# 1911 Encyclopædia Britannica/Condensation of Gases

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​CONDENSATION OF GASES. If the volume of a gas continually decreases at a constant temperature, for which an increasing pressure is required, two cases may occur:—(1) The volume may continue to be homogeneously filled. (2) If the substance is contained in a certain Critical temperature. volume, and if the pressure has a certain value, the substance may divide into two different phases, each of which is again homogeneous. The value of the temperature T decides which case will occur. The temperature which is the limit above which the space will always be homogeneously filled, and below which the substance divides into two phases, is called the critical temperature of the substance. It differs greatly for different substances, and if we represent it by T$_c$, the condition for the condensation of a gas is that T must be below T$_c$. If the substance is divided into two phases, two different cases may occur. The denser phase may be either a liquid or a solid. The limiting temperature for these two cases, at which the division into three phases may occur, is called the triple point. Let us represent it by T$_3$; if the term “condensation of gases” is taken in the sense of “liquefaction of gases”—which is usually done—the condition for condensation is T$_c$ > T > T$_3$. The opinion sometimes held that for all substances T$_3$ is the same fraction of T$_c$ (the value being about ½) has decidedly not been rigorously confirmed. Nor is this to be expected on account of the very different form of crystallization which the solid state presents. Thus for carbon dioxide, CO$_2$, for which T$_c$ = 304° on the absolute scale, and for which we may put T$_3$ = 216°, this fraction is about 0.7; for water it descends down to 0.42, and for other substances it may be still lower.
If we confine ourselves to temperatures between T$_c$ and T$_3$, the gas will pass into a liquid if the pressure is sufficiently increased. When the formation of liquid sets in we call the gas a saturated vapour. If the decrease of volume is continued, the gas pressure remains constant till all the vapour has passed into liquid. The invariability of the properties of the phases is in close connexion with the invariability of the pressure (called maximum tension). Throughout the course of the process of condensation these properties remain unchanged, provided the temperature remain ​constant; only the relative quantity of the two phases changes. Until all the gas has passed into liquid a further decrease of volume will not require increase of pressure. But as soon as the liquefaction is complete a slight decrease of volume will require a great increase of pressure, liquids being but slightly compressible.
The pressure required to condense a gas varies with the temperature, becoming higher as the temperature rises. The highest pressure will therefore be found at T$_c$ and the lowest at T$_3$. We shall represent the pressure at Critical pressure. T$_c$ by p$_c$. It is called the critical pressure. The pressure at T$_3$ we shall represent by p$_3$. It is called the pressure of the triple point. The values of T$_c$ and p$_c$ for different substances will be found at the end of this article. The values of T$_3$ and p$_3$ are accurately known only for a few substances. As a rule p$_3$ is small, though occasionally it is greater than 1 atmosphere. This is the case with CO$_2$, and we may in general expect it if the value of T$_3$/T$_c$ is large. In this case there can only be a question of a real boiling-point (under the normal pressure) if the liquid can be supercooled.
We may find the value of the pressure of the saturated vapour for each T in a geometrical way by drawing in the theoretical isothermal a straight line parallel to the v-axis in such a way that ∫v$_2$ v$_1$pdv will have the same value whether the straight line or the theoretical isothermal is followed. This construction, given by James Clerk Maxwell, may be considered as a result of the application of the general rules for coexisting equilibrium, which we owe to J. Willard Gibbs. The construction derived from the rules of Gibbs is as follows:—Construe the free energy at a constant temperature, i.e. the quantity – fpdv as ordinate, if the abscissa represents v, and determine the inclination of the double tangent. Another construction derived from the rules of Gibbs might be expressed as follows:—Construe the value of pv − ∫pdv as ordinate, the abscissa representing p, and determine the point of intersection of two of the three branches of this curve.
