Page:Zur Thermodynamik bewegter Systeme.djvu/3

$H=U_{0}-\int _{0}^{\beta }\phi \ d\beta ,$

(3)

$\phi =-{\frac {\partial H}{\partial \beta }},$

(4)

$U=H-\beta {\frac {\partial H}{\partial \beta }}.$

(5)

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

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