This question is related to this one. I would ask you to read that question and my answer to the question itself before answering this one.
The problem is the following. In his book Thermodynamics, appendix C, Callen introduces the problem of the adiabatic piston: to find the equilibrium state of an isolated system with an internal movable adiabatic wall.
Callen's argument is the following:
Total volume is fixed:
$$V=V_1+V_2$$
So that
$$dV_1 = - dV_2$$ And the piston is adiabatic, i.e. it allows no heat exchange, so that:
$$dU_{1,2} = -P_{1,2} dV_{1,2}$$
The total entropy of the system is
$$S(U,V)= S_1 (U_1, V_1)+ S_2 (U_2, V_2)$$
hence
$$d S = dS_1 + dS_2 = \frac{dU_1}{T_1} + \frac{P_1}{T_1} dV_1 + \frac{dU_2}{T_2} + \frac{P_2}{T_2} dV_2$$
Since $dU_{1,2} = -P_{1,2} dV_{1,2}$, we see that $dS$ vanishes identically, so that we can say nothing about $P_{1,2}$ and $T_{1,2}$: the entropy maximum principle is thus inconclusive.
On the other hand, there is conservation of energy. We require that $dU=0$ since our system is isolated from the environment, hence
$$dU_1 + dU_2 = 0 \to P_1 d V_1 + P_2 dV_2 = 0$$
But $V=V_1+V_2$ and $V$ is fixed, so that $dV_1 = - dV_2$ and we obtain
$$P_1=P_2$$
I know that this conclusion is correct, i.e. it is a necessary condition for thermodynamic equilibrium. On the other hand, in the article Thermodynamics of systems with internal adiabatic constraints: time evolution of the adiabatic piston by C. Gruber (1998) the author says:
It was then noted that Callen's argument, which was repeated by Leff, could not be correct since the equilibrium condition was derived from the first law, rather than the second law.
My question is: What does the author of this article mean? Why should Callen's argument be flawed?
I stress that I don't want an alternative proof, but just an explanation of what is wrong in Callen's argument, possibly using only thermodynamics and not kinetic theory or fluid dynamics.
Update
An "answer" to this question is actually contained in the article A Thermodynamic Consideration of Mechanical Equilibrium in the Presence of Thermally Insulating Barriers by A. E. Curzon (1968), which is mentioned explicitly in the article I cite in my question. The problem is that to me it in not clear why Curzon argument should be the "right one" while Callen's is the "wrong one"...
This is Curzon's argument:
The system is isolated and its volume is fixed, so that $$\label{1}\tag{1}dU_1+dU_2=0$$ $$\label{2}\tag{2}dV_1+dV_2=0$$ If we assume that the wall separating the two system does not allow exchange of matter, we have $$\label{3}\tag{3}T_{1} dS_{1} = d U_{1} + P_{1} dV_{1}$$ $$\label{4}\tag{4}T_{2} dS_{2} = d U_{2} + P_{2} dV_{2}$$ The entropy of an isolated system with fixed volume at equilibrium is at a maximum: $$dS=dS_1+dS_2=0\tag{5}\label{5}$$ From \ref{1}, \ref{2} and \ref{4} we obtain $$\tag{6}\label{6}T_2 dS_2 = -dU_1 -P_2 dV_1$$ Substituting \ref{6} and \ref{3}in \ref{5}, we obtain $$dS = \left( \frac 1 {T_1} - \frac 1 {T_2} \right) d U_1 + \left(\frac{P_1}{T_1}-\frac{P_2}{T_2} \right) dV_1 = 0 \tag{7}\label{7}$$ Now, since the piston is adiabatic we have for the two subsystems $$\tag{8} \label{8} dS_{1,2} \geq 0$$ (this is a consequence of the Clausius inequality). Equations \ref{5} and \ref{8} can only be consistent if $$dS_1=dS_2=0\label{9}\tag{9}$$ From \ref{3} and \ref{9} we obtain $$dU_1=-P_1 dV_1\tag{10}\label{10}$$ Substituting \ref{10} in \ref{7} we finally obtain $$(P_1-P_2) \frac {dV_1} {T_2} = 0\label{11}\tag{11}$$ Being $dV_1$ arbitrary, we conclude that $$P_1=P_2$$
I am not completely convinced that this can be considered the "correct" proof that $P_1=P_2$ is a necessary condition for equilibrium while Callen's is wrong. Curzon actually addresses the problem of analogous derivations of many authors (for example Kubo), i.e. the fact that they state that the condition $P_1=P_2$ can be derived only if we assume $T_1=T_2$. But Callen does not make such an assumption! In fact, his argument is very similar to Curzon's and it seems to me that Curzon's argument is quite the mathematical trick.
Indeed, from \ref{4} and \ref{9} we would obtain $dU_2 = -P_2 dV_2$ (analogos to \ref{1}): this, together with \ref{1}, \ref{2} and \ref{10}, would give us Callen's argument again!
So, in conclusion, to me it looks like Curzon's argument is quite the same as Callen's: therefore it is not clear what Gruber meant when he wrote that sentence, and I consider the question to be still open.