50. Horstmann's Explanation

A different explanation of the same circumstances given by Horstmann rests at bottom on the second of the hypotheses described in § 48, although Horstmann gives it a quite different form. He throws doubt, in fact, on the exactness of Avogadro's law in the case of vapours and gases which do not strictly obey Boyle's law, and explains it as being only approximately correct. This law is, on our theory, a necessary consequence of the hypothesis that pressure and temperature are caused by rectilinear motion of the molecules alone. Hence Horstmann's assumption is not essentially different from that discussed in § 48, according to which the to-and-fro straight paths of the molecules are supposed to be joined together by curved parts.

By this assumption also the observed anomalies can be explained. The force of the blow with which a molecule. strikes against the wall of the space filled with gas or vapour is greater if the molecules move in right lines free from forces of cohesion than if they are drawn back from the wall and into the interior of the space in curved paths by the attraction of other molecules. The pressure, therefore, of a vapour or gas which contains a given number of molecules will be the less the more frequently the molecules are caused to move in curved paths, and the greater the more frequently and the longer they move in straight paths. If, therefore, the density is increased, the pressure increases, not in the same ratio, but in a less degree, because the increase of the number of collisions causes a diminution of the force of a collision. On the other hand, by an increase of temperature the pressure will increase in greater measure, since the force of a blow increases not only for the reasons already given, but also because the faster moving molecules traverse longer straight paths.

These views, therefore, also suffice to explain simply and naturally not only the greater vapour-density possessed by easily condensed gases and vapours in the neighbourhood of

1 Ann. Chem. Pharm. Suppl. Bd. vi. 1868, p. 53; also in his Habilitationsschrift, Heidelberg 1867.

I

their condensing point, but also the fact that their volumeand pressure-coefficients are greater than those of ideal gases.

51. Claims of the Two Explanations

In the present position of the matter a definite answer to the question, Which of the two explanations of the anomalies deserves preference? is not possible, since each represents the observed behaviour in general, and neither seems to contradict our gaseous theory, which is otherwise confirmed. A distinction between them could only be made after a further pursuit into details of the views that have only been sketched very generally.

In order to carry out the theory of gases with the suggested corrections in the one direction or the other, the calculus of probabilities offers the same method as was used to prove Maxwell's law of speeds. It would be only necessary to consider as unknown not only this law of distribution, but also the form of combination of the molecules, whether in groups or as units. The problem offers the same difficulties as a mathematically formulated chemical statics, which would have to treat of the combination of atoms into molecular groups; and the solution of the one will be the solution of the other.

The contention between the two modes of explanation seems to be capable of more easy decision empirically. It might be recommended to determine the densities of vapours not only by weighing, but also indirectly in the way proposed by Leslie and Bunsen, viz. by observation of their speeds of effusion at different temperatures and under different pressures. Perhaps, too, transpiration- and thermal-measurements may give help in the determination.

At present we shall look on both explanations as equally entitled to consideration, and must consider the probability to be that both circumstances, as discussed in §§ 48-50, work together to cause actual gases and vapours to deviate in their behaviour from the ideal laws.

115

CHAPTER V

MOLECULAR AND ATOMIC ENERGY

52. Dulong and Petit's Law for Gases

THE theories of Clausius and Maxwell, developed in Chapters II. and III., form, in the first place, a kind of molecular mechanics; but as heat is nothing else but the mechanical motion of molecules, these theories are entitled to form the basis of the laws of heat no less than of those of mechanics. One of the immediate conclusions from it is a theorem that exhibits a marked analogy with the law respecting the specific heats of solid bodies, which was discovered by Dulong and Petit.

If the temperature of a gas rises by 1 degree, the kinetic energy (§§ 14, 16)

E = {mG2 = 1mG2(1 + ad),

which its molecules possess on the average, increases by

ImG2a,

a magnitude which, by what has gone before, is the same for

all gases.

On the other hand, the law which Dulong and Petit discovered for the specific heats of solid bodies may, as is well known, be expressed in the form that, in order to heat chemically different bodies to the same extent, the same amount of heat must be communicated to every atom; and for this we generally say more shortly that the atomic heat of all bodies is the same.

If we remember that on our theory heat and energy are identical, the analogy we have mentioned at once comes into view. By a rise of temperature the energy of each ATOM in the SOLID state increases by the same amount, and,

on the contrary, in the GASEOUS state the kinetic energy of each MOLECULE increases by the same amount.

The analogy of the two laws, which, by the bye, can claim only an approximate and limited validity, does not, however, entitle us to consider them identical, and to take as the specific heat the magnitude

mG2a

obtained above as constant for gases. In the next paragraph we shall examine its meaning more closely.

53. Ratio of the Molecular to the Total Energy

From the calculation of the mean speed of the molecules in absolute measure the value of the kinetic energy of molecular motion present in the gas is at once known. On the other hand, the value of the total energy present in the gas can be calculated from its heat-capacity and temperature, since it is equivalent to the heat contained in it. The question arises, whether the two values calculated by these different methods are in harmony with each other, and the resolution of such a doubt rests on the following considerations, which are borrowed from Clausius.

We ought not to expect the two values to be quite identical, so that the calculated kinetic energy should be the exact equivalent of the heat-energy; for the molecules, on their side, consist of atoms that are separately movable. The kinetic energy of those motions which the whole complex of atoms in a molecule together execute need therefore not be the whole energy contained in the gas, but there may, in addition to the forward motion of the molecules as they course to and fro, be other internal motions of the single atoms. The whole energy, calculated from the contained heat, may very well then be greater than the energy of molecular motion.

We find the total heat-energy contained in a gas by assuming that it has been brought into its present condition by being warmed at constant volume from the absolute zero to the temperature . If the volume filled by the gas is

℗.

unity, and therefore the mass of the gas given by the density p, the heat needed is

cpo,

where c is the specific heat of the gas at constant volume. If J is the value of the mechanical equivalent of heat, the equivalent of this amount of heat in terms of mechanical energy is

H = JcpR.

With this value of the whole amount of energy in unit volume of the gas we have to compare the value

K = P,

which we obtained in § 16 for the kinetic energy of molecular motion contained in the same unit volume.

Since in regard to the latter magnitude we know that in correspondence with Gay-Lussac's law it increases proportionally to the absolute temperature when heat is added, we see at once that both H and K are proportional to the absolute temperature . Their ratio is therefore a constant number independent of the temperature of the gas, constant at least if the assumption founded on Regnault's observations for atmospheric air is in general true, viz. that the specific heat c at constant volume does not alter with the temperature. Since also both magnitudes are proportional to the density p, we have with that supposition the proposition: In a perfect gas the kinetic energy of the molecules stands in a constant ratio to the total energy contained in the gas.

sufficient to determine the But the proceeding employed

The formulæ given are numerical value of this ratio. by Clausius,' of first reducing to the same units the two magnitudes to be compared, and of giving them as nearly as possible the same form, is more to be recommended. This was done by Clausius in the same way as J. R. Mayer determined the value of the mechanical equivalent,

Pogg. Ann. c. 1857, p. 377; Abhandl. über Wärmetheorie, 2. Abth. p. 256; Mech. Wärmetheorie, iii. 1889-91, p. 35; transl. Phil. Mag. [4] xiv. 1857, p. 108.

* Ann. Chem. Pharm. xlii. 1842, p. 239; Mechanik der Wärme, 1867, p. 28.