to the molecular energy E nearly as 2: 3, and since all these gases have diatomatic molecules, the mean energy e of an atom is to the molecular energy E nearly as 1 : 3. This class of gases, therefore, possesses the really remarkable property that at equal temperatures not only are the values of their molecular energy equal to each other, but also those of their atomic energy, and, consequently, also their whole heat-energy. These bodies, therefore, obey in the gaseous state the law of Dulong and Petit, to which other substances in the solid state are subject. Not all diatomic gases seem to follow this law equally. Even though HCl, HI, and perhaps HBr, obey it to some extent, yet BrI, CII, Cl, Br, and I, exhibit very considerable deviations from it. But we need not on this account completely deny the validity of this law for diatomic gases. For the substances last named are rather vapours than gases, and it is therefore probable that with them the ratio of the specific heats increases with rising temperature. It is therefore not impossible that for all diatomic gases the ratios C с = 1.4 and = 0.33 e E would be found if the measurements were made at such pressure and temperature that the laws of perfect gases were exactly obeyed. For monatomic gases theory and observation agree (see § 54) in giving C с The idea is accordingly suggested that the value of the ratio of the specific heats, as also those of the different species of energy, depends, not on the material of the atoms, but on their number. The observations quoted seem also to indicate this; at least the numbers for the triatomic gases oscillate about the mean values It has therefore also been attempted to find a general 5 law of dependence of the ratio of the specific heats on the number of atoms in the molecule. After Boedeker1 and Buff had recognised the existence of simple relations between the ratios for different gases, Naumann gave a formula which is based on the assumption that the ratio of e to E that has been found for diatomic gases, viz. 1: 3, holds in general for all gases. On theoretical grounds Boltzmann' put forward the view that the mean energy e of an atom must be equal to the kinetic energy E of the molecule, but he found this assumption not confirmed by observation. Pilling obtained a formula that agrees very well with measurement by starting with the hypothesis that the atoms exert on each other forces which vary inversely as the sixth power of the distance between them. Eddy has made the more general assumption that the action is inversely proportional to some power of the distance; but Richarz has proved by general theorems of mechanics that an assumption of this kind respecting the law of action between atoms is inadmissible, as it does not satisfy the conditions for the stability of the molecular combination. A new memoir by Staigmuller on the kinetic theory of polyatomic gases must also be mentioned. 60. Degrees of Freedom of the Motion 6 In another way Maxwell' and Watson 10 have attempted to answer the question by taking into account the 1 Die gesetzmäss. Beziehungen zw. d. Dicht. d. spec. Wärme u. d. Zusammensetzung der Gase, Göttingen 1857; Gött. Nachr. 1857, p. 165; Ann. d. Chem. civ. 1857, p. 205. 2 Ann. d. Chem. cxv. 1860, p. 301. Ibid. cxlii. 1867, p. 284; Grundriss d. Thermochemie, 1869, p. 44. Wien. Sitzungsber. Ixiii. pt. 2, 1871, pp. 397, 417. › Ueber die Beziehungen d. Wärmecapacität der Gase zu den zwischen Atomen wirkenden Kräften, Inaug.-Diss., Jena 1876. Scient. Proc. Ohio Mech. Inst. 1883, p. 26. Wied. Ann. xlviii. 1893, p. 47. 8 Ibid. lxv. 1898, p. 655. 9 Nature, xi. 1875, p. 357; Scientific Papers, ii. 418; Nature, xvi. 1877, p. 242. 10 Kinetic Theory of Gases. Oxford 1876, pp. 27, 37; 2nd ed. 1893, p. 81. number of ways in which a system of particles is movable. In this method they have been followed by Boltzmann, and then by Roiti,2 and later also by Violi.3 It shall be given here on account of its clearness, although very weighty objections must be raised against its admissibility. We can assert with regard to a single massive point that it has a threefold freedom of movability, since its motion is determined by the values of the three components of its velocity. The above-named English physicists ascribe to it, therefore, three degrees of freedom, which Boltzmann expresses as three kinds of movability (Beweglichkeitsarten). A material