As an approximate half-empirical formula for the calculation of the pressure, −log$_{10}$ p/p$_c$ = f (T$_c$ – T)/T) may be used. It would follow from the law of corresponding states that in this formula the value of f is the same for all substances, the molecules of which do not associate to form larger molecule-complexes. In fact, for a great many substances, we find a value for f, which differs but little from 3, e.g. ether, carbon dioxide, benzene, benzene derivatives, ethyl chloride, ethane, &c. As the chemical structure of these substances differs greatly, and association, if it takes place, must largely depend upon the structure of the molecule, we conclude from this approximate equality that the fact of this value of f being equal to about 3 is characteristic for normal substances in which, consequently, association is excluded. Substances known to associate, such as organic acids and alcohols, have a sensibly higher value of f. Thus T. Estreicher (Cracow, 1896) calculates that for fluor-benzene f varies between 3.07 and 2.94; for ether between 3.0 and 3.1; but for water between 3.2 and 3.33, and for methyl alcohol between 3.65 and 3.84, &c. For isobutyl alcohol f even rises above 4. It is, however, remarkable that for oxygen f has been found almost invariably equal to 2.47 from K. Olszewski’s observations, a value which is appreciably smaller than 3. This fact makes us again seriously doubt the correctness of the supposition that f = 3 is a characteristic for non-association.
It is a general rule that the volume of saturated vapour decreases when the temperature is raised, while that of the coexisting liquid increases. We know only one exception to this rule, and that is the volume of water Critical volume. below 4° C. If we call the liquid volume v$_l$, and the vapour v$_v$, v$_v$ – v$_l$ decreases if the temperature rises, and becomes zero at T$_c$. The limiting value, to which v$_l$ and v$_v$ converge at T$_c$, is called the critical volume, and we shall represent it by v$_c$. According to the law of corresponding states the values both of v$_l$/v$_c$ and v$_v$/v$_c$ must be the same for all substances, if T/T$_c$ has been taken equal for them all. According to the investigations of Sydney Young, this holds good with a high degree of approximation for a long series of substances. Important deviations from this rule for the values of v$_v$/v$_l$ are only found for those substances in which the existence of association has already been discovered by other methods. Since the lowest value of T, for which investigations on v$_l$ and v$_v$ may be made, is the value of T$_3$; and since T$_3$/T$_c$, as has been observed above, is not the same for all substances, we cannot expect the smallest value of v$_l$/v$_c$ to be the same for all substances. But for low values of T, viz. such as are near T$_3$, the influence of the temperature on the volume is but slight, and therefore we are not far from the truth if we assume the minimum value of the ratio v$_l$/v$_c$ as being identical for all normal substances, and put it at about 1/3. Moreover, the influence of the polymerization (association) on the liquid volume appears to be small, so that we may even attribute the value 1/3 to substances which are not normal. The value of v$_v$/v$_c$ at T = T$_3$ differs widely for different substances. If we take p$_3$ so low that the law of Boyle-Gay Lussac may be applied, we can calculate v$_3$/v$_c$ by means of the formula p$_3$v$_3$/T$_3$ = kp$_c$v$_c$/T$_c$, provided k be known. According to the observations of Sydney Young, this factor has proved to be 3.77 for normal substances. In consequence v$_3$/v$_c$ = 3.77p$_c$/p$_3$ T$_3$/T$_c$. A similar formula, but with another value of k, may be given for associating substances, but with another value of k, may be given for associating substances, provided the saturated vapour does not contain any complex molecules. But if it does, as is the case with acetic acid, we must also know the degree of association. It can, however, only be found by measuring the volume itself.
