Liquefaction of Gases

The experiments of Andrews fully explained his own failures, and those of Faraday and earlier investigators, to liquefy the so called permanent gases. The significance of his work, can, how ever, only be appreciated when we consider the theoretical investigations to which it gave rise. Since the work of Regnault had first shown that the known gases departed more or less from the laws of Boyle and Gay-Lussac as expressed by the equation, T being the temperature on the scale of a thermometer filled with the gas itself, approximately equivalent to the temperature on the absolute scale, to which reference will be made later. Many attempts had been made to modify the formula to accord with the facts so far as they were known. These, however, only included the volume and pressure relations of the gas, but in view of the discoveries of Andrews it was now obvious that the theoretical investigation must also take into account the existence of the liquid phase.

Van der Waals's Equation of State.—In 1873 there was pub lished in Leyden a dissertation with the same title as Andrew's Bakerian lecture—"On the continuity of the liquid and gaseous states," by J. D. van der Waals. The title has not, however, the same significance, and is used in a more exact sense than by Andrews. It implies that while the isothermals for temperatures which differ by a very small interval above and below the critical temperature are practically identical in form, the isothermal for the higher temperatures represents the gaseous state only, and that for the lower temperatures relates also to the liquid state. This was the first original paper published by the author who had for some years been engaged upon mathematical investiga tions, based upon the kinetic theory of gases and Laplace's theory of capillarity, and upon this work he based the theory which made him famous. Assuming the molecules of any particular gas to be all exactly similar hard elastic spheres of radius r, the volume occupied by n molecules in volume v is In accordance with the assumption as to the form of the molecule, the gas equation thus becomes where Now, while the molecules in the interior of a mass of gas are subjected to attractions by other molecules which are equal in all directions, so that the resultant attraction is negligible, the resultant attraction on the molecules in the surface layer is directed inwards, and varies as the square of the volume. Allow ing for the effect thus produced the equation now becomes :— This is known as the van der Waals equation.

It may first be pointed out that the equation may also be written in cubic form One of the three roots of this equation must always be real, and the other two may be real or imaginary, so that for every value of p there may be one or three values of v. In a certain region, for the isothermals below the critical point, there are three real values of v for each value of p, so that the curves do not have the simple form indicated by the heavy lines but follow the sinuous lines.

The latter do not exhibit the discontinuity shown by the former, in which, for one particular value of p only, the vapour pressure, there are two values of v, corresponding to the volumes of satu rated vapour and of the liquid. This pressure and the correspond ing volumes are indistinguishable on the van der Waals curve, but at the same time, if the relations of volume, pressure, and temperature under which the liquid and vapour phases are in equilibrium are known, the constants in the van der Waals equa tion can be calculated from them with the same accuracy as from the data obtained from the compressibility of the gas. The mean ing of the continuous curve has been the subject of much specu lation. It has been suggested that the portion of it on the extreme right represents a condition of super-saturated vapour, and that on the left a state in which liquid exists at a pressure below its vapour pressure. The central portion must in any case be with

out physical significance as it represents the pressure and vol ume as increasing together and decreasing together. As a mat ter of fact the van der Waals equation is based upon the consideration of the behaviour of a single phase, which may change continuously into a second phase, but it does not con template discontinuity, or equilibrium between liquid and vapour. The form of the curve has no physical significance at all, but is merely a consequence of the mathematical treatment of the subject.

ap At the point C, the critical point = o and a line drawn parallel to the volume axis is tangential to the isothermal. At this point the three values of V are real and equal, and by a simple algebraical process we can obtain the values of the critical temperature, the critical pressure, and the critical volume in terms of the constants of the equation:— The following are the values of a and b for some of the common gases, when pressures are reckoned in atmospheres, and the volumes in litre-moles: The fundamental ideas which are expressed by the van der Waals equation are undoubtedly true, but it could hardly be expected that from the simple assumptions upon which it was based it would be possible to develop a complete theory of the gaseous and liquid state. The equation does not enable us to connect in a quantitative manner the properties of gases at low pressures. The ratio is not 0.375 pv/T, being generally nearer to 0.27 R/R. It is not possible to trace the isothermals for any distances by means of data calculated from the critical constants. However, the van der Waals equation, even without modification, has been of very great value as a means of calculating the physical constants for gases and liquids approximately.

The Law of Corresponding States.

A second which was made by van der Waals is known as the law of corre sponding states. This is really a mathematical consequence fol lowing on the assumption that the properties of the systems with which we are dealing can be represented by an equation with two constants a and b, R depending merely on the quantity of material in this system, molar or otherwise. If then a series of diagrams are drawn like fig. 4, but representing the properties of different materials, such as carbon dioxide, benzene, etc., it will and therefore all substances in corresponding states should be have alike. For instance, the isothermals of all gases represented in this manner should be identical. The law of corresponding states leads directly to certain useful approximations. The boil ing point Tb of most liquids is approximately a corresponding temperature, since and as it can be shown that the latent heat of liquefaction L should be identical for liquid-gas systems at corresponding tern peratures it follows that It was found by Trouton (Phil. Meg., 1899) that for many liquids this rule holds good, thus :— Trouton's rule does not hold good for liquids at very low tempera turest, nor for associating liquidst.

An interesting instance of the application of the discoveries of van der Waals was that of the Polish physicist Witkowski in the early 8o's. He determined the compressibility of oxygen and hydrogen down to the lowest temperatures which in the case of hydrogen was that of liquid oxygen. From the data, he was then able to calculate the constants in the equation. The critical constant and boiling point of oxygen being known, he was then able to calculate these constants for hydrogen with remarkable accuracy, as later investigations showed.