Page:Zur Thermodynamik bewegter Systeme.djvu/5

assumes (at adiabatic-isochoric acceleration) the value $p$ as given by equation (6).

This theorem can be easier derived in the following way (even though it is less clear from the physical standpoint): At adiabatic state changes, the amount of $U$ only depends on the momentary values of the quantities $\beta$ and $v$. Thus when (at arbitrary velocity) $v$ is adiabatically changed by $dv$, then $U_{0}$ changes by $-p_{0}dv$.^[1] Then the energy increase, which is equal to the work of the external forces here, is:

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

and therefore also

$-pdv-\phi d\beta$

is a complete differential, thus

${\frac {\partial p}{\partial \beta }}=\left({\frac {\partial \phi }{\partial v}}\right)$

Here, $\left({\tfrac {\partial }{\partial v}}\right)$ is to be understood as a differentiation at adiabatic state change; thus if $\phi$ is given as an explicit function of $v$ and $U_{0}$, we have:

$\left({\frac {\partial \phi }{\partial v}}\right)={\frac {\partial \phi }{\partial v}}+{\frac {\partial \phi }{\partial U_{0}}}\left({\frac {\partial U_{0}}{\partial v}}\right)={\frac {\partial \phi }{\partial v}}-p_{0}{\frac {\partial \phi }{\partial U_{0}}}.$

Furthermore, since we have according to (4)

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

the previous equation can be integrated towards $\beta$ and we obtain

$p=p_{0}{\frac {\partial H}{\partial U_{0}}}-{\frac {\partial H}{\partial v}}+\mathrm {const.}$

This constant can also be a function of $U_{0}$ and $v$; it reduces to zero, since we have

$H=U_{0};\quad \left({\frac {\partial H}{\partial v}}\right)_{U_{0}}=0$

for $\beta =0,p=p_{0}$.

1. Compare the following section 2.