E. Mathias has remarked that the following relation exists between the densities of the saturated vapour and of Rule of the rectilinear diameter. the coexisting liquid:—
ρ$_l$ + ρ$_v$ = 2ρ$_c$ $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$1 + a(1 – T/T$_c$)$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$,
and that, accordingly, the curve which represents the densities at different temperatures possesses a rectilinear diameter. According to the law of corresponding states,a would be the same for all substances. Many substances, indeed, actually appear to have a rectilinear diameter, and the value of a appears approximatively to be the same. In a Mémoire présenté à la société royale à Liège, 15th June 1899, E. Mathias gives a list of some twenty substances for which a has a value lying between 0.95 and 1.05. It had been already observed by Sydney Young that a is not perfectly constant even for normal substances. For associating substances the diameter is not rectilinear. Whether the value of a, near 1, may serve as a characteristic for normal substances is rendered doubtful by the fact that for nitrogen a is found equal to 0.6813 and for oxygen to 0.8. At T = T$_c$/2, the formula of E. Mathias, if ρ$_v$ be neglected with respect to ρ$_l$, gives the value 2 + a for ρ$_l$/ρ$_c$.
The heat required to convert a molecular quantity of liquid coexisting with vapour into saturated vapour at the same temperature is called molecular latent heat. It decreases with the rise of the temperature, because at a higher Latent heat. temperature the liquid has already expanded, and because the vapour into which it has to be converted is denser. At the critical temperature it is equal to zero on account of the identity of the liquid and the gaseous states. If we call the molecular weight m and the latent heat per unit of weight r, then, according to the law of corresponding states, mr/T is the same for all normal substances, provided the temperatures are corresponding. According to F. T. Trouton, the value of mr/T is the same for all substances if we take for T the boiling-point. As the boiling-points under the pressure of one atmosphere are generally not equal fractions of T$_c$, the two theorems are not identical; but as the values of p$_c$ for many substances do not differ so much as to make the ratios of the boiling-points under the pressure of one atmosphere differ greatly from the ratios of T$_c$, an approximate confirmation of the law of Trouton may be compatible with an approximate confirmation of the consequence of the law of corresponding states. If we take the term boiling-point in a more general sense, and put T in the law of ​ Trouton to represent the boiling-point under an arbitrary equal pressure, we may take the pressure equal to p$_c$ for a certain substance. For this substance mr/T would be equal to zero, and the values of mr/T would no longer show a trace of equality. At present direct trustworthy investigations about the value of r for different substances are wanting; hence the question whether as to the quantity mr/T the substances are to be divided into normal and associating ones cannot be answered. Let us divide the latent heat into heat necessary for internal work and heat necessary for external work. Let r ′ represent the former of these two quantities, then:—
r = r ′ + p(v$_v$ – v$_l$).
Then the same remark holds good for mr ′/T as has been made for mr/T. The ratio between r and that part that is necessary for external work is given in the formula,
r/p(v$_v$–v$_l$) = T dp/p dT.
By making use of the approximate formula for the vapour tension:−log$_ε$ p/p$_c$ = f ′ (T$_c$ – T/T), we find—
r/p(v$_v$ –v$_l$) = f ′T$_c$/T.
At T = T$_c$ we find for this ratio f ′, a value which, for normal substances is equal to 3/0.4343 = 7. At the critical temperature the quantities r and v$_v$ –v$_l$ are both equal to 0, but they have a finite ratio. As we may equate p(v$_v$ –v$_l$) with pv$_v$ = RT at very low temperatures, we get, if we take into consideration that R expressed in calories is nearly equal to 2/m, the value 2f ′T$_c$ = 14T$_c$ as limiting value for mr for normal substances. This value for mr has, however, merely the character of a rough approximation—especially since the factor f ′ is not perfectly constant.
