Thermodynamics

III. Example. Thermodynamics.

For a suitable choice of coordinates, the laws of reversible thermal processes can be represented in the form (**):

In this, F is what I called the “free energy,” which is a function of the coordinates pa and the absolute temperature ϑ, and L is the vis viva of the visible motions of the heavy masses, and thus a function of the pa and the qa that is homogeneous of degree two in the latter and independent of ϑ.

dQ is the amount of heat that enters the body in the element of time dt – i.e., the work that is done by the environment – and only forces that bring about the motion of heat will be exerted.

We will get this form, with different variables, when we set:

then let s denote a function of S, and further set:

The vis viva of the visible motions L is added in order to insure the completeness that is desirable here.

(6b)

If we now express H and s as functions of the pa and η then the equations above can be written [loc. cit., Abh. I, eq. (1d)]: (6c)

These equations, like the first ones (6), (6a ), have entirely the same form (cf., loc. cit., § 3) as the ones for the motion of a monocyclic system whose kinetic potential is (H – L), and for which η denotes the velocity and s, the moment of motion of the monocyclic motion.

If we let P(η) denote the force that is exerted in the direction of the velocity η then we will have: (6f) P(η) ⋅ η ⋅ dt = − dQ.

The analogy with the Lagrange expressions thus remains true here, as well, and in that way that entropy S of the motion might depend upon the moment of motion s of the monocyclic motion. In this case, this possibility of the coupling of equally-warm systems of bodies to a larger system, and the kinetic theory of gases lead one to assume that:

Thus, for any given system of bodies, the temperature would be set to be proportional to the vis viva of the heat motion, as Clausius and Boltzmann apparently already sought to before me (*).

The later applications of our example are independent of the question of the relationship between the functions S and s.

The motion of heat can apparently be regarded as an especially far-reaching example of the elimination of the coordinates pc , and for that reason H can be a complicated function of ϑ or η.

However, my investigations into the combined monocyclic systems have shown that many combined forms of motions that are already quite similar to the internal molecular motions of warm bodies can also lead to the same laws.