The molecules of a gas undoubtedly attract one another under ordinary circumstances (ex cept in the case of hydrogen, where the force appears to be repulsive at all distances) ; but it is assumed that they are so far apart during the greater part of the time that the attractive forces that exist do not have any great effect upon the motions of the system as a whole. The path of a molecde of gas, between two suc cessive collisions, is called the °free oath" of the molecule, and is believed to 'be sensibly straight, owing to the high velocity that the molecules have on the average, and the (as sumed) fact that the attractive forces are un important at distances comparable in magnitude with the mean °free path " Maxwell showed that the average free path of the molecules of a gas may be calculated by means of the simple formula 3k L = -- SD where L is the average free path in centimeters, k is the coefficient of viscosity of the gas in absolute measure (see Vsscosiry), S is the average speed with which the molecules are moving (in centimeters per second), and D is the mass of one cubic centimeter of the gas (in grammes). At atmospheric pressure and the temperature of melting ice, this formula gives values of L approximately as follows, where the unit, in each case, is the millionth part of a centimeter: Hydrogen, 17; nitrogen, 9.1; oxygen, 9.7; carbon dioxide, 6.4.
The length of the free path of a given gas is increased, when the density of the gas is diminished, in the exact inverse ratio of the change in density. In an exhausted tube con taining hydrogen, for example, at a density one one-millionth of the density assumed above, the mean free path of the molecules would be a million times as great as the value given for hydrogen at the normal density— that is, the free path at this particular exhaustion would be 17 centimeters, so that the molecules would travel, on an average, over six inches between successive collisions. As the kinetic theory of gases assumes that the molecules collide with one another after traveling distances that are negligible in comparison with the dimensions of the whole mass of gas under consideration, its conclusions, when applied to gaseous masses in which this condition is not fulfilled, must be received with proper caution. Maxwell's formula, for example, is itself doubtful when applied to the extreme case in which the density of the hydrogen is only a millionth of the normal value. It is certain, however, that the free paths of gas molecules at such high ex haustions are to be measured in inches, and it is also certain that the pressure is not neces sarily equal in all directions in vacua of this degree of perfection, since it is by means of the incessant collisions that this equality of pressure is brought about at ordinary densities. For these reasons (among others) Sir William Crookes considered that highly attenuated gases, in which the pressure is a millionth of an atmosphere or less, should be considered as constituting a afourth state" of matter, essen tially distinct in its properties from the three states that are commonly recognized. He also devised the radiometer and other instruments to show the reality of the difference of pres sure that can exist in high vacua. In recent years it has become increasingly probable that in certain forms (at any rate) of the apparatus devised by Crookes,— in those forms, namely, in which °cathode rays" are generated by the action of powerfully excited electrodes,— the mechanical effects that are observed are not due directly to the motions of the gas molecules themselves, but rather to the motions of free electrical corpuscles given off by the gas molecules under the influence of the powerful electric discharge. Crookes himself appears to have held views not essentially different from this, though at the time they were stated they were clothed in language that was necessarily rather indefinite, since the corpuscular or elec tron hypothesis had not then taken form. See ELECTRON THEORY; RADIATION.
