Page:Zur Thermodynamik bewegter Systeme.djvu/3



In the differentiation of one of the quantities $$U_0, v, \beta$$, the two others have to be kept constant. We also emphasize, that we understand $$U_0$$ as the energy which it obtains when the body is brought adiabatically and isochorically to rest; the way by which the body has acquired its momentary (moving) state, is completely irrelevant.

1. Computation of the pressure.

We denote the pressure of the resting system by $$p_0$$, that of the moving one by $$p$$. In order to compute the latter, we consider the following circular process:

A. The initial state shall be that of rest; $$U_0, v, p_0$$ shall be the values of the relevant state variables. We change the volume adiabatically from $$v$$ to $$v'=v+dv$$; then the energy assumes the value $$U'_0=U_0-p_0 dv$$.

B. We bring the body to the velocity $$\beta c$$. The energy assumes the value

$U = \Phi(U'_{0},\ v',\ \beta)$

The work of the external forces is

$U - U'_{0} = \Phi(U'_{0},\ v',\ \beta) - U'_{0}$.

C. We change (at constant velocity) the volume adiabatically by $$-dv$$. The external forces perform the compression work $$pdv$$ and the translation work $$\beta d\phi$$ (in order to keep the velocity constant). Thus the increase of energy is

$dU = pdv + \beta d\phi\,$.