Encyclopaedia Britannica > Volume 3, Athens-BOI

39
ATOM
The kinetic energy of a particle is half the product of
its mass into the square of its velocity, and the kirietic
energy of a system is the sum of the kinetic energy of all
its parts.
When an attraction or repulsion exists between two
points, half the product of this stress into the distance
between the two points is called the virial of the stress,
and is reckoned positive when the stress is. an attraction,
and negative when it is a repulsion. The virial of a system
is the sum of the virials of the stresses which exist in it.
If the system is subjected to the external stress of the
pressure of the sides of a vessel in which it is contained,
this stress will introduce an amount of virial -|pV, where
p is the pressure on unit of area and Y is the volume of
the vessel.
The theorem of Clausius may now be stated as follows .
In a material system in a state of stationary motion. the
time-average of the kinetic energy is equal to the time-
average of the virial. In the case of a fluid enclosed in a
vessel
P(m^) = fpV + P2(Rr),
where the first term denotes the kinetic energy, and is half
the sum of the product of each mass into the mean square
of its velocity. In the second term, p is the pressure on
unit of surface of the vessel, whose volume is V, and the
third term expresses the virial due to the internal actions
between the parts of the system. A double symbol of
summation is used, because every pair of parts between
which any action exists must be taken into account. We
have next to show that in gases the principal part of the
pressure arises from the motion of the small parts of the
medium, and not from a repulsion between them.
In the first place, if the pressure of a gas arises from the
repulsion of its parts, the law of repulsion must be inversely
as the distance. For, consider a cube filled with the gas
at pressure p, and let the cube expand till each side is n
times its former length. The pressure on unit of surface
according to Boyle’s law is now , and since the area
of a face of the cube is n1 times what it was, the whole
pressure on the face of the cube is - of its original value.
But since everything has been expanded symmetrically, the
distance between corresponding parts of the air is now n
times what it was, and the force is n times less than it was.
Hence the force must vary inversely as the distance.
But Newton has shown {Principia, bk. i. prop. 93) that
this law is inadmissible, as it makes the effect of the dis¬
tant parts of the medium on a particle greater than that of
the neighbouring parts. Indeed, we should arrive at the
conclusion that the pressure depends not only on the density
of the air but on the form and dimensions of the vessel
which contains it, which we know not to be the case.
If, on the other hand, we suppose the pressure to arise
entirely from the motion of the molecules of the gas, the
interpretation of Boyle’s law becomes very simple. For,
in this case _
y>V = .
The first term is the product of the pressure and the volume,
which according to Boyle’s law is constant for the same
quantity of gas at the same temperature. The second term
is two-thirds of the kinetic energy of the system, and we
have every reason to believe that in gases when the
temperature is constant the kinetic energy of unit of mass
is also constant. If we admit that the kinetic energy of
unit of mass is in a given gas proportional to the absolute
temperature, this equation is the expression of the law of
Charles as well as of that of Boyle, and may be written—
j9Y = R^,
where Q is the temperature reckoned from absolute zero,
and It is a constant. The fact that this equation expresses
with considerable accuracy the relation between the volume,
pressure, and temperature of a gas when in an extremely
rarified state, and that as the gas is more and more com¬
pressed the deviation from this equation becomes more
apparent, shows that the pressure of a gas is due almost
entirely to the motion of its molecules when the gas is rare,
and that it is only when the density of the gas is consider-
ably increased that the effect of direct action between the
molecules becomes apparent.
The effect of the direct action of the molecules on each
other depends on the number of pairs of molecules which
at a given instant are near enough to act on one another.
The number of such pairs is proportional to the square of
the number of molecules in unit of volume, that is, to the
square of the density of the gas. Hence, as long as the
medium is so rare that the encounter between two molecules
is not affected by the presence of others, the deviation from
Boyle’s law will be proportional to the square of the
density. If the action between the molecules is on the
whole repulsive, the pressure will be greater than that given
by Boyle’s law. If it is, on the whole, attractive, the
pressure will be less than that given by Boyle’s law. It
appears, by the experiments of Begnault and others, that
the pressure does deviate from Boyle s law when the
density of the gas is increased. In the case of carbonic
acid and other gases which are easily liquefied, this devia¬
tion is very great. In all cases, however, except that of
hydrogen, the pressure is less than that given by Boyle s
law, showing that the virial is on the whole due to
attractive forces between the molecules.
Another kind of evidence as to the nature of the action
between the molecules is furnished by an experiment made
by Dr Joule. Of two vessels, one was exhausted and the
other filled with a gas at a pressure of 20 atmospheres;
and both were placed side by side in a vessel of water,
which was constantly stirred. The temperature of the
whole was observed. Then a communication was opened
between the vessels, the compressed gas expanded to
twice its volume, and the work of expansion, which at
first produced a strong current in the gas, was soon con¬
verted into heat by the internal friction of the gas. When
all was again at rest, and the temperature uniform, the
temperature was again observed. In Dr Joule’s original
experiments the observed temperature was the same as
before. In a series of experiments, conducted by Dr Joule
and Sir W. Thomson on a different plan, by which the
thermal effect of free expansion can be more accurately
measured, a slight cooling effect was observed in all the
gases examined except hydrogen. Since the temperature
depends on the velocity of agitation of the molecules, it
appears that when a gas expands without doing external
work the velocity of agitation is not much affected, but
that in most cases it is slightly diminished. Now, if the
molecules during their mutual separation act on each other,
their velocity will increase or diminish according as the
force is repulsive or attractive. It appears, therefore, from
the experiments on the free expansion of gases, that the
force between the molecules is small but, on the whole,
attractive.
Having thus justified the hypothesis that a gas consists
of molecules in motion, which act on each other only
when they come very close together during an encounter,
but which, during the intervals between their encounters
which constitute the greater part of their existence, are
describing free paths, and are not acted on by any mole¬
cular force, we proceed to investigate the motion of such a
system.
The mathematical investigation of the properties of such