In liquids, the molecules are supposed to be so near together that the attractive forces that they exert upon one another are powerful at all times. The kinetic theory of liquids is imperfectly understood, but it is considered certain that collisions occur among the mole cules just as they do in gases and that the colliding molecules rebound from one another like perfectly elastic bodies. In liquids, how ever, there is nothing strictly analogous to the °free path)) in gases; for the liquid molecules are always exposed to attractive forces of con siderable magnitude and hence in the intervals between successive collisions they describe paths that are everywhere markedly curved. There is, doubtless, as great a variety of velocities among the molecules of a liquid as among those of a gas, but the law of distribution of velocities among liquid molecules has not yet been deter mined, on account of the mathematical difficul ties that are involved and which have thus far proved insuperable. Admitting the fact that the velocities of the molecules are unequal, let us consider what would happen at a free surface, of the liquid, assuming for the moment that above this free surface there is a boundless vacuum. A molecule that is well within the liquid is attracted, on the whole, equally in all directions. A molecule at the surface, how ever, is attracted only downward. Hence it is evident that when a molecule, in the course of its wanderings, comes to the surface, the pos sibility of its escape from the liquid depends upon the magnitude of the vertical component of its velocity. If this vertical component is sufficient to carry the molecule beyond the range of sensible attraction of the liquid, the molecule will pass away indefinitely into the space above. On the other hand, if the vertical component of its velocity is not sufficient to carry the mole cule beyond the range of sensible attraction of the liquid, it will rise into the vacuous space only a short distance, its upward velocity grow ing less and less, under the influence of the downward attractive forces, until it vanishes altogether; after which the molecule will begin to fall back and it will finally plunge into the liquid again. From the slowness with which
free evaporation takes place, we must conclude that by far the greater part of the molecules that start upward fall back into the liquid. Those that do escape by reason of their great velocities carry off more than their equable share of the kinetic energy of the molecules of the liquid and this causes the average kinetic energy of the liquid, per molecule, to grow continually less. In other words, free evapora tion causes a reduction of the temperature of the mother liquid. When the liquid is enclosed in a containing vessel of finite volume, the phenomena are somewhat different, after the evaporation has proceeded for a time. If the space above the liquid is vacuous at the outset, the evaporation, at the first instant, takes place precisely as before. Of the molecules that come to the surface of the liquid, those that are mov ing most rapidly in a vertical direction fly off as in the case previously considered; but they can no longer pass away indefinitely into space. They are now retained in the vessel, in which they will accumulate, constituting a gas or vapor whose density will go on increasing until a certain limit is reached.. The molecules com posing this vapor will travel in every direction, precisely as they do in other gaseous bodies. Many of them, therefore, will plunge hack into the liquid again and become an integral part of it once more. Now the number of molecules that leave the mother liquid in a given time will be quite independent of the density over head ; but the number that fly back into it again, in a given time, will be greater, the greater the density of the vapor. At the beginning of the evaporation the vapor will be quite rare and the number of molecules that fly off in any given time will greatly excped the number that re turn during that time. The density of the vapor will, therefore, increase. After a certain inter val (short as measured by ordinary standards), the density of the vapor will become so great that the number of molecules plunging back into the liquid in a given time will become sensi bly equal to the number that fly off from it in the same time. When this adjustment becomes perfect, the density of the vapor will no longer increase. It is then said to be and its density will remain constant until the tem perature of the system is altered. If the tem perature be now raised, all the molecules will accelerated and hence more molecules will plunge from the the vapor into the liquid in a given time than before, and more molecules will also come to the surface of the liquid from the interior. Furthermore, of the increased number of molecules that emerge from the interior of the liquid, a larger proportion than before will have velocities exceeding the critical velocity that a molecule must have in order to escape from the attraction of its fellows. Hence, on the whole, the density of the vapor will in crease, approaching a new limit at which the number of incoming and outgoing molecules will again become equal. It follows, therefore, that for any vapor in contact with its liquid there is a definite density corresponding to each temperature. The existence of a critical point (q.v.) may be explained in a similar manner, by considering the average kinetic energy that a molecule must have in order that it may be able to pass away from the attraction of other molecules in its immediate vicinity. A stone thrown upward by the hand does not proceed far before the attractive force of the earth an nuls its velocity and causes it to fall back again. A rifle will project a ball far higher, but the ball will eventually fall back, just as the stone did. With a good modern cannon we can throw a projectile several miles into the air— and stilt it falls back. But we might conceivably project one with such a speed that it would leave the earth forever. It may be shown, in fact, that if the retarding action of the air is omitted from consideration, an initial vertical speed of 36,700 feet•per second would be quite sufficient. With this much premised, conceive two molecules of a gas to be in contact, and let a sudden impulse be given to one of them, to drive it away from the other one. If the impulse is small enough, the disturbed molecule will only travel a short distance, and will then fall back to its original position; but we may give it such a speed that the attractive force of the fixed molecule will fail to bring it back, and in this case it will travel onward indefinitely. Now, just as in the case .of the cannon-ball and the earth, there must be some intermediate initial speed that will be just sufficient to separate the two mole cules under consideration. We may call this the velocity? and we may say that if the molecules of a given gas are moving so that, on an average, when two of them collide they have a relative velocity greater than this critical value, the gas in question cannot be liquefied by pressure alone; for even if its molecules were forced, almost into absolute contact with one another, their velocities would be sufficient to separate them again indefinitely, as soon as the pressure was removed. From this, and from the relation between temperature and molecular velocity in gases (see GASES, Kitirric THEORY OF), it follows that for every gas there is a temperature above which the gas cannot be liquefied by any pressure whatever.