All the phenomena which accompany the condensation of gases into liquids may be explained by the supposition, that the condition of aggregation which we call liquid differs only in quantity, and not in quality, from that which we call gas. We imagine a gas to consist of separate Nature of a liquid. molecules of a certain mass μ, having a certain velocity depending on the temperature. This velocity is distributed according to the law of probabilities, and furnishes a quantity of vis viva proportional to the temperatures. We must attribute extension to the molecules, and they will attract one another with a force which quickly decreases with the distance. Even those suppositions which reduce molecules to centra of forces, like that of Maxwell, lead us to the result that the molecules behave in mutual collisions as if they had extension—an extension which in this case is not constant, but determined by the law of repulsion in the collision, the law of the distribution, and the value of the velocities. In order to explain capillary phenomena it was assumed so early as Laplace, that between the molecules of the same substance an attraction exists which quickly decreases with the distance. That this attraction is found in gases too is proved by the fall which occurs in the temperature of a gas that is expanded without performing external work. We are still perfectly in the dark as to the cause of this attraction, and opinion differs greatly as to its dependence on the distance. Nor is this knowledge necessary in order to find the influence of the attraction, for a homogeneous state, on the value of the external pressure which is required to keep the moving molecules at a certain volume (T being given). We may, viz., assume either in the strict sense, or as a first approximation, that the influence of the attraction is quite equal to a pressure which is proportional to the square of the density. Though this molecular pressure is small for gases, yet it will be considerable for the great densities of liquids, and calculation shows that we may estimate it at more than 1000 atmos., possibly increasing up to 10,000. We may now make the same supposition for a liquid as for a gas, and imagine it to consist of molecules, which for non-associating substances are the same as those of the rarefied vapour; these, if T is the same, have the same mean vis viva as the vapour molecules, but are more closely massed together. Starting from this supposition and all its consequences, van der Waals derived the following formula which would hold both for the liquid state and for the gaseous state:—
(p + a/v²) (v – b) = RT.
It follows from this deduction that for the rarefied gaseous state b would be four times the volume of the molecules, but that for greater densities the factor 4 would decrease. If we represent the volume of the molecules by β, the quantity b will be found to have the following form:—
b = 4β $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$1 – γ$_1$ (4β/v) + γ$_2$ (4β/v)2 &c.$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$
Only two of the successive coefficients γ$_1$, γ$_2$, &c., have been worked out, for the determination requires very lengthy calculations, and has not even led to definitive results (L. Boltzmann, Proc. Royal Acad. Amsterdam, March 1899). The latter formula supposes the molecules to be rigid spheres of invariable size. If the molecules are things which are compressible, another formula for b is found, which is different according to the number of atoms in the molecule (Proc. Royal Acad. Amsterdam, 1900–1901). If we keep the value of a and b constant, the given equation will not completely represent the net of isothermals of a substance. Yet even in this form it is sufficient as to the principal features. From it we may argue to the existence of a critical temperature, to a minimum value of the product pv, to the law of corresponding states, &c. Some of the numerical results to which it leads, however, have not been confirmed by experience. Thus it would follow from the given equation that p$_c$v$_c$/T$_c$ = 3/8 pv/T, if the value of v is taken so great that the gaseous laws may be applied, whereas Sydney Young has found 1/3.77 for a number of substances instead of the factor 3/8. Again it follows from the given equation, that if a is thought to be independent of the temperature, T$_c$/p$_c$( dp/dT)$_c$ = 4, whereas for a number of substances a value is found for it which is near 7. If we assume with Clausius that a depends on the temperature, and has a value a ′273/T, we find T$_c$/p$_c$ (dp/dT)$_c$ = 7.
That the accurate knowledge of the equation of state is of the highest importance is universally acknowledged, because, in connexion with the results of thermodynamics, it will enable us to explain all phenomena relating to ponderable matter. This general conviction is shown by the numerous efforts made to complete or modify the given equation, or to replace it by another, for instance, by R. Clausius, P. G. Tait, E. H. Amagat, L. Boltzmann, T. G. Jager, C. Dieterici, B. Galitzine, T. Rose Innes and M. Reinganum.
If we hold to the supposition that the molecules in the gaseous and the liquid state are the same—which we may call the supposition of the identity of the two conditions of aggregation—then the heat which is given out by the condensation at constant T is due to the potential energy lost in consequence of the coming closer of the molecules which attract each other, and then it is equal to a(1/v$_l$ – 1/v$_v$). If a should be a function of the temperature, it follows from thermodynamics that it would be equal to (a – Tda/dT) (1/v$_l$ – 1/v$_v$). Not only in the case of liquid and gas, but always when the volume is diminished, a quantity of heat is given out equal to a(1/v$_1$ – 1/v$_2$) or (a – Tda/dT) (1/v$_1$ – 1/v$_2$).
