MOLECULE 615 Now to find this latter chance we observe that it is the chance of the s group being in their required limits of position and motion, when the internal forces between the r arid s group become forces between the s group and fixed centres. If the total kinetic energy of the r group in their given state be T r , and that of the r + s group be T^,, the total kinetic energy of the s group must be T^, - T r . Also if the total potential energy of the r + s group under the influence of all forces be Xr+t> this is made up of (1) x r > the potential energy of the r group to fixed centres, and of its internal forces ; (2) x,> similarly taken for the s group ; and (3) r x,, the potential energy of the r and s group forces. And when the r group is fixed the potential energy of the s group is reduced to (2) and (3), or is Xr+> ~ X r - Therefore the chance of the s group having its variables within the required limits when the r group is fixed must be Therefore or Therefore E r ) ^ (x) = e Cx = c t - E r ) = Cx ~ }>x suppose. And the chances of the r group having its variables between the limits q l and q l + dq l ...p r and p r + dp r must, in the state of per manent or stable motion, be proportional to e-^rdq^.dpf, which was to be proved. Supposing now that the aggregate of molecules under considera tion consists of a number of sets of similar molecules, the number of molecules in one of these sets being N, where N is very large, and suppose that each of these N molecules possesses a degrees of freedom defined by the coordinates qi...q ff with the momenta Pi.-.pf and that its mass is m. Three of these coordinates may be taken as the rectangular coordinates of its centre of mass, in which case the corresponding momenta will be mu, mv, mw, where u, v, and w are the component velocities of translation of that centre of mass. Then in this case, if q...q ff p---Pa be the remaining coordinates and momenta of the molecule, the chance of the mole cule s variables being within the limits x and x + dx...p ff and will be proportional to - * (X+f) - & < 2 e dx dy dz dq 4 ...dp 4 e 2 du dv dw...(], where T, the kinetic energy of the molecule, is equal to where / is a quadratic function of the p s, having as coefficients known functions of the q s. If we integrate the expression (I) for all possible values of x, y, z, q i ...q ff Pi-.-Pa we obtain an expression of the form i 771 Be" ~2~ dudvdw ........................ (II), Avhere B is independent of u, v, and w, and c 2 =u z + i? + w z . From theformof (II) it follows, exactly as in the casesof the elastic spheres, that the chances of all directions of the velocity of translation of a molecule are equal, that the mean velocity and mean square velocity of translation of each molecule are respectively, and that the mean kinetic energy of translation is , and the same for a molecule of any set. Again, if T be the mean total kinetic energy of the molecule, then ///... T.e- h &+ T >dx...dp ff - (Ill); and if we evaluate this expression, paying attention to the form of T as a quadratic function of the p s mentioned above, we shall find for (III) the expression . 2h It follows from this result that each additional degree of freedom of the molecule increases the mean total kinetic energy of the mole cule by the quantity , which is the mean kinetic energy of trans- ,n/ lation parallel to any one of the axes, and that the total kinetic energy is proportional to the number of such degrees of freedom. If, again, we integrate the expression (I) for all values of the momenta, we obtain an expression of the form Ce~ K X dx dy dz dq 4 ...dq f (IV), Avhere x is the potential energy of the molecule due to fixed centre and to interatomic forces in the position defined by x, y, z, q...q ff . The dimensions of the molecule are so small that we may regard forces from each fixed centre on different parts of the molecule as parallel and equal and functions of the distance of the centre of mass from that fixed centre, so that, if the part of x arising from these fixed centre forces be called x, Xi w ill he a function of x, y, z, and of these variables only, the remaining part of x (arising from interatomic forces), which may be called x 2 > will be a function of the ff 3 variables q...<ia If in (IV) we write Xi + X-> f r X> an( l then integrate for all values of q^.-.q^ we obtain an expression of the form Dc- ft Xi dx dy dz (V), where D is independent of x, y, z, and therefore p the density of the N molecule matter in the neighbourhood of the point x, y, z, is From these results all the propositions proved above with reference to the aggregate of elastic spheres or monatomic molecules, as to the correspondence of the physical properties of such an aggregate with those of gases as indicated by the gaseous laws, may be deduced also for this aggregate of polyatomic molecules. So that if T be equal to , or the mean kinetic energy of agitation of any one of the aggregate of moving molecules, if v be the volume occupied by unit of mass, r the number of molecules in unit of volume, and m the mass of each molecule, we have, exactly as in the case referred to, mr=, P v = l, and pi- = rT. We also get the ordinary hydrostatical equations dp _ y. dp _ Y dp v from this expression for p combined with the equation remembering that dy dz whence the coincidence of the physical properties of this aggregate of polyatomic moving molecules with those of a gas, on the assump tion that the temperature represents the mean kinetic energy of agitation, is at once apparent. It can be shown also that the aggregate of moving molecules, such as we conceive a gas to be, possesses another very important physical property which, by its analogy to the second law of thermo dynamics, affords additional evidence of the relation between the phenomena of heat and those of aggregates in some kind of motion, the property in question being that, if in any aggregate of moving molecules the mean kinetic energy of any one of them be called T, and if SQ be an increment of energy imparted to the aggregate from without, then is a perfect differential. If to this aggregate we apply a certain small quantity SQ of heat or energy from without, and if ST be the increase of the mean kinetic energy of agitation when the volume is unaltered, then this constancy of volume prevents any of the energy 5$ from being absorbed in doing external work ; but it is conceivable that the increase of T may cause such a change in the average state of the molecule as to produce a variation Sx in the mean potential energy of the molecule, Sx being proportional to 5r. Therefore ,.,-, dr But therefore If the volume vary by 8v, the pressure being constant, then we must add external work, orpSv, to the energy absorbed, so that if the whole external energy now applied be S Q, and the increase of tempera ture ST be the same in both cases, we have . 3 dr SQ
But if p be constant, then as before