For aëronauts it is very useful to know, besides this, the weight of the air at different heights, and I therefore insert the adjoining table, constructed on the basis of Glaisher's data, for the temperature and moisture of the atmospheric strata in clear weather. All the figures are given in the metrical system—1,000 millimetres = 39·37 inches, 1,000 kilograms = 2204·3375 lbs., 1,000 cubic metres = 35,316·6 cubic feet. The starting temperature at the earth's surface is taken as = 15° C., its moisture 60 p.c., pressure 760 millimetres. The pressures are taken as indicated by an aneroid barometer, assumed to be corrected at the sea level and at lat. 45° C. If the height above the level of the sea equal z kilometres, then the weight of 1 cubic metre of air may be approximately taken as 1·222 - 0·12z + 0·00377z2 kilogram.

PressureTemperatureMoistureHeightWeight of the air
760mm. 15° C.60p.c. 0metres1222kilos. 1,000 cubic metres
700„ 11·0°„64„ 690„1141„
650„ 7·6°„64„1200„1073„
600„ 4·3°„63„1960„1003„
550„ - 1·0°„62„2660„ 931„
500„ - 2·4°„58„3420„ 857„
450„ - 5·8°„52„4250„ 781„
400„ - 9·1°„44„5170„ 703„
350„-12·5°„36„6190„ 624„
300„-15·9°„27„7360„ 542„
250„-19·2°„18„8720„ 457„

Although the figures in this table are calculated with every possible care from average data, yet they can only be taken approximately, for in every separate case the conditions, both at the earth's surface and in the atmosphere, will differ from those here taken. In calculating the height to which a balloon can ascend, it is evident that the density of gas in relation to air must be known. This density for ordinary coal gas is from 0·6 to 0·35, and for hydrogen with its ordinary contents of moisture and air from 0·1 to 0·15.

Hence, for instance, it may be calculated that a balloon of 1,000 cubic metres capacity filled with pure hydrogen, and weighing (the envelope, tackle, people, and ballast) 727 kilograms, will only ascend to a height of about 4,250 metres.

[24] If a cracked flask be filled with hydrogen and its neck immersed under water or mercury, then the liquid will rise up into the flask, owing to the hydrogen passing through the cracks about 3·8 times quicker than the air is able to pass through these cracks into the flask. The same phenomenon may be better observed if, instead of a flask, a tube be employed, whose end is closed by a porous substance, such as graphite, unglazed earthenware, or a gypsum plate.

[25] According to Boyle and Mariotte's law, for a given gas at a constant temperature the volume decreases by as many times as the pressure increases; that is, this law requires that the product of the volume v and the pressure p for a given gas should be a constant quantity: pv = C, a constant quantity which does not vary with a change of pressure. This equation does very nearly and exactly express the observed relation between the volume and pressure, but only within comparatively small variations of pressure, density, and volume. If these variations be in any degree considerable, the quantity pv proves to be dependent on the pressure, and it either increases or diminishes with an increase of pressure. In the former case the compressibility is less than it should he according to Mariotte's law, in the latter case it is greater. We will call the first case a positive discrepancy (because then d(pv)/d(p) is greater than zero), and the second case a negative discrepancy (because then d(pv)/d(p) is less than zero). Determinations made by myself (in the seventies), M. L. Kirpicheff, and V. A. Hemilian showed that all known gases at low pressures—i.e. when considerably rarefied—present positive discrepancies. On the other hand, it appears from the researches of Cailletet, Natterer, and Amagat that all gases under great pressures (when the volume obtained is 500–1,000 times less than under the atmospheric pressure) also present positive discrepancies. Thus under a pressure of 2,700 atmospheres air is compressed, not 2,700 times, but only 800, and hydrogen 1,000 times. Hence the positive kind of discrepancy is, so to say, normal to gases. And this is easily intelligible. If a gas followed Mariotte's law, or if it were compressed to a greater extent than is shown by this law, then under great pressures it would attain a density greater than that of solid and liquid substances, which is in itself improbable and even impossible by reason of the fact that solid and liquid substances are themselves but little compressible. For instance, a cubic centimetre of oxygen at 0° and under the atmospheric pressure weighs about 0·0014 gram, and at a pressure of 3,000 atmospheres (this pressure is attained in guns) it would, if it followed Mariotte's law, weigh 4·2 grams—that is, would be about four times heavier than water—and at a pressure of 10,000 atmospheres it would be heavier than mercury. Besides this, positive discrepancies are probable because the molecules of a gas themselves must occupy a certain volume. Considering that Mariotte's law, strictly speaking, applies only to the intermolecular space, we can understand the necessity of positive discrepancies. If we designate the volume of the molecules of a gas by b (like van der Waals, see Chap. I., Note [34]), then it must be expected that p(v - b) = C. Hence pv = C + bp, which expresses a positive discrepancy. Supposing that for hydrogen pv = 1,000, at a pressure of one metre of mercury, according to the results of Regnault's, Amagat's, and Natterer's experiments, we obtain b as approximately 0·7 to 0·9.