If, however, when the volume is diminished at a given temperature, and also during the transition from the gaseous to the liquid state, combination into larger molecule-complexes takes place, the total internal heat may be considered as the sum of that which is caused by the combination Associating substances.of the molecules into greater molecule-complexes and by their approach towards each other. We have the simplest case of possible greater complexity when two molecules combine to one. From the course of the changes in the density of the vapour we assume that this occurs, e.g. with nitrogen peroxide, NO$_2$, and acetic acid, and the somewhat close agreement of the ​ observed density of the vapour with that which is calculated from the hypothesis of such an association to double-molecules, makes this supposition almost a certainty. In such cases the molecules in the much denser liquid state must therefore be considered as double-molecules, either completely so or in a variable degree depending on the temperature. The given equation of state cannot hold for such substances. Even though we assume that a and b are not modified by the formation of double-molecules, yet RT is modified, and, since it is proportional to the number of the molecules, is diminished by the combination. The laws found for normal substances will, therefore, not hold for such associating substances. Accordingly for substances for which we have already found an anormal density of the vapour, we cannot expect the general laws for the liquid state, which have been treated above, to hold good without modification, and in many respects such substances will therefore not follow the law of corresponding states. There are, however, also substances of which the anormal density of vapour has not been stated, and which yet cannot be ranged under this law, e.g. water and alcohols. The most natural thing, of course, is to ascribe the deviation of these substances, as of the others, to the fact that the molecules of the liquid are polymerized. In this case we have to account for the following circumstance, that whereas for NO$_2$ and acetic acid in the state of saturated vapour the degree of association increases if the temperature falls, the reverse must take place for water and alcohols. Such a difference may be accounted for by the difference in the quantity of heat released by the polymerization to double-molecules or larger molecule-complexes. The quantity of heat given out when two molecules fall together may be calculated for NO$_2$ and acetic acid from the formula of Gibbs for the density of vapour, and it proves to be very considerable. With this the following fact is closely connected. If in the pv diagram, starting from a point indicating the state of saturated vapour, a geometrical locus is drawn of the points which have the same degree of association, this curve, which passes towards isothermals of higher T if the volume diminishes, requires for the same change in T a greater diminution of volume than is indicated by the border-curve. For water and alcohols this geometrical locus will be found on the other side of the border-curve, and the polymerization heat will be small, i.e. smaller than the latent heat. For substances with a small polymerization heat the degree of association will continually decrease if we move along the border-curve on the side of the saturated vapour in the direction towards lower T. With this, it is perfectly compatible that for such substances the saturated vapour, e.g. under the pressure of one atmosphere, should show an almost normal density. Saturated vapour of water at 100° has a density which seems nearly 4% greater than the theoretical one, an amount which is greater than can be ascribed to the deviation from the gas-laws. For the relation between v, T, and x, if x represents the fraction of the number of double-molecules, the following formula has been found (“Moleculartheorie,” Zeits. Phys. Chem., 1890, vol. v):
| log | x(v – b) | = 2 | E$_1$ – E$_2$ | + C, |
| log | (1 – x)$^2$ | = 2 | R$_1$T | + C, |
from which
| T | ( | dv | ) | | = −2 | E$_1$ – E$_2$ | , |
| v – b | ( | dT | ) | $_x$ | = −2 | R$_1$T | , |
which may elucidate what precedes.
