No Free Meal With Plasma

From the post "" dated

$\cfrac{pV}{T_n}= -\cfrac{pV}{T}=ln(T^{ -A }_n)+C$

$C=(pV)_{\small{n=1}}$

with $T^{+}$,

$\cfrac{pV}{T_p}= \cfrac{pV}{T}=ln(T^{ +A }_n)+C=-ln(T^{ -A }_n)+C$  --- (*)

and

$\cfrac{pV}{T_n}= -\cfrac{pV}{T}=-\cfrac{T^{+}}{\tau_o}.\cfrac{V}{A_o}$

which is a constant.

with $T^{+}$,

$\cfrac{pV}{T_p}= \cfrac{pV}{T}=\cfrac{T^{+}}{\tau_o}.\cfrac{V}{A_o}$

Both positive and negative temperature have the same equations, but does $T^{+}$ does negative work?  No, a spinning $T^{+}$ particles does not produce a $g$ field that act against gravity, it does produce an $E$ field.  Unless in the presence of an existing $E$ field, $T^{+}$ does not perform positive work.  $T^{-}$ was removed from the system to decrease work done.  Adding $T^{-}$ to the system will increase work done, and reduces temperature.  $T^{-}$ itself does positive work against gravity.  So, equation

$\Delta pV=-A\Delta T_{p}$

is not valid! And so equation (*) for $T^{+}$ is not valid.

We plot $\cfrac{pV}{T}$ vs $T$,  where $T=T_{n}+T_{p}$,

Work done per temperature is a constant over temperature.  The increase in $\cfrac{pV}{T}$ after $T=1+T_{p}$ is due to the removal of $\cfrac{T^{-}}{\tau_o}.\cfrac{V}{A_o}$ which is expected to act against $\cfrac{T^{+}}{\tau_o}.\cfrac{V}{A_o}$.

This graph suggests that on cooling a hot gas by adding $T^{-}$ particles, $\cfrac{pV}{T}$ increases, but cooling by removing $T^{+}$ particles, $\cfrac{pV}{T}$ remains constant.

There is no free gain.  And removing negative temperature particles requires work.