614 MOLECULE energy of this mixture depends, as we know, upon its temperature, and the energy of these moving spheres is entirely kinetic, and may be conceived, therefore, to be a function of the mean vis viva. Let us then assume that in this medium of moving spheres we have a representation of a mass of gases, and that what is called the temperature of the gaseous mass is nothing else than the r or mean kinetic energy of each moving sphere. Then, with this assumption, the three parts (a, |3, 7) of inference (3) above correspond to the gaseous laws connected with the names of Boyle, Charles, and Avogadro respectively, and inference (2) corresponds with the law of Dalton concerning gaseous mixtures. We may also deduce the ordinary hydrostatical equations of equilibrium from the formulae which we have obtained. For, since these equations give us p = -i-r, and p = mBe ~ , mh, we get and similarly dx - dx = -mX, dx dy dz where X, Y, and Z are the component impressed forces, or the negatives of the space variations of x along the coordinate axes. So far, therefore, the physical properties of a perfect gas or mixture of such gases correspond, in all respects, with the physical properties of a medium consisting of a set of elastic spheres, or of a mixture of sets of such spheres, with the sole assumption that the physical property termed temperature, in the case of the gas, corresponds to, or is represented by, the mean kinetic energy of each of the spheres, and that each sphere represents the chemical atom. There are, however, physical properties of gases which this theory fails to explain. The most important of these is the ratio of the specific heats at constant volume and constant temperature respect ively. The specific heat of gas expanding while being heated under a constant pressure is greater than that of gas heated with a con stant volume, as when it is contained in a rigid vessel, for the obvious physical reason that in the former case a portion of the heat is converted into mechanical work, namely, that performed by the expansion under the constant pressure. This ratio of the specific heat of gas under constant pressure to the specific heat with con stant volume has been determined for many gases with great ac curacy, chiefly from observations of the velocity of sound in such gases, in which velocity the value of this ratio bears a very im portant part. Now, on the assumption of the gas being constituted of a number of elastic spheres in rapid but irregular motion among each other, and the physical property of temperature being represented or measured by the mean vis viva of each sphere, the ratio of these specific heats must be exactly 1. For, if v be the volume occupied by a unit of mass of this moving sphere medium, and r the number of spheres to the unit mass, and if p be the density, it follows that rm=pv=l. Also we know ih&tp, the pressure referred to unit surface, is given by the equation 2 pv = -Q- rr, where r is the mean vis viva. If now r increase from r to r + Sr, while v remains constant, the increase of intrinsic energy must be, from definition, rdr. Also if there be a similar change in r without the restriction of v being constant, but supposing p to be constant, there is external mechanical work performed equal to pSv, where ov is the increase of volume. Also s 2 5 pdv = ror 3 and therefore the whole energy required to be supplied from without must be in this case 2 , ror + ror. 3 Or the ratio of the energies to be supplied from without, in order that the mean vis viva of the moving sphere medium should be increased by the same amount in the two cases respectively, becomes 2 If therefore the gaseous mass be adequately represented by the moving sphere medium, the ratio of the specific heats must be Ijj. Mercury vapour is the only gas for which the ratio has so large a value as this. Several of the more permanent gases have the ratio equal to 1 408, while in others it falls as low as 1 26. The value for mercury vapour, as determined by Kundt and Warburg (Poggcndorjf, clvii. 353), is between 1 695 and 1 631, the mean of all the observations being somewhat under 1 6. If any value above 1 6 be insisted on it will be impossible to retain the theory as above enunciated. In point of fact we may say, in anticipation of what has yet to come, that there is no modification of the kinetic theory as hitherto treated which could give a higher value for the ratio in question than 1. It follows from what has been proved that either all known gases and vapours, except the vapour of mercury, and perhaps cadmium, must be polyatomic, or else that the attempts to explain the consti tution of gases by the kinetic theory must be abandoned. We must therefore proceed further to investigate the physical pro perties of a medium consisting of compound atoms or molecules built up of atoms in any definite arrangement, such molecules being in a condition of irregular motion among themselves, such as we have supposed in the cases of the spherical atoms hitherto con sidered. It will be observed, on reference to the cases of the spheres hitherto investigated, that, whether there be forces to fixed centres in action on the medium or not, the chance of any sphere having the coordi nates of its centre and its component velocities between x and x + dx, y and y + dy,z and z + dz, u and u + du, v and v + dv, w and iv + die, is proportional to e ~ hE dxdydz du dv dw, where E is the total energy, kinetic and potential, of the sphere in the state of position and motion defined by x, y, z, u, v, w. We may generalize this proposition, and prove that when the sphere is replaced by a molecule of any shape and constitution, so as to be defined as to position and motion by r generalized coordi nates qi-..q r with their corresponding momenta p l . . .p r , the chances of the molecule having its defining variables between the limits q l and <li + dq 1 ...p r &ndp r + dp r , or, what is the same thing, the number of such molecules at any time with variables thus limited, whether there be forces to fixed centres or not, and whether interatomic forces or intermolecular forces are or are not in action on the mole cular aggregate, is proportional to e -hE r dq^^dpr, where h is a constant, the same for all molecules, and E r is the total energy, kinetic and potential, of the molecule in the free state as to position and motion, the potential energy being that of the fixed centre forces on the molecule, together with that of its inter atomic forces, in the given position. The problem before us may be stated thus : A number of similar molecules possessing in the whole n degrees of freedom, where n is very large, are in motion in a region of space bounded by a material envelope, under the action either of forces to fixed centres (called external forces) or of forces between different molecules and different parts of the same molecule, as well as by forces between the fixed boundary and the contained molecules, all of them conservative, so that the total energy, kinetic and potential, of the aggregate remains always the same ; it is required to find the chance of a group of any one or more molecules possessing in the whole r degrees of freedom, defined by the coordinates qi-..q r and momenta p...p r , where r is small compared with n, having its variables between the limits q t and q 1 + dq l . . .p r andp r + dp r . We might start with the assumption made above in the case of the spheres under central forces, that this chance must be of the form fr (0i, <t>2> kc.)dq 1 ...dp r , where fa = a lt <f>. 2 a. 2 , &c. , are obtained by the elimination of t between the equations of motion of the r group under the fixed centre and boundary forces and those between its component atoms, because there is nothing in the conception of a molecule beyond that of a system with a number of degrees of freedom, and under internal forces ; and in this case, considering the generality of the assumption as to the external forces, it would be impossible to con ceive the existence of any general equation, independent of the time, between the variables, except that of the conservation of energy, so that the chance in question becomes $ (E r ) dq^dp* where E r is above defined, and it remains to determine the form of //. If we considered a second group of one or more molecules con taining s degrees of freedom (where s may or may not be equal to r, but, like r, is much smaller than 7;), and defined by the coordinates and momenta q r+l ...q r ^ f , p^-^^.p^.^, then the two groups together contain r + s degrees of freedom defined by the variables qi---Pri-,, and since r + s is small compared with n, the chance of this group having its variables between q^ and q i + dq l ...p r+s and P r j t -, must be But this chance must be equal to the chance of the r group being fixed in the state q lt qi + dq^.^p^ p r + dp r , multiplied by the chance of the remaining s group being in the state q
Prf, Pr+i + dpr+a where the r group are so fixed.