CHAPTER V

THE MOLECULAR THEORY OF GASES

The more obvious physical phenomena of gases were, of course, well known by the middle of the nineteenth century; and the so-called gaseous laws—the laws of Boyle, Dalton, and Gay Lussac—were universally accepted by chemists and physicists at that time as fundamental. That the first two laws were only approximations to truth in a mathematical sense was also well known; and the experimental labours of Regnault and Magnus had not only established limits within which they were inexact, but had, to some extent, indicated the cause of their departure from the ideal condition. The hypothesis of Avogadro, as already stated, was practically ignored at this period, or at least its value was unappreciated until the time of Gerhardt and Laurent, and more particularly Cannizzaro, who in 1858 pointed out its real meaning and made it the keystone in the edifice of modern chemistry.

One of the most significant achievements of the last half-century has been the demonstration that these gaseous laws are interdependent. Their further study, and in particular the study of their variation from exact mathematical expression, has led to a conception of the real nature of a gas which not only comprehends and knits together these laws, but affords a rational explanation of them. If the laws of Boyle and Dalton concerning the relations between pressure, temperature, and gaseous volume are, and must be from the very nature of the case, only approximations, it follows that the same is equally true of the laws of Gay Lussac and Avogadro, since these are dependent on the others. The definite experimental proof that gases do not actually combine in the precise ratios demanded by the law of Gay Lussac has been forthcoming only during the last twenty years. It has been found that, instead of oxygen and hydrogen combining in the exact ratio of one volume of oxygen to two volumes of hydrogen to form water, as stated by Gay Lussac, one volume of oxygen combines with, according to Scott, 2.00245 vols.; according to Leduc, 2.0024 vols.; according to Morley, 2.00268 vols. of hydrogen. What is true of the volume-ratios in which oxygen and hydrogen are actually found to combine, under ordinary conditions, is no doubt equally true of analogous instances, such as the union of hydrogen and chlorine to form hydrogen chloride. It follows also that the extent of the variation from the mathematical expression of Gay Lussac’s law must get smaller and smaller as the combining gases approach the condition of the ideal gas—as they do, for example, under very low pressures. The precise degree of departure from Gay Lussac’s law is therefore in a sense accidental, and is dependent upon the conditions under which combination takes place.

The more exact study of the physical phenomena of gases, and in particular the clearer recognition of the causes which determine the extent of their departure from the ideal gaseous laws, have afforded valuable assistance in ascertaining the atomic weights of certain elements independently of chemical considerations. The processes of physical measurement have been so refined within recent years that physical methods of arriving at molecular—and, inferentially, at atomic—weights, in the case of all elements and compounds which can be brought into a condition approaching that of the ideal gas, are to be preferred to the gravimetric methods of analysis or synthesis as affording the most probable values of the true atomic weights of the elements. The work of Lord Rayleigh, Leduc, and of Guye and his pupils on the densities of the gases has furnished us with a series of values for the atomic weights of a number of the elements which, in point of accuracy, are as superior to the values of Stas as the values of Stas were superior to those of his predecessors. Daniel Berthelot pointed out in 1898 that the true molecular weight of a gas can be deduced from its density and its observed variations from Boyle’s law under atmospheric pressure and at very low pressures. Incidentally, the study of gaseous phenomena has served to place the theory of atoms upon a far more stable foundation than it occupied half a century ago. How halting was the adhesion which even some of the most eminent chemists then gave to this theory was well exemplified by the remarkable lecture given before the Chemical Society of London in 1869, in which Williamson—one of the most sturdy champions of Dalton’s doctrine—set forth its true value.

That a gas may be looked upon as an association of particles—hard elastic spheres—moving backwards and forwards in right lines with great velocity, and possessing in the aggregate a very small proportion of the space through which they travel, was first conceived by Daniel Bernoulli in 1738. By means of this hypothesis he explained the direct proportionality between the density and pressure of a gas. If the gas consists of moving particles, and the pressure which it exerts on the sides of the containing vessel is due to the impacts of these particles, it is obvious that by halving the original volume of the containing space we halve the space through which the particles travel, and therefore double the number of their impacts in a given time; in other words, by compressing the gas to half its initial volume we double the pressure it exerts, which is nothing else than the law of Boyle. This conception of the nature of a gas is known as the _kinetic theory of gases_; it was further developed by Waterston in 1845, still more fully by Clausius in 1857, and was subsequently placed in its present position by Maxwell and Boltzmann.

