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 CONDENSATION fore, Tc amounts to 133° absolute. The lowest T which, may be expected for air in a highly rarefied space may be evaluated at 60° absolute—a value which is higher than the Tc for hydrogen. Without new contrivances it would, accordingly, not be possible to reach the critical temperature of H2. The method by which we try to obtain successively lower temperatures by making use of successive gases is called the “ cascade method.” It is not self-evident that by sufficiently diminishing the pressure on a liquid, it may be cooled to such a degree that the temperature will be lowered to T3, if the initial temperature was equal to Tc, or but little below it, and we can even predict with certainty that this will not be the case for all substances. It is possible, too, that long before the triple point is reached the whole liquid will have evaporated. The most favourable conditions will, of course, be attained when the influx of heat is reduced to a minimum. As a limiting case we imagine the process to be adiabatic. Now the question has become, Will an isentropic line, which starts from a point of the border-curve on the side of the liquid not far from the critical point, remain throughout its descending course in the heterogeneous region, or will it leave the region on the side of the vapour ? As early as 1878 the author of this article (Verdagen Kon. Alcad. Amsterdam) pointed out that the former may be expected to be the case only for substances for which cp/cv is large, and the latter for those for which it is small; in other words, the former will take place for substances the molecules of which contain few atoms, and the latter for substances the molecules of which contain many atoms. Ether is an example of the latter class, and if we say that the quantity h (specific heat of the saturated vapour) for ether is found to be positive, we state the same thing in other words. It is not necessary to prove this theorem further here, as the molecules of the gases under consideration contain only two atoms and the total evaporation of the liquid is not to be feared. In the practical application of this cascade-method some variation is found in the gases chosen for the successive stages. Thus chloric methyl, ethylene, and oxygen are used in the cryogenic laboratory of Leyden, while Professor Dewar has used air as the last term. Carbonic acid is not to be recommended on account of the comparatively high value of T3. In order to prevent loss of gas a system of “circulation” is employed. This method of obtaining low temperatures is decidedly laborious, and requires very intricate apparatus, but it has the great advantage that very constant low temperatures may be obtained, and can be regulated arbitrarily within pretty wide limits. In order to lower the temperature of a substance down to T3, it is not always necessary to convert it first into the liquid state by means of another substance, as expansion.' was assume(l in the last method for obtaining low temperatures. Its own expansion is sufficient, provided the initial condition be properly chosen, and provided we take care, even more than in the former method, that there is no influx of heat. Those conditions being fulfilled, we may, simply by adiabatic expansion, not only lower the temperature of some substances down to T3, but also convert them into the liquid state. This is especially the case with substances the molecules of which contain few atoms. Let us imagine the whole net of isothermals for homogeneous phases drawn in a pv diagram, and in it the border-curve. Within this border-curve, as in the heterogeneous region, the theoretical part of every isothermal must be replaced by a straight line. The isothermals may therefore be divided into two groups, viz., those which keep outside the heterogeneous region, and those which cross this region. Hence an isothermal,

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belonging to the latter group, enters the heterogeneous region on the liquid side, and leaves it at the same level on the vapour side. Let us imagine in the same way all the isentropic curves drawn for homogeneous states. Their form resembles that of isothermals in so far as they show a maximum and a minimum, if the entropy-constant is below a certain value, while if it is above this value, both the maximum and the minimum disappear, the isentropic line in a certain point having at the same and =0 for this particular value of the constant. This point, which we might call the critical point of the isentropic lines, lies in the heterogeneous region, and therefore cannot be realized, since as soon as an isentropic curve enters this region its theoretical part will be replaced by an empiric part. If an isentropic curve crosses the heterogeneous region, the point where it enters this region must, just as for the isothermals, be connected with the point where it leaves the region by another curve. When Cp/cv = k (the limiting value of ep/c„ for infinite rarefaction is meant) approaches unity, the isentropic curves approach the isothermals and vice versd. In the same way the critical point of the isentropic curves comes nearer to that of the isothermals. And if Ic is not much greater than 1, e.g., ^<1,08, the following property of the isothermals is also preserved, viz., that an isentropic curve, which enters the heterogeneous region on the side of the liquid, leaves it again on the side of the vapour, not of course at the same level, but at a lower point. If, however, k is greater, and particularly if it is so great as it is with molecules of one or two atoms, an isentropic curve, which enters on the side of the liquid, however far prolonged, always remains within the heterogeneous region. .But in this case all isentropic curves, if sufficiently prolonged, will enter the heterogeneous region. Every isentropic curve has one point of intersection with the border-curve, but only a small group intersect the border-curve in three points, two of which are to be found not far from the top of the border-curve and on the side of the vapour. Whether the sign of h (specific heat of the saturated vapour) is negative or positive, is closely connected with the preceding facts. For substances having k great, h will be negative if T is low, positive if T rises, while it will change its sign again before Tc is reached. The values of T, at which change of sign takes place, depend on k. The law of corresponding states holds good for this value of T for all substances which have the same value of k. Now the gases which were considered as permanent are exactly those for which k has a high value. From this it would follow that every adiabatic expansion, provided it be sufficiently continued, will bring such substances into the heterogeneous region, i.e., they can be condensed by adiabatic expansion. But since the final pressure must not fall below a certain limit, determined by experimental convenience, and since the quantity which passes into the liquid state must remain a fraction as large as possible, and since the expansion never can take place in such a manner that no heat is given out by the walls or the surroundings, it is best to choose the initial condition in such a way that the isentropic curve of this point cuts the border-curve in a point on the side of the liquid, lying as low as possible. The border-curve being rather broad at the top, there are many isentropic curves which penetrate the heterogeneous region under a pressure which differs but little from Availing himself of this property, Olszewski has determined p0 for hydrogen at 20atmospheres. Isentropic curves, which lie on the right and on the left of this group, will show a point of condensation at a lower pressure. Olszewski has investigated this for those lying on the right, but not for those on the left.