cluding the following law, first established by Clausius (see § 29), and in taking it as definition of equality of temperature, viz. two different gases are at the same temperature when the mean kinetic energy of the molecules of both kinds is the same.

If both gases are also under the same pressure, and have therefore equal amounts of kinetic energy in unit volume, the further conclusion of Avogadro's law, discussed in § 31, holds good, viz. that two different gases at the same temperature and pressure contain in equal volumes equal numbers of molecules.

21*. Polyatomic Molecules

The foregoing considerations can, strictly speaking, claim applicability only to gases whose molecules have no internal motions; for atomic motion was left out of account, and our conclusions are thus justified only for gases whose molecules consist each of a single atom.

For systems of polyatomic molecules the investigation is certainly somewhat more complicated; but for these media too the distribution of the motion among the individual molecules may be found by the same method used before, and also the law of distribution of speeds among the constituents of the molecules, i.e. among the atoms.

Let a molecule m consist of a number of similar or different atoms m, m2, . . . which, in addition to the molecular velocities u, v, w, execute special atomic motions with the velocities U1, v1, w1, U2, V2, W2, ... The magnitudes of these last velocities must satisfy the equations

0 = S.mu, 0 = S.mv, 0 = S.mw,

when the summation denoted by S is extended over all the atoms forming a molecule. The equations

aΣ.m= 2.mu, b2.m = 2.mv, cΣ.m = Σ.mw

also hold good, in which a, b, c, as before, denote the velocities of flow of the gas, and the summation Σ is extended over all molecules.

But the equation for the kinetic energy takes in this case an essentially different shape. While the motion of the molecules. investigated before was not constrained by molecular forces, the

new motion of the atoms which now comes in is not free; it takes place under the influence of forces which maintain the combination of the atoms into a molecule, and whose joint effect we call affinity. In addition, then, to the kinetic energy, which is made up of the sum of the kinetic energies of the atoms and molecules, we must bring the potential energy also into the theorem, which, therefore, takes the form

N(E + €) = {2. {m(u2 + v2 + w2) + S.m(u2 + v2 + w2)} + ZS.. To the mean energy E of the to-and-fro motion of a molecule, which I will call the molecular energy, there is here added on the left-hand side the atomic energy & which is present inside the molecule, in the form partly of heat-motions of atoms, and partly of chemical affinity, while on the right-hand side these magnitudes are taken into consideration as the kinetic energy of the molecules and atoms, and as chemical work performed by heat. The last magnitude is introduced by a function which represents the part ф of the work which is done on a single atom contained in the complex of the molecule. So that S. expresses the amount of chemical work in the molecule, and 28.p the whole amount of chemical work in the medium.

The value of this chemical energy we may easily express by the attractive forces between the atoms if these forces belong to the class named by Helmholtz central forces. Thus, if one atom exerts on another distant by r from it a force f(r), the work required to increase this distance by dr is

f(r)dr;

to overcome the affinity, therefore, in the infinitely small displacement of an atom, an amount of energy or of heat

as the value of the energy spent, the constant & here denoting the smallest distance from the atom in question to which the attracted

atom can come.

The new more general formulæ, like the earlier simpler ones, hold good for all possible states of motion, and may therefore undergo variation in the same way as the others. But in this

operation we must consider not only the velocities u, v, w of the molecules and the atomic velocities u, v, w as variable, but also the distances r and the function p, since the mode in which the atoms are bound together into molecules and arranged within them is not given. The number of molecules N therefore cannot be taken as absolutely fixed, but we shall have the product

N(E)H,

where H stands for a given constant, which in § 53 was thus denoted.

The variation consequently gives

0 = 2.m(uòu + vèv + wòw) + ZS. {m(udu + vềv + wềw) + dp} 0= 02.mou, 0= 2.mèv, 0= Σ.mòw,

=

The variation do here occurring is not independent of the other variations. For by addition of heat not only does the kinetic energy rise in amount, but also the relaxing of the bonds, which Clausius calls disgregation, increases in continually corresponding measure, till at last it leads to dissociation. The regular connection between these phenomena is to be introduced into the calculation.

Since an increase of the velocity with which the centroid of a molecule moves is conceivable without the internal connection between its component parts needing in any way to be altered,

cannot depend on u, v, w. On the contrary, must be considered a function of u, v, w; for an increase in the atomic motions must cause the distances r between the molecules to increase in consequence both of the collisions between the atoms and of their centrifugal force. Hence we must put

where the sum is to be taken over all the atoms of the molecule which contains the atom subjected to the influence of affinity.

If now, as before, we denote the probability that an atom possesses the molecular velocities u, v, w, and also the special velocities u, v, w by F= F(u, v, w, u, v, w),

the formula defining the state of equilibrium of the medium for which we are looking is

Since the variations here contained must satisfy the above conditions, we obtain by the method of elimination already used the following differential equations for the determination of the function F:

Here k, a, ẞ, y are the same constant magnitudes as before; but a, b, c are to be considered constant only so far that for all atoms in one and the same molecule they have the same value; they might possibly have different values for different molecules, and, so far at least, should be taken as dependent on the state of the molecule, that is, as functions of u, v, w.

We easily see, however, that the above equations would contradict each other if a, b, c changed with u, v, w from molecule to molecule.

By differentiating, for example, the first and fourth of the above equations, we obtain inconsistent values of the second differential coefficient with respect to the two variables u, u, which are independent of each other, thus:

and similarly with the others; whence we see that the magnitudes a, b, c must in general be constant in respect to u, v, w also. The integration of the equations is now easy to carry out, and we obtain

and A denotes a constant which may be different for each kind of atom. This function F may, as before, be broken up into the product of several simple functions, for we may put

Ae ̄k = U(u) V(v) W(w)F(u, v, w),

of which the first three have the form

where by B may be understood a constant which is the same for all molecules and atoms, and the fourth is

where

F(u, v, w) Be-*x

x = m {(u − a)2 + (v − b)2 + (w — c)2} + 2p

and B denotes a constant which may have a special value for each kind of atoms.

The three former functions have the same meaning as before in § 14*, so that, for instance, U(u)du denotes the probability that the atoms of a molecule move parallel to the x-axis with a velocity between u and u + du. The determination of the constants can therefore be carried out exactly as before; in this case, too, we have

a)2

1 = B√ due-ku - aja
B["_due="

with similar equations for the components v and w. Hence follow B = (T1km)

a = a, ß=b, y = c.

We obtain, therefore, exactly the same equations as before,