That gases actually do move, and at rates depending on their specific nature, was rendered probable—apart from this explanation of Boyle’s law—by many phenomena observed by chemists and physicists in the eighteenth and early part of the nineteenth century. It was known from the observations of Leslie in 1804 that specifically light gases moved or diffused faster than heavy gases. Attempts to determine these rates were made by Schmidt in 1820, and by Graham in 1846, both of whom found that the rate of movement of a gas was independent of its chemical nature, and was determined solely by its mass: _gases move at rates inversely proportional to the square roots of their densities_. The following table given by Graham shows the experimentally ascertained relative rates for a number of gases compared with the rates demanded by the “_law of gaseous diffusion_.” Column one gives the name of the gas; column two, the observed rate of diffusion; and column three, the square root of the density of the gas (air = 1):

_Gas._ _Time of diffusion._ _√density._

Air 1 1 Hydrogen 0.276 0.263 Marsh gas 0.753 0.745 Ethylene 0.987 0.985 Nitrogen 0.986 0.986 Oxygen 1.053 1.051 Carbon dioxide 1.203 1.237

Nitrogen and ethylene are, chemically, totally dissimilar gases, but they have the same density and hence the same rate of movement. As Graham showed, it is possible to separate more or less completely a mixture of gases, if the constituents are of different densities, by taking advantage of their different rates of movement. Such an _atmolytic_ method was employed by Rayleigh and Ramsay to prove that atmospheric nitrogen contained argon.

The fact that all gaseous substances, however different their chemical nature, conform in the main to certain simple “laws” indicates the probability that their mechanical structure is similar and comparatively simple. The so-called gaseous “laws”—the laws of Boyle, Dalton, Gay Lussac, Avogadro, and Graham—are to-day explained on the assumption that a gas consists of an aggregation of molecules, moving incessantly in straight lines and with great rapidity. The rate of movement of the particles is variable by reason of their mutual encounters; at the same instant some are moving rapidly, others more slowly. As already explained, to this ceaseless movement of the molecules is to be ascribed the pressure they exert; the pressure which a gas exerts on any containing surface is the aggregate effect of the impact of its molecules. The law of Boyle states that the product of the volume V and pressure P of a given mass of gas is invariable so long as the temperature is unchanged: PV = constant. It was found by Regnault, Magnus, Natterer, and Amagat that all gases, with the exception of hydrogen, show a departure from Boyle’s law in the sense that PV is less than theory demands. In the case of hydrogen PV is greater than theory. This exception, however, is only apparent. Every gas, if maintained above a certain temperature, shows, after a certain pressure has been reached, a deviation in the same sense as that exhibited by hydrogen.

The deviations from Boyle’s law are probably due to two causes: (1) to the effect of cohesion among the molecules, whereby the volume, and hence PV, is less than theory requires; (2) the molecules are not mathematical points—they have a certain volume; hence, with increasing pressure, PV is greater than theory demands. The effect of the molecule having a certain magnitude will be clear from the following figure: Let M be a molecule moving backwards and forwards within a certain space, _a b_:

| M | _a_ | | _b_ | • | | |

Assume, now, we halve the containing space:

| M | _a_ | | _b_ | • | | |

It will be seen that M, since its volume is unchanged, will have less than half the original distance to travel or, in other words, it will strike the boundaries of the containing space _more_ than twice as frequently in the same interval of time as before; hence P, and therefore PV, becomes greater than Boyle’s law demands.

It will be noticed, then, that the two causes tending to bring about deviations from Boyle’s law act in contrary directions. In the greater number of gases the effect due to cohesion at ordinary pressure is greater than the effect due to the actual space occupied by the molecules. In the case of hydrogen at ordinary temperature the contrary is the case; if, however, hydrogen is strongly cooled, it shows variations similar to those exhibited by other gases at ordinary temperatures. By heating these gases the effect due to cohesion—to the mutual attraction of the molecules—becomes less and less; in such circumstances these gases show departures from theory in the same sense that hydrogen does at ordinary temperatures.

