I have a couple of questions regarding the grand potential at equilibrium.

I am well aware that in the grand canonical ensemble, particles and energy can be exchanged. I also understand why the grand potential can be expressed as


First question: I read that when connecting a system to a large reservoir, the differentials can be written as


Where does this expression come from, and why do we subtract, not add, the conjugate variables? The source justifies it as $\text{d}V_1=-\text{d}V_2$ and so on but I don't see why.

Second question: Knowing what can be exchanged in the GCE, I understand that $T_1=T_2$ and $\mu_1=\mu_2$, so $\text{d}T=0$ and $\text{d}\mu=0$, and that for minimisation, $\text{d}\Phi_G=0$. This means either $p_1=p_2$ or $\text{d}V$. Which is it? This feels rather silly to ask, but feel like either one is a possible consequence of equilibrium.


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