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 $$-p_{0}dv$$. 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:

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

Furthermore, since we have according to (4)

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

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

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

This constant can also be a function of $$U_0$$ and $$v$$; it reduces to zero, since we have

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

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