The effect of mutual attraction among the molecules is to make the volume of the gas less than the theoretical value; the cohesive force may therefore be regarded as equal in effect to a certain additional pressure; that is, (P + A) V = constant, in which A is the measure of the force of cohesion. A, of course, must have relation to the number of molecules mutually attracted: _A is proportional to the square of the number of the molecules_. But the number of the molecules in the unit volume is proportional to the density of the gas, and in a given mass of gas the density is inversely proportional to the volume. Hence A is inversely proportional to the square of the volume—A = _a_/V2, hence (P + _a_/V2) V = constant. Now let us trace the effect of the second cause of variation from the mathematical exactitude of Boyle’s law. The fact that the molecules are not mathematical points means that V in the foregoing expression is not identical with the space in which the molecules move. That space is V - _b_, in which _b_ is the measure of the aggregate volume of the molecules. Hence the true expression becomes (P + _a_/V2) (V - _b_) = constant.

The law of Dalton (Charles) also receives its simplest explanation by the kinetic theory of gases; and, moreover, the departures from the mathematical truth of the statement follow as a necessary consequence of the facts that the molecules have sensible magnitudes and are mutually attracted. We can measure the effect of heat upon a gas in two ways. We can either keep the pressure of the gas constant, and measure the increase in volume; or we can prevent the gas from expanding, and measure the elastic force or pressure it exerts. If the law of Dalton were mathematically true, it would follow that, _if the volume of the gas were maintained constant during the heating, its pressure would increase in the same proportion as the volume would have increased if the gas had been allowed to expand, but maintained at a constant pressure_. In other words, the expansion-coefficient and the pressure-coefficient should be the same. Experiment shows, however, that they are not identical.

The following table gives the results of a number of measurements by Regnault:

_Expansion _Pressure (Pressure (Volume Constant)._ Constant)._

Hydrogen .003661 .003667 Air .003670 .003665 Carbon dioxide .003710 .003688 Sulphur dioxide .003903 .003845

Variations in the same sense have since been observed by Jolly and Chappuis. With the exception of hydrogen, and probably also helium, all the gases show greater values for the coefficient of expansion than for the coefficient of pressure, and the differences are greater the greater the coefficient of expansion of the gas.

Since, as we have already stated, the law of Boyle is directly related to the law of Dalton, both being dependent on molecular movements, the same course of reasoning used to account for the variations in the case of Boyle’s law applies equally to the case of Dalton’s law. The “law” of Avogadro follows also as a necessary consequence of this explanation of the laws of gaseous pressure and temperature. If all gases show approximately the same increase in pressure when heated under constant volume, and if the increased pressure is due only to the increased energy with which the molecules strike the sides of the containing vessel, it follows that all gases must contain the same number of molecules in unit volume. But as, from the very nature of the case, the laws of Boyle and Dalton cannot be mathematically true, it follows that the laws of Avogadro and Gay Lussac must be only approximations in the same sense.

The law of Graham, connecting the rate of diffusion of a gas with its density, follows also as a necessary consequence of this explanation of the laws of Boyle, Dalton, Gay Lussac, and Avogadro. If the number of molecules in the unit volume of any gas, whatever be its nature and whatever be their mass, is approximately the same, it follows that the mean velocity of the molecules must be variable; their mean velocities must be in the inverse ratio of the square roots of their densities.

The mean velocity with which the molecules of a gas move can be calculated if we know the pressure it exerts, the weight of a definite volume, and the value of the acceleration due to gravity. The square of this velocity in metres per second of time at 0°C. is given by the expression U² = 3_pg_/_q_ in which _p_ = pressure per square metre = 10,333 kilograms; _g_ = the gravitation constant = 9.81; _q_ = weight of a cubic metre of the gas at 0°C. and one atmosphere of pressure.

For hydrogen we have

U² = 3 × 10,333 × 9.81/0.0899

whence U = 1842 metres per second; for oxygen we have U² = 3 × 10,333 × 9.81/1.430, whence U = 461. These numbers accord with those demanded by Graham’s law. The density of H being taken as 1, that of oxygen is 16 and √16 = 4; the numbers 1842 and 461 are in the ratio of 4 to 1.

