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_{.mw-parser-output .grc{font-family:SBL BibLit,SBL Greek,DejaVu Sans,DejaVu Serif,FreeSerif,FreeSans,Athena,Gentium Plus,Gentium,Palatino Linotype,Arial Unicode MS,Lucida Sans Unicode,Lucida Grande,Code2000,sans-serif}.mw-parser-output .polytonic{font-family:"SBL BibLit","SBL Greek",Athena,"Foulis Greek","Gentium Plus",Gentium,"Palatino Linotype","Arial Unicode MS","Lucida Sans Unicode","Lucida Grande",Code2000}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