over water at about 20° C., was one-sixtieth less viscous than dry air at the same temperature. Kundt and Warburg found for water vapour, at 21° C. and 16 mm. pressure, the value
μ=O000975, a little more than half that of air.
The results obtained by Meyer and Springmühl and by Puluj from their transpiration experiments agree well with those already given. In such experiments, however, the slipping of the gas over the solid surface has in certain circumstances a measurable effect. This slipping is measured by a certain coefficient, called the Gleitungs∙Coeffi,cient by Helmholtz and Von Piotrowski. When this coefficient becomes appreciable, the gas in contact with the solid surface, instead of being at rest relatively to that surface, will be gliding over it with a finite velocity v. The circumstances of the motion will be very nearly the same if we remove a layer of the solid surface and replace it by fluid, the new surface of fluid in contact with the new solid surface being at rest. The thickness which must be so removed is the measure of the coefficient of slipping. Kundt and Warburg, in their experiments with glass tubes, found this coefficient for dry air at about 20° C. to be
8/p centimetres, where p is the pressure in dynes per square centimetre, which is nearly the same as in millionths of an atmosphere. The value for hydrogen on glass is 15∕p. Hence at ordinary pressures and moderate exhaustions this coefficient is very small, becoming appreciable only at low pressures.
The relation between viscosity and temperature is indicated at once by Maxwell’s and Meyer’s formulae given above. According to Maxwell, the viscosity is proportional to the absolute temperature. If in the kinetic theory the forces between the molecules are disregarded, that is, if the molecules are assumed to rebound after collision like elastic spheres, the relation deduced is that the viscosity varies as the square root of the absolute temperature. Hence the mutual molecular forces must be taken into account. Maxwell’s experimental law would require any two molecules to repel each other with a force varying inversely as the fifth power of the distance. According to Meyer, however, the viscosity varies according to a power of the absolute temperature less than unity, but greater than one-half. His results in this respect are corroborated by those of Kundt and Warburg, Puluj, and other later experimenters. The ’ 77 power is probably not far from the truth. Hence we may give as the final value for the viscosity of dry air the expression μ = ∙000185(1 + ’00280).
The following table gives the values for the different gases, as determined by the different investigators, the viscosity of air being taken as unity. [table]
We do not here enter into the question of the thermodynamics of gases; enough to say that the relations between viscosity, diffusion, and thermal conductivity deduced by Maxwell from the kinetic theory have received remarkable corroboration from the experiments of Loschmicht, Stefan, Kundt and Warburg, and others. A discussion of the dynamical properties of gases would not. however, appear complete without mention of Crookes’s so-called radiometer, even though these phenomena of high vacua are ultimately thermodynamic.
The typical form of the radiometer is a glass bulb, in which is hung a delicately poised arrangement of vanes. These, usually four in number, are fixed at the extremities of two light horizontal cross-rods, which are supported so as to be capable of easy rotation about a central vertical axis. The vanes or disks are set in vertical planes passing through the axis; and each has its one side bright, and the other blackened. For any rotation the motion of each vane is exactly alike; that is, either the bright faces all move first, or the dark faces do so. If the pressure of the gas inside the bulb is reduced to a very low exhaustion, the vanes under the action of light or heat will begin to rotate. The mere bringing the radiometer out of a dark region into daylight is enough to set up this rotation. In ordinary circumstances the dark faces are apparently repelled, and the vanes move round with their bright faces in advance.
The phenomenon is really a thermal one, as was demonstrated experimentally by Tait and Dewar. Further, although it is most evident in high vacua (provided they are not too high), it can be produced in very moderate exhaustions by a suitable arrangement, as was long ago pointed out by Fresnel. Thus, if under the receiver of an ordinary air-pump a light disk be delicately poised near a parallel fixed surface, it will be apparently strongly repelled by this surface if the opposing surfaces are brought to different temperatures. This may be effectively done by means of a ray of sunlight. In this experiment, the essential condition is (as shown by Tait and Dewar) that the surfaces be at a distance comparable to the mean free path of the gaseous molecules. In Crookes’s radiometer the free path is very long, and hence there is apparent repulsion between the blackened surfaces and the walls of the bulb. The reason simply is that, under the action of the radiant energy directed in upon the vanes, the dark faces, absorbing more energy, become warmer than the bright faces. Hence an inequality of temperature is produced in the highly rarefied gas, and this brings into existence a stress which displaces the vanes.