The amount of heat required to raise the temperature of the unit mass of a gas through a definite interval depends, as Laplace first pointed out, upon whether the gas is allowed to expand or not; in other words, the specific heat of a gas varies as the heating is at constant volume or at constant pressure. If, having raised the temperature of the unit mass, and so expanded it, we then compress it until it occupies its initial volume, a further rise of temperature takes place without any external heat having been applied. This rise of temperature is, in fact, due to the liberation of the amount of heat required merely to expand the gas without increasing its temperature. The quantity of heat needed to raise the temperature of a gas through a definite interval is therefore greater when it is allowed to expand than when its volume is kept constant; in other words, the specific heat at constant pressure is greater than the specific heat at constant volume. The ratio of the two specific heats can be calculated: on the assumption that the energy imparted to the molecules simply accelerates their mean rectilinear velocity, and that no energy is absorbed in doing internal work among them, it is found that, when the gas is permitted to expand, the amount of heat required is 1.67 times greater than that needed when its volume is kept constant. This ratio has been experimentally determined for a number of gases. For oxygen under normal conditions it is 1.408, for hydrogen 1.414, for carbon dioxide 1.264, for methane 1.269—all numbers notably below the value 1.67. The direct experimental determination of this ratio by thermometric measurements is a matter of some difficulty. It was, however, demonstrated by Dulong that it can be ascertained with comparative ease from observations on the velocity of sound in the gas—the velocity being probably a direct function of this ratio. As carried out experimentally, the method consists in sending a sound-wave through the gas contained in a glass tube along the horizontal length of which is strewn a quantity of a light powder such as the spores of lycopodium or finely divided silica. The glass tube is fitted at one end with a glass rod; by rubbing this a series of longitudinal vibrations is set up and communicated to the gas whereby the light powder is thrown up into little heaps along the tube, the distance between the heaps being equal to half a wave length. By comparative measurements with air and the gas under examination, data are obtained from which the ratio of the specific heats can be deduced.

By experiments conducted on this principle Kundt and Warburg found that mercury vapour gave numbers agreeing with the theoretical ratio 1.67. Now, its vapour density shows that mercury vapour is a monatomic gas; it actually fulfils the conditions prescribed for a gas which theory indicates should give the value 1.67. All the energy imparted to its molecules on heating simply accelerates their translational velocity. On the other hand, all the gases above named as giving values below 1.67 are diatomic gases; in their case the energy imparted to them is employed partly in augmenting the translational velocity of the molecules, and partly in bringing about internal changes within them. By experiments made in like manner Ramsay and Travers succeeded in showing that the inert gases of the atmosphere are monatomic.

No attempt can be made here to explain the various methods by which it has been sought to obtain an estimate of the absolute size of gaseous molecules or to determine their number in a definite volume. By observations on their viscosity, rates of diffusion, conductivity for heat, variations from the law of Boyle, dielectric constants, electric charges, etc., Maxwell, O. E. Meyer, Loschmidt, Lothar Meyer, Van der Waals, Mossotti, Planck, Sir J. J. Thomson, and others, have arrived at estimates of the magnitude and number of molecules in a gas. These estimates necessarily vary with the hypotheses made in deducing them. It would serve no useful purpose to give the results, since the figures convey no impression to the mind of the minuteness of molecules, or even as to the extraordinary number of them in, say, so small a volume as one cubic centimetre. As an example, it has been calculated that there are about 640 trillions of hydrogen molecules in one milligram of the gas.[3]

[3] O. E. Meyer, _The Kinetic Theory of Gases_, 1899.