Thus the increase of pv with the increase of pressure must be considered as the normal law of the compressibility of gases. Hydrogen presents such a positive compressibility at all pressures, for it presents positive discrepancies from Mariotte's law, according to Regnault, at all pressures above the atmospheric pressure. Hence hydrogen is, so to say, a perfect gas. No other gas behaves so simply with a change of pressure. All other gases at pressures from 1 to 30 atmospheres present negative discrepancies—that is, they are then compressed to a greater degree than should follow from Mariotte's law, as was shown by the determinations of Regnault, which were verified when repeated by myself and Boguzsky. Thus, for example, on changing the pressure from 4 to 20 metres of mercury—that is, on increasing the pressure five times—the volume only decreased 4·93 times when hydrogen was taken, and 5·06 when air was taken.

The positive discrepancies from the law at low pressures are of particular interest, and, according to the above-mentioned determinations made by myself, Kirpicheff, and Hemilian, and verified (by two methods) by K. D. Kraevitch and Prof. Ramsay (London, 1894), they are proper to all gases (even to those which are easily compressed into a liquid state, such as carbonic and sulphurous anhydrides). These discrepancies approach the case of a very high rarefaction of gases, where a gas is near to a condition of maximum dispersion of its molecules, and perhaps presents a passage towards the substance termed ‘luminiferous ether’ which fills up interplanetary and interstellar space. If we suppose that gases are rarefiable to a definite limit only, having attained which they (like solids) do not alter in volume with a decrease of pressure, then on the one hand the passage of the atmosphere at its upper limits into a homogeneous ethereal medium becomes comprehensible, and on the other hand it would be expected that gases would, in a state of high rarefaction (i.e. when small masses of gases occupy large volumes, or when furthest removed from a liquid state), present positive discrepancies from Boyle and Mariotte's law. Our present acquaintance with this province of highly rarefied gases is very limited (because direct measurements are exceedingly difficult to make, and are hampered by possible errors of experiment, which may be considerable), and its further development promises to elucidate much in respect to natural phenomena. To the three states of matter (solid, liquid, and gaseous) it is evident a fourth must yet be added, the ethereal or ultra-gaseous (as Crookes proposed), understanding by this, matter in its highest possible state of rarefaction.

[26] The law of Gay-Lussac states that all gases in all conditions present one coefficient of expansion 0·00367; that is, when heated from 0° to 100° they expand like air; namely, a thousand volumes of a gas measured at 0° will occupy 1367 volumes at 100°. Regnault, about 1850, showed that Gay-Lussac's law is not entirely correct, and that different gases, and also one and the same gas at different pressures, have not quite the same coefficients of expansion. Thus the expansion of air between 0° and 100° is 0·367 under the ordinary pressure of one atmosphere, and at three atmospheres it is 0·371, the expansion of hydrogen is 0·366, and of carbonic anhydride 0·37. Regnault, however, did not directly determine the change of volume between 0° and 100°, but measured the variation of tension with the change of temperature; but since gases do not entirely follow Mariotte's law, the change of volume cannot be directly judged by the variation of tension. The investigations carried on by myself and Kayander, about 1870, showed the variation of volume on heating from 0° to 100° under a constant pressure. These investigations confirmed Regnault's conclusion that Gay-Lussac's law is not entirely correct, and further showed (1) that the expansion per volume from 0° to 100° under a pressure of one atmosphere, for air = 0·368, for hydrogen = 0·367, for carbonic anhydride = 0·373, for hydrogen bromide = 0·386, &c.; (2) that for gases which are more compressible than should follow from Mariotte's law the expansion by heat increases with the pressure—for example, for air at a pressure of three and a half atmospheres, it equals 0·371, for carbonic anhydride at one atmosphere it equals 0·373, at three atmospheres 0·389, and at eight atmospheres 0·413; (3) that for gases which are less compressible than should follow from Mariotte's law, the expansion by heat decreases with an increase of pressure—for example, for hydrogen at one atmosphere 0·367, at eight atmospheres 0·369, for air at a quarter of an atmosphere 0·370, at one atmosphere 0·368; and hydrogen like air (and all gases) is less compressed at low pressures than should follow from Mariotte's law (see Note [25]). Hence, hydrogen, starting from zero to the highest pressures, exhibits a gradually, although only slightly, varying coefficient of expansion, whilst for air and other gases at the atmospheric and higher pressures, the coefficient of expansion increases with the increase of pressure, so long as their compressibility is greater than should follow from Mariotte's law. But when at considerable pressures, this kind of discrepancy passes into the normal (see Note [25]), then the coefficient of expansion of all gases decreases with an increase of pressure, as is seen from the researches of Amagat. The difference between the two coefficients of expansion, for a constant pressure and for a constant volume, is explained by these relations. Thus, for example, for air at a pressure of one atmosphere the true coefficient of expansion (the volume varying at constant pressure) = 0·00368 (according to Mendeléeff and Kayander) and the variation of tension (at a constant volume, according to Regnault) = 0·00367.

[27] Permanent gases are those which cannot be liquefied by an increase of pressure alone. With a rise of temperature, all gases and vapours become permanent gases. As we shall afterwards learn, carbonic anhydride becomes a permanent gas at temperatures above 31°, and at lower temperatures it has a maximum tension, and may be liquefied by pressure alone.