By far the majority of substances have a value of T$_c$ above the ordinary temperature, and diminution of volume (increase of pressure) is sufficient to condense such gaseous substances into liquids. If T$_c$ is but little above the ordinary temperature, a great increase of pressure is Condensation of substances with low T$_c$. in general required to effect condensation. Substances for which T$_c$ is much higher than the ordinary temperature T$_0$, e.g. T$_c$ > 5/3 T$_0$, occur as liquids, even without increase of pressure; that is, at the pressure of one atmosphere. The value 5/3 is to be considered as only a mean value, because of the inequality of p$_c$. The substances for which T$_c$ is smaller than the ordinary temperature are but few in number. Taking the temperature of melting ice as a limit, these gases are in successive order: CH$_4$, NO, O$_2$, CO, N$_2$ and H$_2$ (the recently discovered gases argon, helium, &c., are left out of account). If these gases are compressed at 0° centigrade they do not show a trace of liquefaction, and therefore they were long known under the name of “permanent gases.” The discovery, however, of the critical temperature carried the conviction that these substances would not be “permanent gases” if they were compressed at much lower T. Hence the problem arose how “low temperatures” were to be brought about. Considered from a general point of view the means to attain this end may be described as follows: we must make use of the above-mentioned circumstance that heat disappears when a substance expands, either with or without performing external work. According as this heat is derived from the substance itself which is to be condensed, or from the substance which is used as a means of cooling, we may divide the methods for condensing the so-called permanent gases into two principal groups.
In order to use a liquid as a cooling bath it must be placed in a vacuum, and it must be possible to keep the pressure of the vapour in that space at a small value. According to the boiling-law, the temperature of the liquid must descend to that at which the maximum tension of the Liquids as means of cooling. vapour is equal to the pressure which reigns on the surface of the liquid. If the vapour, either by means of absorption or by an air-pump, is exhausted from the space, the temperature of the liquid and that of the space itself depend upon the value of the pressure which finally prevails in the space. From a practical point of view the value of T$_3$ may be regarded as the limit to which the temperature falls. It is true that if the air is exhausted to the utmost possible extent, the temperature may fall still lower, but when the substance has become solid, a further diminution of the pressure in the space is of little advantage. At any rate, as a solid body evaporates only on the surface, and solid gases are bad conductors of heat, further cooling will only take place very slowly, and will scarcely neutralize the influx of heat. If the pressure p$_3$ is very small, it is perhaps practically impossible to reach T$_3$; if so, T$_3$ in the following lines will represent the temperature practically attainable. There is thus for every gas a limit below which it is not to be cooled further, at least not in this way. If, however, we can find another gas for which the critical temperature is sufficiently above T$_3$ of the first chosen gas, and if it is converted into a liquid by cooling with the first gas, and then treated in the same way as the first gas, it may in its turn be cooled down to (T$_3$)$_2$. Going on in this way, continually lower temperatures may be attained, and it would be possible to condense all gases, provided the difference of the successive critical temperatures of two gases fulfils certain conditions. If the ratio of the absolute critical temperatures for two gases, which succeed one another in the series, should be sensibly greater than 2, the value of T$_3$ for the first gas is not, or not sufficiently, below the T$_c$ of the second gas. This is the case when one of the gases is nitrogen, on which hydrogen would follow as second gas. Generally, however, we shall take atmospheric air instead of nitrogen. Though this mixture of N$_2$ and O$_2$ will show other critical phenomena than a simple substance, yet we shall continue to speak of a T$_c$ for air, which is given at −140° C., and for which, therefore, T$_c$ 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 T$_c$ for hydrogen. Without new contrivances it would, accordingly, not be possible to reach the critical temperature of H$_2$. 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 T$_3$, if the initial temperature was equal to T$_c$, 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 isentropic. 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 van der Waals (Verslagen Kon. Akad. Amsterdam) pointed out that the former may be expected to be the case only for substances for which c$_p$/c$_v$ 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 methyl chloride, ethylene and oxygen are used in the cryogenic laboratory of Leiden, while Sir James Dewar has used air as the last term. Carbonic acid is not to be recommended on account of the comparatively high value of T$_3$. 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 T$_3$, it is not always necessary to convert it first into the liquid state by means of another substance, as was assumed in the last method for obtaining low temperatures. Its own expansion is sufficient, provided the initial Cooling by expansion. 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 T$_3$, 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, 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 time dp/dv and d $^2$p/dv $^2$ = 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 c$_p$/c$_v$ = k (the limiting value of c$_p$/c$_v$ for infinite rarefaction is meant) approaches unity, the isentropic curves approach the isothermals and vice versa. In the same way the critical point of the isentropic curves comes nearer to that of the isothermals. And if k is not much greater than 1, e.g. k < 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 T$_c$ 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 p$_c$. Availing himself of this property, K. Olszewski has determined p$_c$ for hydrogen at 15 atmospheres. 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.
