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ATOM
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a system of molecules in motion is tlie foundation of mole
cular science. Clausius was the first to express the
relation between the density of the gas, the length of the
free paths of its molecules, and the distance at which
they encounter each other. He assumed, however, at least
in his earlier investigations, that the velocities of all the
molecules are equal. The mode in which the velocities are
distributed was first investigated by the present writer,
who showed that in the moving system the velocities of
the molecules range from zero to infinity, but that the
number of molecules whose velocities lie within given
limits can be expressed by a formula identical with that
which expresses in the theory of errors the number of
errors of observation lying within corresponding limits.
The proof of this theorem has been carefully investigated
by Boltzmann,1 who has strengthened it where it appeared
weak, and to whom the method of taking into account the
action of external forces is entirely due.
The mean kinetic energy of a molecule, however, has a
definite value, which is easily expressed in terms of the
quantities which enter into the expression for the distribu¬
tion of velocities. The most important result of this investi¬
gation is that when several kinds of molecules are in motion
and acting on one another, the mean kinetic energy of a mole¬
cule is the same whatever be its mass, the molecules of
greater mass having smaller mean velocities. Now, when
gases are mixed their temperatures become equal. Hence
we conclude that the physical condition which determines
that the temperature of two gases shall be the same is that
the mean kinetic energies of agitation of the individual mole¬
cules of the two gases are equal. This result is of great
importance in the theory of heat, though we are not yet
able to establish any similar result for bodies in the liquid
or solid state.
In the next place, we know that in the case in which the
whole pressure of the medium is due to the motion of its
molecules, the pressure on unit of area is numerically
equal to two-thirds of the kinetic energy in unit of volume.
Hence, if equal volumes of two gases are at equal pressures
the kinetic energy is the same in each. If they are also
at equal temperatures the mean kinetic energy of each
molecule is the same in each. If, therefore, equal volumes
of two gases are at equal temperatures and pressures, the
number of molecules in each is the same, and therefore,
the masses of the two kinds of molecules are in the same
ratio as the densities of the gases to which they belong.
This statement has been believed by chemists since the
time of Gay-Lussac, who first established that the weights
of the chemical equivalents of different substances are
proportional to the densities of these substances when in
the form of gas. The definition of the word molecule,
however, as employed in the statement of Gay-Lussac’s law
is by no means identical with the definition of the same
word as in the kinetic theory of gases. The chemists
ascertain by experiment the ratios of the masses of the
different substances in a compound. From these they
deduce the chemical equivalents of the different substances,
that of a particular substance, say hydrogen, being taken
as unity. The only evidence made use of is that furnished
by chemical combinations. It is also assumed, in order to
account for the facts of combination, that the reason why
substances combine in definite ratios is that the molecules
of the substances are in the ratio of their chemical equiva¬
lents, and that what we call combination is an action
which takes place by a union of a molecule of one substance
to a molecule of the other.
This kind of reasoning, when presented in a proper form
and sustained by proper evidence, has a high degree of
cogency. But it is purely chemical reasoning; it is not
dynamical reasoning. It is founded on chemical experi¬
ence, not on the laws of motion.
Our definition of a molecule is purely dynamical. A
molecule is that minute portion of a substance which moves
about as a whole, so that its parts, if it has any, do not part
company during the motion of agitation of the gas. The
result of the kinetic theory, therefore, is to give us informa¬
tion about the relative masses of molecules considered as
moving bodies. The consistency of this information with
the deductions of chemists from the phenomena of com¬
bination, greatly strengthens the evidence in favour of the
actual existence and motion of gaseous molecules.
Another confirmation of the theory of molecules is
derived from the experiments of Dulong and Petit on the
specific heat of gases, from which they deduced the law
which bears their name, and which asserts that the specific
heats of equal weights of gases are inversely as their com¬
bining weights, or, in other words, that the capacities for
heat of the chemical equivalents of different gases are
equal. We have seen that the temperature is determined
by the kinetic energy of agitation of each molecule. The
molecule has also a certain amount of energy of internal mo¬
tion, whether of rotation or of vibration, but the hypothesis
of Clausius, that the mean value of the internal energy
always bears a proportion fixed for each gas to the energy
of agitation, seems highly probable and consistent with
experiment. The whole kinetic energy is therefore equal
to the energy of agitation multiplied by a certain factor.
Thus the energy communicated to a gas by heating it is
divided in a certain proportion between the energy of agita¬
tion and that of the internal motion of each molecule. For
a given rise of temperature the energy of agitation, say of a
million molecules, is increased by the same amount what¬
ever be the gas. The heat spent in raising the temper ature
is measured by the increase of the whole kinetic energy.
The thermal capacities, therefore, of equal numbers of
molecules of different gases are in the ratio of the factors
by which the energy of agitation must be multiplied to
obtain the whole energy. As this factor appears to.be
nearly the same for all gases of the same degree of atomicity,
Dulong and Petit’s law is true for such gases.
Another result of this investigation is of considerable
importance in relation to certain theories,2 which assume the
existence of aethers or rare media consisting of molecules
very much smaller than those of ordinary gases. According
to our result, such a medium would be neither more nor
less than a gas. Supposing its molecules so small that
they can penetrate between the molecules of solid substances
such as glass, a so-called vacuum would be full of this rare
gas at the observed temperature, and at the pressure, what¬
ever it may be, of the setherial medium in space. The
specific heat, therefore, of the medium in the so-called
vacuum will be equal to that of the same volume of any
other gas at the same temperature and pressure. Now, the
purpose for which this molecular aether is assumed in these
theories is to act on bodies by its pressure, and for this
purpose the pressure is generally assumed to be very great.
Hence, according to these theories, we should find the
specific heat of a so-called vacuum very considerable com¬
pared with that of a quantity of air filling the same space.
We have now made a certain definite amount of progress
towards a complete molecular theory of gases. We know
the mean velocity of the molecules of each gas in metres
per second, and we know the relative masses of the molecules
of different gases. We also know that the molecules of
one and the same gas are all equal in mass. For if they
2 See Gustav Hansemann, Die Atome und ihre Bewecjurujen. 1871.
(H. G. Mayer.)
1 Sitzungsberichte der K. K. Akad., Wien, 8th Oct. 1868.