* * * * *

In the preceding volume a short account has been given of the history of the early attempts to effect the liquefaction of the gases. These resulted in their division into the two classes of _liquefiable_ and _permanent_ gases. One of the most notable achievements of the latter half of the last century was to sweep away this arbitrary distinction. The fundamental condition needed to effect the liquefaction of a gas, although surmised by Faraday, was first clearly indicated by Andrews about 1863. He showed that, in order to liquefy a gas, its temperature must be lowered to a point peculiar to each gas, when, on the application of sufficient pressure, it will become a liquid. Thus, in the case of gaseous carbon dioxide, Andrews found that, if its temperature were maintained above 31° C., no amount of pressure would cause it to liquefy; if the temperature were lowered just below this point—termed the _critical point_—a pressure of 75 atmospheres would effect its liquefaction. On the other hand, if the temperature of the liquid carbon dioxide be slowly raised to about 31°, the surface of demarcation between the liquid and the gas becomes gradually fainter and eventually disappears. Carbon dioxide may thus be made to pass from the state of liquid to that of gas without any sudden alteration of volume. If a given volume of the gas, say at 50°, be exposed to gradually increasing pressure, say up to 150 atmospheres, the volume is gradually diminished with the increment of pressure, but no sudden contraction indicating liquefaction occurs. If the gas under the high pressure be allowed to cool down to the ordinary temperature, no sudden contraction is observed to follow. The carbon dioxide, at the outset a gas, in the end becomes a liquid by a gradual and continuous transition, unaccompanied by any abrupt change of volume. These observations show that what we style the liquid and gaseous states are simply separated manifestations of the same condition of matter. There is a definite temperature for every gaseous substance at which it ceases to be liquefiable under pressure; and the reason that Faraday failed to liquefy certain gases was that he was unable, with the means at his command, to lower their temperatures sufficiently and so reach their critical points; hence the enormous pressures which he and other investigators applied were unavailing. These facts were definitely made known by Andrews in 1869, were theoretically developed by Van der Waals in 1873, and practically applied to the liquefaction of oxygen in 1877, independently and almost simultaneously, by Pictet, of Geneva, and Cailletet, of Châtillon-sur-Seine. Pictet exposed oxygen, under great pressure, to the cold produced by the rapid evaporation of liquid carbon dioxide; Cailletet brought about the same result by suddenly diminishing the tension of the strongly compressed oxygen, the rapid expansion of the gas effecting the reduction of its temperature below the critical point. Other workers took up the subject, notably Wroblewski and Olszewki in Poland, Dewar in England, and Kammerlingh Onnes in Holland; and the liquefaction of all the gases has now been accomplished.

The following table shows the _absolute_ boiling-(B.P.) and melting-points (M.P.), critical points (C.P.), and pressures (C.Press.), together with the density (D) at their boiling-points of a number of liquefied gases:

C. B. P. M. P. C. P. Press. D. _degrees_ _degrees_ _degrees_ _m._

Helium 4.5 —— —— —— 0.15 Hydrogen 20 15 35 11.6 0.06 Oxygen 90.5 below 50 154 44 1.131 Nitrogen 77.5 60 124 20.9 0.791 Methane 108.3 —— 191 42.4 0.416 Ethylene 169.5 104 282 44 0.571 Fluorine 186 40 —— —— 1.11 Chlorine 239.6 —— —— —— 1.507 Ammonia 234.5 197.5 404 85.9 —— Neon 30.40 —— below 65 —— —— Argon 86.90 —— 155.6 40.2 1.212 Krypton 121.33 —— 210.5 41.2 2.155 Xenon 163.9 —— 287.8 43.5 3.52

The principle of the Cailletet method of effecting the liquefaction of oxygen had been theoretically and experimentally studied by Joule and Lord Kelvin many years previously. It was extended by Siemens and has been applied by Linde and Hampson to the construction of machinery for the production of liquid air on a large scale, without the use of any intermediate refrigerant.

It is now readily possible to procure considerable quantities of liquid air, and even of liquid hydrogen. By the evaporation of liquid hydrogen temperatures approaching the _absolute zero_—that is, 273° C. below the melting-point of ice—can now be reached. Incidentally there has been developed a special field of inquiry relating to the behaviour of substances at low temperatures.

The pioneers in this field have been Dewar in England and Kammerlingh Onnes in Holland. Research at low temperatures, indeed, has been the main feature of the work of the Royal Institution of Great Britain during the last twenty years. It has included observations at temperatures approaching the absolute zero, on the electrical resistivity of metals and alloys, on the behaviour of so-called insulators, on changes in the cohesive force of metals, on the dielectric constants of frozen electrolytes, on the influence of cold on magnetisation and on magnetic permeability, and on the optical behaviour of bodies, on vital phenomena at low temperatures, and on the influence of cold on chemical change.

Dewar has succeeded in liquefying and solidifying large quantities of hydrogen, and has studied its properties at low temperatures. Liquid hydrogen is transparent and colourless. It is a non-conductor of electricity, and gives no absorption spectrum. It freezes into an ice-like solid, devoid of metallic properties. Dewar has made use of the property possessed by charcoal of occluding gases, especially at low temperatures, in the production of high vacua, and in the separation of gases; and he has also determined the molecular heat of absorption by charcoal of various gases. He has employed liquid air, liquid nitrogen, and liquid hydrogen as calorimetric agents, and has determined by means of them the heat capacities of a number of substances at very low temperatures. Lastly, his ingenious contrivance of silvered vacuum protected vessels, now introduced into commerce under the name of “Thermos flasks,” has greatly facilitated the manipulation of liquefied gases for experimental purposes.