From the equation of state (p + a/v $^2$)(v − b) = RT, the equation of the isentropic curve follows as (p + a/v $^2$)(v − b)$^k$ = C, and from this we may deduce T(v − b)$^{k−1}$ = C′. This latter relation shows in how high a degree the cooling depends on the amount by which k surpasses unity, the change in v – b being the same.
What has been said concerning the relative position of the border-curve and the isentropic curve may be easily tested for points of the border-curve which represent rarefied gaseous states, in the following way. Following the border-curve we found before f ′T$_c$/T for the value of T/pdp/dT. Following the isentropic curve the value of T/pdp/dT is equal to k/k – 1. If k/k – 1 < f ′ T$_c$/T, the isentropic curve rises more steeply than the border-curve. If we take f ′ = 7 and choose the value of T$_c$/2 for T—a temperature at which the saturated vapour may be considered to follow the gas-laws—then k/(k – 1) = 14, or k = 1.07 would be the limiting value for the two cases. At any rate k = 1.41 is great enough to fulfil the condition, even for other values of T. Cailletet and Pictet have availed themselves of this adiabatic expansion for condensing some permanent gases, and it must also be used when, in the cascade method, T$_3$ of one of the gases lies above T$_c$ of the next.
A third method of condensing the permanent gases is applied in C. P. G. Linde’s apparatus for liquefying air. Under a high pressure p$_1$ a current of gas is conducted through a narrow spiral, returning through another spiral which surrounds the first. Between the end of the first spiral Linde’s apparatus.and the beginning of the second the current of gas is reduced to a much lower pressure p$_2$ by passing through a tap with a fine ​orifice. On account of the expansion resulting from this sudden decrease of pressure, the temperature of the gas, and consequently of the two spirals, falls sensibly. If this process is repeated with another current of gas, this current, having been cooled in the inner spiral, will be cooled still further, and the temperature of the two spirals will become still lower. If the pressures p$_1$ and p$_2$ remain constant the cooling will increase with the lowering of the temperature. In Linde’s apparatus this cycle is repeated over and over again, and after some time (about two or three hours) it becomes possible to draw off liquid air.
The cooling which is the consequence of such a decrease of pressure was experimentally determined in 1854 by Lord Kelvin (then Professor W. Thomson) and Joule, who represent the result of their experiments in the formula
| T$_1$−T$_2$ = γ | p$_1$ – p$_2$ | . |
| T$_1$−T$_2$ = γ | T$^2$ | . |
In their experiments p$_2$ was always 1 atmosphere, and the amount of p$_1$ was not large. It would, therefore, be certainly wrong, even though for a small difference in pressure the empiric formula might be approximately correct, without closer investigation to make use of it for the differences of pressure used in Linde’s apparatus, where p$_1$ = 200 and p$_2$ = 18 atmospheres. For the existence of a most favourable value of p$_1$ is in contradiction with the formula, since it would follow from it that T$_1$ – T$_2$ would always increase with the increase of p$_1$. Nor would it be right to regard as the cause for the existence of this most favourable value of p$_1$ the fact that the heat produced in the compression of the expanded gas, and therefore p$_1$/p$_2$, must be kept as small as possible, for the simple reason that the heat is produced in quite another part of the apparatus, and might be neutralized in different ways.