system consisting of a multitude of particles possesses as many degrees of freedom or kinds of movability as the number of variables which must be given for the complete determination of its state of motion; if made up of atoms, therefore, a gaseous molecule has a greater number of degrees of freedom, the number depending upon that of the atoms. This mode of attacking the question is marked by simplicity and clearness, and it therefore leads to very simple results. If heat W is added to a gas whose molecules consist each of a single atom, while its volume is kept constant, the energy, which is only kinetic, will increase equally in the direction of each of the three given degrees of freedom, and thus in each by the amount 3W measured in heat-units. If this gas is heated at constant pressure, so as to attain the same temperature as before, there must, in addition to the heat W, be given to it, to overcome the external forces, a further amount of heat W', which bears to W the ratio 2:3; for in this case the ratio of the specific heats is Wien. Sitzungsber. lxxiv. 1877, p. 553; Pogg. Ann. clx. 1877, p. 175; transl. Phil. Mag. [5] iii. 1877, p. 320. 2 Atti dell' Acc. dei Lincei [3] i. 1877, p. 774; Nuovo Cimento [3] ii. 1877, p. 61. "Atti dell' Acc. dei Lincei [3] vii. 1883, p. 112; Nuovo Cimento, xiv. 1884, p. 183. If now a gas which possesses q degrees of freedom is heated just as much, each of the degrees of freedom will require a corresponding share of heat, and therefore W heat-units are necessary for the heating at constant volume, if we have to consider only translatory kinetic energy. If, besides, there is other energy in question, a further amount of heat hq W The whole heat, therefore, is needed, where h is a constant. } (a + h) W; on the contrary, for heating at constant pressure the heat (q + hq) W + W' = (1 + hq + 2) W is needed. The specific heats, therefore, must be in the ratio where h is a constant and q the number of degrees of freedom and therefore an integer, the value of which is the greater the more atoms there are in a molecule. Before this formula can be tested by experiment, the mode of dependence of on the number of atoms n must be determined. In most cases, since the degree of movableness of the atoms is in general unknown, this can only be done by the aid of hypotheses; and for several cases such necessary hypotheses have been made. Of these we shall here pursue those which Boltzmann has investigated. For monatomic molecules q = 3, and, since C/ c = {}, h If the molecules consist each of two atoms, Boltzmann puts q = 5, since he assumes that the atoms do not alter their distance apart, but are bound fast together; the position of a molecule is then completely given by the three coordinates of its centroid and by two angles, which determine the direction of the joining line of the two atoms. The formula therefore becomes -which has been found for atmospheric air and other diatomic gases-if for these gases also may be put h We should arrive at this same result also if we considered the molecules to be rigid bodies, shaped like any figures of revolution we like; for their place and position in space are also determined by five coordinates. But against this Pirogoff' has raised the justifiable objection that a figure of revolution can rotate not only about its axis of symmetry, but also about another axis at right angles to the first as well. A figure of revolution and also a diatomic molecule would thus have more than five kinds of free movability. In spite of this objection, we may suppose that inside a diatomic molecule no shifting of the atoms towards each other occurs when the heating is kept within the limits within which the observations have been made. For this speaks the fact that for diatomic gases-at least for atmospheric air and carbon monoxide-the value of the ratio varies with the temperature only very inconsiderably. If, therefore, the molecules of diatomic gases behave as solid bodies, in which the parts suffer no relative motion, only such motions of the atoms as consist in a common rotation of all the atoms about their centroid can occur in addition to the translatory motion of the molecule as a whole. In monatomic molecules, as we have seen, such rotations do not also occur; in their case there is nothing but the rectilinear translatory motion of the centroid. 1 Fortschr. d. Physik. 1886, 2. Abth. p. 247. |