# 1911 Encyclopædia Britannica/Condensation of Gases

**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 ⅓.
Moreover, the influence of the polymerization (association) on
the liquid volume appears to be small, so that we may even
attribute the value ⅓ 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} = *k**p*_{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} = 3.77*p*_{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} 1 + *a*(1 – TT_{c}),

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 | = | T dp | . |

p(v_{v} – v_{l}) |
p dT |

By making use of the approximate formula for the vapour
tension:−log_{ε}*p**p*_{c} = *f* ′ (T_{c} – TT), we find—

r | = f ′ | T_{c} | . |

p(v_{v} – v_{l}) |
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 2*f* ′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
Nature of

a liquid.
we call gas. We imagine a gas to consist of separate
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**v*²) (*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 1 – γ_{1}(4β*v*) + γ_{2} (4β*v*)² &c.

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}
= 38 *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**d*T)_{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* ′ · 273T, we find
T_{c}*p*_{c} (*dp**d*T)_{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* – T*da**d*T) (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* – T*da**d*T) (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 Associating substances.as the sum of that which is caused by the combination 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, |

(1 – x)² | R_{1}T |

from which

T | ( | dv | ) | = −2 | E_{1} – E_{2} | , | |

v – b | dT | _{x} | 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
Condensa-

tion of

substances with low

T_{c}.
ordinary temperature, a great increase of pressure is
in general required to effect condensation. Substances
for which T_{c} is much higher than the ordinary temperature
T_{0}, *e.g.* T_{c} > 53 T_{0}, occur as liquids, even without increase of
pressure; that is, at the pressure of one atmosphere. The
value 53 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
Liquids as means of cooling.
descend to that at which the maximum tension of the
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.
Cooling by expansion.
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 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*p**dp**d*T. Following the isentropic curve
the value of T*p**dp**d*T 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 Linde’s apparatus.surrounds the first. Between the end of the first spiral 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} | . |

T² |

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} |
δT_{1} | _{p2} |

If T_{2} is to have a minimum value, we have

( | δχ_{1} | ) | = 0 or ( | δχ_{1} | ) | = 0. | ||

δp_{1} | _{T1} |
δv_{1} | _{T1} |

From this follows

( | δε_{1} | ) | + [ | δ(p_{1}v_{1}) | ] | = 0. | ||

δv_{1} | _{T1} |
δv_{1} | _{T1} |

As (δε_{1}/δ*v*_{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} = 27 p_{c}[1 – √ |
4 | T_{1} |
] [3 √ | 4 | T_{1} |
– 1]. |

27 | T_{c} |
27 | T_{c} |

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
8*p*_{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} = 2.5*p*_{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.)