Liquid in the spheroidal state illustrates the same principle. That a drop of water may be supported over a hot surface without touching it requires an upward pressure. In other words, the vertical stress in the vapour and gas which separate the drop from the surface must be greater than the ordinary gaseous pressure all round the drop. This stress exists because of the difference of temperature between the drop and the surface, so that the pressure in the thin layer of vapour and gas is slightly greater in the vertical than in any horizontal direction.
A general notion of the manner in which this stress is sustained may be obtained from the following consideration. According to the kinetic theory of gases, the mean speed of the molecules is a function of the temperature—the higher the temperature the greater the speed. Hence molecules, impinging upon a surface at a higher temperature, and in a direction more nearly perpendicular to it, will rebound from that surface with increased momentum. The simultaneous motion of the surface, as if repelled, is then somewhat analogous to the recoil of a cannon when fired. The whole investigation of the question is, however, by no means simple. Maxwell has discussed it with characteristic lucidity in his latest contribution to the dynamical theory of gases. He finds that, when inequalities of temperature exist at a given point in a gas, the pressure is not the same in all directions. Its value in any given direction, in so far as it depends upon the temperature inequality, is proportional to the space-rate of change of the space-variation of the temperature in that direction—that is, to the second differential coefficient of the temperature with respect to the given direction. Hence the pressure will be greatest along the line for which this differential coefficient is a maximum. It appears that the pressure so called into existence by a possible temperature inequality is very minute at ordinary hydrostatic pressures, but becomes considerable when the pressure of the gas is made very small. If the inequality of temperature throughout the gas is due to the presence of small bodies, whose temperatures differ from the temperature of the gas at a distance from them, then the small bodies will be acted upon by the stresses set up, provided they are of the same order of smallness as the mean free path of the molecules. In the case of two such small bodies, there will be apparent repulsion between them if the bodies are warmer than the air at a distance from them, and attraction if they are colder. If one is warmer and the other colder, the action may be either attractive or repulsive, according to the relative sizes of the bodies and their exact temperatures. These results are obtained by considering only the stresses normal to the solid surfaces. When the tangential stresses are taken into account, then it appears that inequality of temperature, when the flow of heat becomes steady, cannot produce other than equilibrium in the material system immersed in the gas. Hence Maxwell believes that the explanation of Crookes’s phenomenon must depend ultimately upon the slipping of the gas over the solid surface. If such slipping be permitted, its effect will be to diminish the tangential stresses acting on the solid surface without affecting the normal stresses; and hence the equilibrium will be destroyed. In attempting to express the conditions to be satisfied by the gas at the solid surface. Maxwell is led to the consideration of the phenomenon discovered by Osborne Reynolds and named thermal transpiration. This phenomenon consists of a sliding of the gas over the surface of an unequally heated solid from the colder to the hotter parts. Maxwell considers the particular case of the slow steady flow of gas along a capillary tube of circular section, the temperature of which varies steadily from point to point. The amount of gas which passes through any section depends both upon the rate of change of pressure and the rate of change of temperature in passing along the axis of the tube. If the pressure is uniform there will be a flow of gas from the colder to the hotter end. If there is no flow of gas, the pressure will increase from the colder to the hotter end. The case of uniform temperature is the ordinary case of transpiration through capillary tubes, as discussed experimentally by Graham, Meyer, Puluj, and Kundt and Warburg. The experimental investigation of the first two cases seems at present hopeless, on account of the minuteness of the quantities to be measured. Reynolds experimented, not on capillary tubes, but on the passage of the gas through a porous plate, the temperatures being different on the two sides.