Closer examination of the process shows that if p$_2$ is given, a most favourable value of p$_1$ must exist for the cooling itself. If p$_1$ is taken still higher, the cooling decreases again; and we might take a value for p$_1$ for which the cooling would be zero, or even negative.
If we call the energy per unit of weight ε and the specific volume v, the following equation holds:—
ε$_1$ + p$_1$v$_1$ – p$_2$v$_2$ = ε$_2$,
or
ε$_1$ + p$_1$v$_1$ = ε$_2$ + p$_2$v$_2$.
According to the symbols chosen by Gibbs, χ$_1$ = χ$_2$.
As χ$_1$ is determined by T$_1$ and p$_1$, and χ$_2$ by T$_2$ and p$_2$, we obtain, if we take T$_1$ and p$_2$ as being constant,
| ( | δχ$_1$ | ) | | dp$_1$ = ( | δχ$_2$ | ) | | dT$_2$. |
| ( | δp$_1$ | ) | $_{T1}$ | dp$_1$ = ( | δT$_1$ | ) | $_{p2}$ | dT$_2$. |
If T$_2$ is to have a minimum value, we have
| ( | δχ$_1$ | ) | | = 0 or ( | δχ$_1$ | ) | | = 0. |
| ( | δp$_1$ | ) | $_{T1}$ | = 0 or ( | δv$_1$ | ) | $_{T1}$ | = 0. |
From this follows
| ( | δε$_1$ | ) | | + [ | δ(p$_1$v$_1$) | ] | | = 0. |
| ( | δv$_1$ | ) | $_{T1}$ | + [ | δv$_1$ | ] | $_{T1}$ | = 0. |
As (δε$_1$/δv$_1$)$_T$ is positive, we shall have to take for the maximum cooling such a pressure that the product pv decreases with v, viz. a pressure larger than that at which pv has the minimum value. By means of the equation of state mentioned already, we find for the value of the specific volume that gives the greatest cooling the formula
| RT$_1$b | = | 2a | , |
| (v$_1$ – b)$^2$ | = | v$_1$$^2$ | , |
and for the value of the pressure
| p$_1$ = 27p$_c$[1 – √ | 4 | T$_1$ | ] [3 √ | 4 | T$_1$ | – 1]. |
| p$_1$ = 27p$_c$[1 – √ | 27 | T$_c$ | ] [3 √ | 27 | T$_c$ | – 1]. |
If we take the value 2T$_c$ for T$_1$, as we may approximately for air when we begin to work with the apparatus, we find for p$_1$ about 8p$_c$, or more than 300 atmospheres. If we take T$_1$ = T$_c$, as we may at the end of the process, we find p$_1$ = 2.5p$_c$, or 100 atmospheres. The constant pressure which has been found the most favourable in Linde’s apparatus is a mean of the two calculated pressures. In a theoretically perfect apparatus we ought, therefore, to be able to regulate p$_1$ according to the temperature in the inner spiral.
The critical temperatures and pressures of the permanent gases are given in the following table, the former being expressed on the absolute scale and the latter in atmospheres:—
| | T$_c$ | p$_c$ | | T$_c$ | p$_c$ |
| CH$_4$ | 191.2° | 55 | CO | 133.5° | 35.5 |
| NO | 179.5° | 71.2 | N$_2$ | 127° | 35 |
| O$_2$ | 155° | 50 | Air | 133° | 39 |
| Argon | 152° | 50.6 | H$_2$ | 32° | 15 |
The values of T$_c$ and p$_c$ for hydrogen are those of Dewar. They are in approximate accordance with those given by K. Olszewski. Liquid hydrogen was first collected by J. Dewar in 1898. Apparatus for obtaining moderate and small quantities have been described by M. W. Travers and K. Olszewski. H. Kamerlingh Onnes at Leiden has brought about a circulation yielding more than 3 litres per hour, and has made use of it to keep baths of 1.5 litre capacity at all temperatures between 20.2° and 13.7° absolute, the temperatures remaining constant within 0.01°. (See also Liquid Gases.) (J. D. v. d. W.)