# Relation for Temperature and Pressure starting from chemical potential

Assume we have a system filled with a specific substance. Inside the system there is a small box with an opening. Usually this would mean $T,p,\mu$ would all be the same inside the box and outside. Let's assume there is a semipermeable wall at the entrance of the box, which only lets a specific phase of the substance through. The index $_1$ means outside of the box, $_2$ means inside the box. Because of the semipermeable wall it is possible for $T_1 \neq T_2, p_1 \neq p_2$ but it still has to hold that $\mu_1 = \mu_2$ (in our case at least).

The question now is: Derive an expression for the difference in pressure and temperature starting from the difference in chemical potential.

$0 = d\mu = -SdT + Vdp$ where $\mu$ is the chemical potential, $S$ is the entropy. This equation implies that $SdT = Vdp$

I'm not quite sure how to proceed from here. Can someone give me a hint?

Cheers

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